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--git a/09FST4oBgHgl3EQfWDi_/content/tmp_files/2301.13779v1.pdf.txt b/09FST4oBgHgl3EQfWDi_/content/tmp_files/2301.13779v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a1ed4de43b5570de4e31c3e350567850997bfad --- /dev/null +++ b/09FST4oBgHgl3EQfWDi_/content/tmp_files/2301.13779v1.pdf.txt @@ -0,0 +1,1471 @@ +FLAME: A small language model for spreadsheet formulas +Harshit Joshi1 , Abishai Ebenezer1 , Jos´e Cambronero2∗ , Sumit Gulwani2∗ , +Aditya Kanade3∗ , Vu Le2∗ , Ivan Radiˇcek4∗ , Gust Verbruggen5∗ +1Microsoft, India +2Microsoft, USA +3Microsoft Research, India +4Microsoft, Croatia +5Microsoft, Belgium +{t-hjoshi, t-aebenezer, jcambronero, sumitg, kanadeaditya, levu, ivradice, gverbruggen}@microsoft.com +Abstract +The widespread use of spreadsheet environments +by billions of users presents a unique opportunity +for formula-authoring assistance. Although large +language models, such as Codex, can assist in +general-purpose languages, they are expensive to +train and challenging to deploy due to their large +model sizes (up to billions of parameters). More- +over, they require hundreds of gigabytes of train- +ing data. We present FLAME, a T5-based model +trained on Excel formulas that leverages domain +insights to achieve competitive performance with a +substantially smaller model (60M parameters) and +two orders of magnitude less training data. We cu- +rate a training dataset using sketch deduplication, +introduce an Excel-specific formula tokenizer for +our model, and use domain-specific versions of +masked span prediction and noisy auto-encoding +as pretraining objectives. We evaluate FLAME on +formula repair, formula auto-completion, and a +novel task called syntax reconstruction. +FLAME +(60M) can outperform much larger models, such +as Codex-Davinci (175B), Codex-Cushman (12B), +and CodeT5 (220M), in 6 out of 10 settings. +1 +Introduction +Despite a much larger user base, spreadsheet environments +do not have access to nearly the same range of productivity +tools as available for general programming environments. The +latter typically have code completion, refactoring, linting, and +a wide range of extensions for additional functionality, like +generating tests, inserting code snippets, and summarizing +code. Many of these advanced programming assistance tools +are driven by advances in large language models trained on +code (LLMCs). Codex [Chen et al., 2021a] is used for code +completion [GitHub, 2021] and repair [Joshi et al., 2022], +AlphaCode [Li et al., 2022a] solves competitive programming +problems, [Li et al., 2022b] built a code review system, and +many other models show great performance in code related +tasks [Xu et al., 2022; Fried et al., 2022; Nijkamp et al., 2022]. +∗Listed in alphabetical order +Formula Autocompletion +Last Mile Repair +Syntax Reconstruction +0.90 +0.85 +0.80 +0.75 +FLAME +(16M) +FLAME +(60M) +CodeT5 +(220M) +Codex Cushman +(12B) +Codex Davinci +(175B) +Model Parameters (log-scale) +Performance +0.40 Z +Figure 1: A summary of model comparisons in fine-tuned setting for +different formula assistance tasks. We show the results under a top-5 +cutoff on a public excel benchmark. Note that all Codex-Davinci +results are few-shot, and Autocompletion is zeroshot for all systems +except CodeT5. For Autocompletion, results represent the fraction of +benchmarks successfully (based on a sketch match metric) completed +given 90% of the prefix. +To capture the complexity and variety of code and com- +ments in different languages, these models need billions of +parameters—the smallest variant of Codex, used by GitHub +Copilot, has 12 billion parameters. As a result, these models +are trained for long periods on corpora containing millions of +programs. For example, Incoder 6.7B used 159GB of code +over a period of 24 days on 248 V100 GPUs. In addition to +training costs, inference on large models is expensive due to +extensive hardware requirements. For example, using Codex- +Davinci to process 1000 tokens, including the prompt, costs +$0.02 USD [OpenAI, 2023]. In a spreadsheet environment +used by billions, these costs quickly add up. +In this paper, we present FLAME, a Formula LAnguage Model +for Excel trained exclusively on Excel formulas. FLAME is +based on T5-small [Raffel et al., 2020] and has only 60 mil- +lions parameters, yet it can compete with much larger models +(up to 175B parameters) on three formula authoring tasks: +last-mile repair, formula auto-completion and syntax recon- +struction. Syntax reconstruction is a novel task where all de- +limiters are removed from a formula, resulting in a flat stream +arXiv:2301.13779v1 [cs.PL] 31 Jan 2023 + +=SUMIF(B1:B5, A1:A5, "Yes") +=SUMIF(B1:B5, "Yes", A1:A5) +Last Mile Repair +=AVERAGEIFS(B4:M4 +=AVERAGEIFS(B4:M4, B4:M4, ">0") +Formula Autocompletion +Syntax Reconstruction +IFERROR VLOOKUP A2 Sheet2 $A$1:$E$22 5 0 "Not available" +=IFERROR(VLOOKUP(A2, Sheet2!$A$1:$E$22, 5, 0), "Not available") +Figure 2: We consider three downstream tasks: Last Mile Repair, +Formula Autocompletion, and Syntax Reconstruction. Red and green +colors denote the input and the expected output, respectively. Yellow +text denotes the buggy part of the formula in the repair task, where +the user has swapped the correct order of arguments resulting in a +type error. Each task shows a case that FLAME successfully solves. +of tokens, and the model must recover the original formula. +Figure 1 shows a high-level summary of results as a function +of model size on a public dataset, where FLAME can outper- +form larger models in all three tasks. Figure 2 provides real +examples, solved by FLAME, for each of these tasks. +There are three main challenges involved in training a model +for Excel formulas: obtaining diverse training data, tokenizing +their unique structure, and pretraining with objectives that +teach the model about this distinctive structure. Spreadsheets +contain many duplicate formulas due to copying down for- +mula cells. We reduced our corpus from 927M formulas down +to 6.1M by comparing formulas based on syntax, creating +540MB of training data. We combine formulas insights with +byte pair encoding (BPE) to train an Excel-specific tokenizer. +In addition to two generic objectives (tail-masking and de- +noising auto-encoding), we introduce two new pretraining +objectives designed for formulas: language-aware masked +span prediction and user-inspired denoising. +We extensively evaluate FLAME on three downstream tasks, +showing that our proposed solutions to the modeling chal- +lenges significantly improve the performance of FLAME over +T5-based models and can compete with much larger models. +Specifically, we find that FLAME can outperform other models +in 6 out of 10 settings in our evaluation. +We make the following contributions: +• We present FLAME, the first language model designed +exclusively for Excel formulas (§3). To this end, we +introduce domain-specific dataset curation (§3.2), tok- +enization (§3.3), and pretraining objectives (§3.4). +• We extensively evaluate FLAME on three formula assis- +tance tasks: last-mile repair, formula autocompletion, +and syntax reconstruction (§4.3). +• We compare our performance to two variants of Codex +(latest version of Cushman and Davinci) and CodeT5, +and finetune Cushman for downstream tasks (§4.1). We +show that FLAME can outperform larger models in 6 out +of 10 settings (§5.1). +• We analyze the contribution of different design choices +for FLAME (§5.2,§5.3) +2 +Related Work +Language models for code +Multiple popular language +model architectures have been successfully adapted to code. +CodeBERT [Feng et al., 2020] trained BERT (encoder) on nat- +ural language and code. CodeT5 [Wang et al., 2021] trained +T5 (encoder-decoder) on a similar corpus. Codex [Chen et +al., 2021a], PolyCoder [Xu et al., 2022], or CodeGen [Ni- +jkamp et al., 2022] are all trained variants of GPT (decoder). +These models are trained on multiple programming languages +and use pretraining objectives to understand or generate code +and natural language, but do not adapt them for specific lan- +guages. In contrast, FLAME exploits a single domain to use +domain-specific objectives, such as span masking that respects +programming language tokens, to learn a better representation. +Evaluating code models +Many tasks have been presented +to evaluate code models, and CodeXGLUE [Lu et al., 2021] +bundles most of these. These tasks are categorized by the +modality (text/code) of their input and output. FLAME is trained +on formulas exclusively and is focused on formula tasks. We +now describe related work for these tasks. +Formula repair +A popular code authoring task is repairing +small mistakes. DeepFix [Gupta et al., 2017], BIFI [Yasunaga +and Liang, 2021], Dr.Repair [Yasunaga and Liang, 2020], and +TFix [Berabi et al., 2021] use deep learning to perform syntax, +compilation, or diagnostics repair in general-purpose program- +ming languages. LaMirage [Bavishi et al., 2022] generates +repair engines for low-code languages and coins the term last- +mile repair for these types of fixes. RING [Joshi et al., 2022] +uses Codex to fix last-mile errors across multiple languages, +but it requires additional information, such as examples of +repairs and compiler messages. +Formula autocompletion +The generative nature of LLMCs +makes them serve as code-completion engines. This feature +has been shipped in commercial products, such as GitHub +Copilot in Visual studio Code [GitHub, 2021] and IntelliCode +in Visual Studio [Svyatkovskiy et al., 2020]. Spreadsheet- +Coder [Chen et al., 2021b] is a model designed for predicting +simple formulas from context in the spreadsheet. +Syntax reconstruction +Syntax reconstruction, where all de- +limiters in a formula are removed, resembles component-based +program synthesis, where partial programs are combined into +a program that satisfies a specification. Components are pro- +vided by a user [Jha et al., 2010], generated by a model [Rah- +mani et al., 2021], or defined by an API [Feng et al., 2017]. +3 +FLAME: Approach +We now describe the FLAME architecture and how it overcomes +the three key challenges (data, tokenization, and training) in +pretraining a general language model for formulas. +3.1 +Architecture +To facilitate both formula understanding and generation, +FLAME follows an encoder-decoder architecture based on T5 +[Raffel et al., 2020]. Encoder models like CodeBERT [Feng +et al., 2020] show remarkable code understanding capabilities. +Decoder models like CodeGen [Nijkamp et al., 2022] and + +User-inspired Denoising +INDEX(summary!N:N; MATCH(A350; +summary!$D:$D, 0, 0)) +INDEX(summary!N:N; MATCH(A350; +summary!$D:$D; 0)) +Change Function Arity +Comma to Semi colon +17 user-inspired noise operators +INDEX(summary!N:N, MATCH(A350, +summary!$D:$D, 0)) +INDEX(summary!N:N, MAT +Tail Masking +INDEX(summary!N:N, (A350, +summary!$D:$D, 0) +low mask rate, low average span length +INDEX2(summary!N:N, yeMATCH(A350, +summary!$[D:$D, 0)) +Random Noising +INDEX(!N:N, MATCHA350, +summary!:$D, 0) +high mask rate, low average span length +Language-Aware Span Masking +different combinations of high and low +masking rate and average span lengths +Figure 3: Four pretraining objectives used by FLAME. For each batch, we randomly (with weighted probability) choose one of the four objectives. +Generic objectives (tail masking and random noise) are shown with a yellow header, while formula-specific variants (language-aware span +masking and user-inspired noise) are shown with a green header. We depict inserted tokens with red and deleted tokens with blue. +Codex [Chen et al., 2021a] perform well on code generation. +Encoder-decoder models seek to blend these strengths. +3.2 +Training Data +We start from a dataset of 927M formulas drawn from a corpus +of 1.8M publicly available Excel workbooks.1 Each workbook +contains one or more worksheets, and each worksheet contains +zero or more formulas. Formulas in spreadsheets are often +repeated with minor cell reference changes across rows or +columns. For example, a user can drag a formula to another +cell to repeat a computation on neighboring cell values. +We compute formula sketches to preserve a single instance +of each unique formula per workbook. In a formula sketch, +numeric constants, string constants and cell references are +replaced by their token type. For example, the sketch of +=SUM(A1:A10) is =SUM(cell:cell). After applying sketch +deduplication, we are left with 6.1M formulas. Note that ap- +plying this globally to the corpus, rather than per workbook, +results in only 591K formulas. We found this globally dedu- +plicated corpus to be insufficient for training as it skews the +distribution of formulas —see evaluation (§5.2) for details. +3.3 +Tokenizing Formulas +Tokenization is an essential part of language models [Domingo +et al., 2018]. A popular method for tokenization is byte pair +encoding (BPE) [Sennrich et al., 2016]. BPE iteratively joins +consecutive tokens that appear together most frequently until +a target vocabulary size is reached. However, this procedure +can have adverse effects on formulas. For example, SUM and ( +are combined to get SUM(, which can reduce expressiveness +and hurt performance for tasks like repair. +Our tokenizer considers punctuation, whitespace, built-in +function names, and digits as individual tokens [Chowdhery +et al., 2022] and applies BPE [Radford et al., 2019] to the re- +maining parts of formulas, like string constants. Excel is case +insensitive (with the exception of string contents) so we con- +vert all input tokens to lowercase to map differently capitalized +tokens to a single token. For example, without lowercasing, +the same function SUM and sum will map to different tokens. +Example 1. A formula +=SUMIF(B1:B5, "Not available", A1:A5) +1These workbooks were collected as part of a large Excel corpus +planned for public release by a separate group of authors. +is tokenized as += sumif ( b 1 : +b 5 , ␣ " not ␣ available " +, ␣ a 1 : +a 5 ) +with space tokens denoted by ␣. +3.4 +Pretraining Objectives for Training +In this section, we describe the combination of generic and +Excel-specific pretraining objectives, as summarized in Fig- +ure 3, that we use to train FLAME. +Masking objectives +We use two forms of masking to pre-train FLAME, an Excel- +specific variant of masked span prediction (MSP), and a +generic tail masking objective. +Language-aware masked span prediction +In contrast to +traditional MSP, spans must respect Excel lexer bounds. For +example, when an Excel cell reference BC18 is divided into +four tokens B C 1 8, we ensure that either all or none of its +constituent tokens is masked. Consecutive masked tokens are +represented with a single token. Inspired by Mixture- +of-Denoisers [Tay et al., 2022], we mask spans of tokens using +combinations of high (35%) and low (15%) masking rates, and +big (6 tokens) and small (2 tokens) average span lengths. +Generic tail masking +We perform tail masking at the char- +acter level and allow partial masks of complete tokens. We +keep the leading {30%,40%,··· ,70%} tokens of the input +sequence and append a token. +Noisy Auto-encoding +Previous work in natural language processing has used denois- +ing auto-encoding during pretraining [Lewis et al., 2020]. We +incorporate two such objectives in FLAME. +Random Noise +We introduce generic noise by randomly +inserting, deleting, or updating tokens in the input sequence. +The insertion and update operators randomly sample a token +from the vocabulary. +Excel-specific user-inspired noise +We introduce noise op- +erators that mirror mistakes that real users might make when +writing Excel formulas. For example, users often write formu- +las with the incorrect function arity for in-built functions such +as SUMIF. We implement 17 noise operators (Appendix A) + +based on a combination of help forum and code analysis. We +randomly choose one of these noise operators when introduc- +ing noise into an input sequence. +Note that for all pretraining objectives, FLAME needs to +generate a complete formula (rather than just mask values). +Combining pretraining objectives +Rather than applying all pretraining objectives on every batch +and then combining losses, we pick a single objective for +each batch. We use the following probabilities {MSP: 50%, +tail masking: 20%, user-inspired denoising: 20%, random +denoising: 5%} for choosing the objective to be applied, and +with a 5% probability, we leave the sequence intact. +4 +Experimental Setup +We now describe our experimental setup. We start with the +baseline models we compare against (§4.1), the training setup +(§4.2), and then detail each downstream task in our evaluation, +along with their corresponding datasets (§4.3). +4.1 +Baselines and Configurations +We compare FLAME to the following much larger language +models, summarized in Table 1: +• CodeT5: a 220 million parameter T5-based encoder- +decoder model trained on both natural language and code. +We present fine-tuned results. +• Codex-Cushman: a 12 billion parameter autoregressive, +decoder-only, GPT-3-based model trained on both natural +language and code. We present both zeroshot and fine- +tuned results. +• Codex-Davinci: a 175 billion parameter autoregressive, +decoder-only, GPT-3-based model trained on both natural +language and code. We present zeroshot and few-shot +results. We do not have resources to fine-tune Davinci. +For Codex-based baselines, we use nucleus sampling [Holtz- +man et al., 2019] (temperature=0.7) and sample 50 sequences +per task. We sort these sequences based on their average token +log probabilities following [Joshi et al., 2022]. We detail the +prompts in Appendix B. For CodeT5, we use beam search with +a beam width of 50, and we consider the top 50 sequences. +4.2 +Training Details +We pretrain FLAME for 10 epochs and finetune CodeT5 and +FLAME on a cluster with 16 AMD MI200s, 96 cores and 900 +GB RAM. We finetune FLAME for 2 epochs for each down- +stream task and finetune CodeT5 for 25 epochs with a patience +of 5 epochs. We carry out all Codex experiments on a cluster +with 8 V100s, 40 cores, and 672 GB RAM. For Codex fine- +tuning we use low-rank adaptation (LoRA) [Hu et al., 2021]. +Refer to Appendix C for more details. +4.3 +Downstream Tasks +We consider three different downstream tasks. +System +Architecture +Number of parameters +Codex-Cushman +Decoder +12 billion +Codex-Davinci +Decoder +175 billion +CodeT5 (base) +Encoder-Decoder +220 million +FLAME (ours) +Encoder-Decoder +60 million +Table 1: Architecture and size comparison of baselines and FLAME +Last-mile Repair +Last-mile repair refers to repairs that require few edits and fix +syntax and simple semantic errors, such as wrong function call +arity. In this setting, FLAME is given the buggy formula as the +input sequence, and the task is to generate the user’s intended +(and syntactically correct) formula without any last-mile error. +Example 2. The user has used the wrong call arity for +ISERROR. Red highlights the error in the buggy formula, and +green denotes the required edit to match the groundtruth. +Buggy Formula: =IF(ISERROR(G6 *1.2, "" ) ) +Groundtruth Formula: =IF(ISERROR(G6 *1.2 ) , "") +Fine Tuning +We create a finetuning dataset for all systems +by taking 200K well-formed formulas from Excel help forums. +We then randomly apply our user-inspired noise operators to +generate broken versions. +Evaluation Metric +We compute an exact match with re- +spect to the ground truth repair. We consider the top 1 and top +5 candidates produced by each system per formula and report +the exact match fraction. +Benchmarks +We evaluate all systems on two benchmarks. +We use the collection of 273 labeled Excel formulas used +in recent last-mile repair literature [Joshi et al., 2022]. The +authors sourced these formulas from Excel help forums. We +refer to this benchmark set as Forum. +We also reserve a split of randomly sampled 500 formulas +derived using the same procedure as our finetuning dataset to +create a Test benchmark set. +Autocompletion +Code completion is a popular task for language models trained +on code, both due to its autoregressive nature and the practical +value of code completion as a feature in developers’ workflows. +In this setting, FLAME is given a formula prefix, and the task is +to generate the complete formula. +Example 3. Formula Autocompletion +Formula Prefix: =B2<=EDATE( +Formula Completion: =B2<=EDATE(TODAY(),-33) +Fine Tuning +We curated a finetuning dataset for autocom- +pletion by splitting 189k formulas and sampling a prefix length +of {0.2,··· ,0.7,0.8} fraction of tokens. +Evaluation Metric +When completing formulas, some parts +can be hard to predict due to lack of context [Guo et al., +2021], such as cell references, sheet names, string literals, and +numerics. Therefore, in addition to exatch match, we also +consider sketch match for autocompletion with respect to the +ground truth. Precisely, for sketch match, we use the same +sketch procedure described in §3. This uses the Excel lexer + +Model +Last Mile Repair +Syntax Reconstuction +Forum +Test +Forum +Test +T@1 +T@5 +T@1 +T@5 +T@1 +T@5 +T@1 +T@5 +Cushman +0.79 +0.88 +0.87 +0.93 +0.70 +0.80 +0.84 +0.91 +Davinci (FS) +0.76 +0.89 +0.54 +0.77 +0.62 +0.77 +0.61 +0.73 +CodeT5 (220M) +0.70 +0.84 +0.84 +0.90 +0.70 +0.84 +0.82 +0.89 +CodeT5 (60M) +0.72 +0.83 +0.82 +0.89 +0.65 +0.81 +0.83 +0.89 +FLAME +0.76 +0.89 +0.83 +0.91 +0.75 +0.89 +0.84 +0.89 +Table 2: Fine-tuned performance for Last Mile Repair and Syntax reconstruction tasks. Codex-Davinci uses few-shots and is denoted by an +FS suffix). FLAME outperforms larger models at last-mile repair in the Forum benchmark at top-5, and comes in second at top-1. In syntax +reconstruction, FLAME outperforms all models at both cutoffs in the Forum benchmark. Bold denotes best performing model and Underline +represents second best. +Models +Exact Match +Sketch Match +0.25 +0.50 +0.75 +0.90 +0.99 +0.25 +0.50 +0.75 +0.90 +0.99 +Cushman +0.0 +0.04 +0.27 +0.61 +0.86 +0.12 +0.26 +0.47 +0.71 +0.86 +Davinci +0.0 +0.03 +0.31 +0.64 +0.85 +0.10 +0.25 +0.53 +0.76 +0.85 +CodeT5 +0.0 +0.02 +0.10 +0.27 +0.21 +0.03 +0.09 +0.20 +0.39 +0.22 +FLAME +0.01 +0.06 +0.34 +0.70 +0.93 +0.10 +0.24 +0.55 +0.84 +0.94 +Table 3: Zeroshot autcompletion performance of FLAME, Codex-Cushman and Codex-Davinci, and fine-tuned CodeT5 (as denoted by FT +suffix). Given {0.25,0.50,0.75,0.90,0.99} fraction of formula prefix, we report the proportion of formulas completed in the top 5. We observe +that FLAME outperforms all the large language models in the exact match setting and most (3/5) of the sketch match settings. Bold denotes best +performing model and Underline represents second best. +to tokenize a formula and preserves built-in function names +but replaces all other tokens with their token type. We then +compare the sketches of the formulas for a match. For instance, +in Example 3, predicting the numeric −33 is highly contextual, +so in a sketch we match with its token type, Numeric. +Benchmarks +We evaluate autocompletion on a single bench- +mark, consisting of the 273 ground truth formulas from the +Forum last-mile repair benchmark. For each formula, given +exact match or sketch match metric, we predict completions +at 0.25, 0.5, 0.75, 0.90 and 0.99 fractions of formula prefix. +Syntax Reconstruction +We introduce a new task that we term syntax reconstruction. +The input to this task consists of Excel formulas which we +have processed to remove any delimiters, resulting in a flat +stream of lexer tokens. Excel delimiters are defined to be the +following set of tokens: {( ) ! +, ; { } [ ] .}. The +model is then tasked with generating the original formula with +appropriate delimiters. +Example 4. Syntax Reconstruction given the excel tokens. +Tokens: MAX 0 MOD C10 - B10 1 - D10 +Reconstruction: MAX(0,MOD(C10-B10,1)-D10) +Since, by definition, syntax reconstruction cannot introduce +tokens into the output that are not delimiters or not in the orig- +inal input token stream, FLAME employs constrained decoding +to greedily remove invalid candidates from the search space. +Our tokenizer design, particularly splitting on punctuation, +makes this decoding strategy easier to implement. +Fine Tuning +We curate a finetuning dataset by sampling +200k formulas from the publicly available Excel corpus that +we used for FLAME’s pretraining. We keep the subset that con- +tains at least one delimiter (139k) and remove all delimiters. +Evaluation Metric +We compute an exact match with re- +spect to the ground truth and consider the top 1 and top 5 +candidates produced by each system per formula. +Benchmarks +We derive a benchmark set from the last- +mile repair benchmarks by removing the delimiters for every +groundtruth formula. We refer to this benchmark as Forum. +Finally, we also consider a Test split that reflects the same +preparation as the fine tuning dataset. +5 +Evaluation +We explore the following research questions in our evaluation: +• RQ1: How does FLAME perform on formula intelligence +tasks compared to substantially larger language models? +• RQ2: How do pretraining design decisions such as data +curation, model size, pretraining objectives, and tokenizer +affect FLAME’s downstream performance? +• RQ3: How do various decoding strategies affect different +downstream-task performances for FLAME? +5.1 +RQ1: Larger Language Models +We now compare FLAME to substantially larger language mod- +els on our three formula intelligence tasks. +Last Mile Repair and Syntax Reconstruction +We finetune FLAME, CodeT5, and Codex-Cushman for last- +mile repair and syntax reconstruction, and use few-shot +prompts with three shots for Codex Davinci. Although one of + +Model +Last Mile Repair +Syntax Reconstuction +Forum +Test +Forum +Test +T@1 +T@5 +T@1 +T@5 +T@1 +T@5 +T@1 +T@5 +Cushman +0.55 +0.85 +0.41 +0.63 +0.27 +0.53 +0.23 +0.46 +Davinci +0.60 +0.82 +0.51 +0.75 +0.51 +0.65 +0.31 +0.45 +FLAME +0.71 +0.88 +0.74 +0.85 +0.41 +0.53 +0.50 +0.58 +Table 4: Zeroshot last-mile repair and syntax reconstruction performance of FLAME and Codex models. FLAME outperforms all the larger +models in Last Mile Repair task and solves more benchmarks than Codex-Cushman for the Syntax Reconstruction task. Bold denotes best +performing model and Underline represents second best. +Model +Zeroshot +Finetuned +LMR +SR +AC (EM) +AC (SM) +LMR +SR +Forum +Test +Forum +Test +0.75 +0.90 +0.75 +0.90 +Forum +Test +Forum +Test +FLAME (60M) +0.71 +0.74 +0.41 +0.50 +0.34 +0.70 +0.55 +0.84 +0.76 +0.83 +0.75 +0.84 +FLAME (16M) +0.68 +0.64 +0.23 +0.42 +0.24 +0.59 +0.54 +0.76 +0.73 +0.78 +0.73 +0.78 +Global Deduplication +0.57 +0.56 +0.16 +0.2 +0.15 +0.45 +0.41 +0.59 +0.68 +0.76 +0.73 +0.81 +T5 (Generic objectives and tokenizer) +0.11 +0.12 +0.02 +0.05 +0.07 +0.22 +0.25 +0.37 +0.62 +0.82 +0.49 +0.74 +Table 5: We compare multiple pretraining design decisions: model size, pretraining data curation, domain-specific pretraining objectives and +tokenizer. We consider at top-1 for Last-Mile Repair (LMR) and Syntax Reconstruction (SR) and top-5 for Autocompletion (AC) with Exact +Match (EM) and Sketch Match (SM). For details refer to Appendix D. Smaller model performs worse across the board. Curating data with +global deduplication reduces performance by up to 30 points. Removing domain-specific objectives and tokenizer impacts performance most. +our pretraining objectives closely resembles last-mile repair +(noisy auto-encoding) we find that finetuning FLAME helps +direct it towards a particular task. +We summarize the results in Table 2 and observe that on +the Forum last-mile repair benchmark FLAME outperforms all +models at top-5, and is second best to Codex-Cushman at top- +1. In the Test benchmark, we find that FLAME is second-best +to Codex-Cushman at top-5 and is close to CodeT5’s second- +best performance at top-1. In the Test benchmark, Davinci’s +performance is substantially worse than the fine-tuned models. +On further analysis, we found that all models solve 73% of +the Forum benchmark. FLAME solves 4% of the benchmarks +that no other model solves and fails on 1% of the benchmarks +that all other models fix. FLAME also generates syntactically +correct formulas for 98% of the benchmarks in top 5. In +Figure 4, we show examples where FLAME gets the correct +fix, and other models do not, and vice versa. We note that in +some cases, FLAME’s fixes appear to be more natural, but fail +to match the user’s ground truth repair. +For syntax reconstruction Forum, we find that FLAME outper- +forms other models across the top-1 and top-5. Interestingly, +CodeT5 also solves more syntax reconstruction tasks than +both Codex models. We hypothesize that since syntax recon- +struction is a new task, as compared to the more traditional +repair problem, after fine-tuning, encoder-decoder models per- +form better than decoder-only models, as shown by [Tay et +al., 2022]. In Test, we find that FLAME performs similar to +Codex-Cushman (same at top-1 and -2 points lower at top-5). +We find that 54% of the Forum syntax reconstruction bench- +marks are solved by all the models, 1% is solved only by +FLAME, and there are no benchmarks that all other models +solve but FLAME doesn’t. We attribute this performance to our +=IF('Jan 13'!B2="", 'Feb 13'!B2="", 'Mar 13'!B2="", 'Apr 13'!B2="", yes, no) +=IF(AND('Jan 13'!B2="", 'Feb 13'!B2="", 'Mar 13'!B2="", 'Apr 13'!B2=""), "yes", "no") +Buggy Formula +Ground Truth Fix +FLAME +Codex-Cushman +Codex-Davinci +CodeT5 +X +X +X +=VLOOKUP($Z25,$X$25:$Y:31,2,FALSE) +=VLOOKUP($Z25,$X$25:$Y31,2,FALSE) +Buggy Formula +Ground Truth Fix +FLAME +Codex-Cushman +Codex-Davinci +CodeT5 +X +=VLOOKUP($Z25,$X$25:$Y$31,2,FALSE) +FLAME +Example 1 +Example 2 +Figure 4: Repair tasks with diverging performance. In Example 1, the +user did not use the AND function and missed double quotes around +string literals yes and no. FLAME fixes this (in top-5), while other +models fail. In Example 2 FLAME’s top candidate is syntactically valid +but does not match the user’s fix, while other models’ predictions do. +pretraining design choices. First, FLAME learns to generate +syntactically correct code as a result of its noisy auto-encoding +pretraining objective. Second, FLAME learns the natural distri- +bution of formulas by generating complete sequences during +pretraining, rather than just mask values and sentinel tokens. +Zeroshot Performance +FLAME’s pretraining objectives al- +low us to consider zeroshot performance for both last-mile +repair and syntax reconstruction. In Table 4, we observe that +FLAME outperforms Codex models for last-mile repair across +all benchmarks. We attribute this to the closeness of our noisy +auto-encoding pretraining objectives and the last-mile repair +task. We find that in the syntax reconstruction task, FLAME out- +performs Codex-Cushman. We believe this is because syntax +reconstruction can be considered an extreme case of repair. + +Formula Autocompletion +The autoregressive nature of Codex models and FLAME’s pre- +training objectives allows us to evaluate their zeroshot per- +formance2 for formula auto-completion. Note that we fine- +tune CodeT5 for this task as it is pretrained on smaller span +lengths (1 to 5 tokens) and generates special mask tokens (e.g., +) in a zeroshot setting. We compute exact match and +sketch match metrics with top-5 results. +In Table 3, we observe that FLAME performs better than all +the larger models on the exact match metric and 3 out of 5 pre- +fix lengths for sketch match. We note that Codex-Cushman and +Codex-Davinci fail to complete 14% and 15% of the bench- +marks with 0.99 fraction of the prefix, respectively, whereas +FLAME fails to complete 6% of the benchmarks. We observe +significantly lower performance by CodeT5, likely due to +the lack of longer masks spans during pretraining. Surpris- +ingly, Codex-Davinci performs slightly worse than the smaller +Codex-Cushman for 3 out of 5 prefix lengths. Inspection of +completions shows that Codex-Davinci tends to generate more +tokens than required when completing these benchmark tasks. +We also observe cases where models succeed with a shorter +prefix but fail given a longer prefix. +5.2 +RQ2: Pretraining design decisions +We investigate FLAME’s data curation, model size, the use of +domain-specific pretraining objectives, and domain-specific +tokenizer, and present results in Table 5. +Training data curation +Previous work [Lee et al., 2021; Kandpal et al., 2022] have +shown that deduplication can improve the performance of +language models and reduce the memorization of training +data. Therefore, we curate a pretraining dataset by performing +workbook-level sketch-based formula deduplication. Alterna- +tively, one might consider performing global (pooled across +all workbooks) sketch-based deduplication. This alternative +results in a pretraining set of 591K formulas. Table 5 shows +that training on this smaller corpus results in a lower perfor- +mance model . We find that FLAME’s zeroshot performance +falls by 14 points and finetuned performance falls by 18 points +for last-mile repair in Forum benchmarks. +Model size +We trained two variants of FLAME with 16M and 60M parame- +ters. Table 5 compares FLAME-16M and FLAME-60M. We find +that performance declines slightly across tasks/benchmarks +when we reduce the model size to 16M. However, note that +FLAME-16M can still outperform larger models such as Codex +in 5 out of 10 zeroshot and finetuned settings, highlighting the +efficacy of our design choices for FLAME. +Pretraining objectives and Tokenizer +To evaluate the effectiveness of our domain-specific pretrain- +ing objectives and tokenizer, we pretrained a 60M parameters +T5 model with generic pertaining objectives and tokenizer. +Specifically, this model uses tail-masking, masked span pre- +diction without accounting for lexer token boundaries, and +2We finetuned Codex-Cushman and FLAME but observed worse +performance, possibly from over-fitting. +MAX C2 Sum C3:C4 SUM C5:C7 1 +MAX(C2, Sum(C3:C4),SUM(C5:C7),1) +MAX(C2,Sum!C3:C4,SUM(C5:C7),1) +Tokens +Formula +T5 (Generic Pretraining and Tokenizer) +Figure 5: Failing case of syntax reconstruction. Due to the different +capitalization of Sum and SUM, the model treats them as different +tokens, converting them to an identifier and a function, respectively. +random denoising objectives. Additionally, it uses the CodeT5 +tokenizer trained on our pretraining data. Table 5 shows that +this variant performs worse across all tasks and benchmarks, +both in a zeroshot and finetuned setting. We attribute the huge +drop, up to 62 points, in last-mile repair tasks in zeroshot to +our user-inspired denoising pretraining objective. Moreover, +we hypothesize that FLAME’s good syntax reconstruction per- +formance can be attributed to the domain-specific tokenizer. +Figure 5 illustrates how the generic tokenizer treats tokens +with different capitalizations, resulting in incorrect generation. +5.3 +RQ3: Decoding strategy +In Table 6, we evaluate FLAME using four different decoding +strategies, Beam Search, Group Beam Search [Vijayakumar et +al., 2016], Nucleus Sampling [Holtzman et al., 2019] and Top +K Sampling [Fan et al., 2018]. We find FLAME to perform bet- +ter with group beam search decoding (group size of 2) for all +the formula intelligence tasks. However, for autocompletion +with sketch match, nucleus sampling showed superior perfor- +mance. We believe this is because autocompletion requires +more diverse results, particularly at shorter prefixes. Refer to +Appendix E for autocompletion table. +Decoding Method +LMR (Forum) +SR (Forum) +T@1 +T@5 +T@1 +T@5 +Beam Search +0.76 +0.88 +0.75 +0.89 +Group Beam +0.76 +0.89 +0.75 +0.89 +Nucleus Sampling +0.72 +0.85 +0.7 +0.84 +Top K +0.67 +0.86 +0.67 +0.84 +Table 6: Performance by decoder strategy for last mile repair (LMR) +and syntax reconstruction (SR). Beam and Grouped Beam Search +have similar performance, and outperform Nucleus, Top K Sampling. +6 +Conclusions and Future Work +We present FLAME, a small (60M parameter) language model +for spreadsheet formulas, which captures domain-specific +properties in its data curation, tokenization, and pretraining +objectives. We implemented FLAME for Excel formulas and +evaluate on three downstream tasks: last-mile repair, autocom- +pletion, and a novel task that we term syntax reconstruction. +We compare with the much larger models CodeT5, Codex- +Cushman, and Codex-Davinci. 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Codet5: Identifier-aware unified pre- +trained encoder-decoder models for code understanding +and generation. arXiv preprint arXiv:2109.00859, 2021. +[Xu et al., 2022] Frank F Xu, Uri Alon, Graham Neubig, and +Vincent Josua Hellendoorn. A systematic evaluation of +large language models of code. +In Proceedings of the +6th ACM SIGPLAN International Symposium on Machine +Programming, pages 1–10, 2022. +[Yasunaga and Liang, 2020] Michihiro Yasunaga and Percy +Liang. Graph-based, self-supervised program repair from +diagnostic feedback. In International Conference on Ma- +chine Learning, pages 10799–10808. PMLR, 2020. +[Yasunaga and Liang, 2021] Michihiro Yasunaga and Percy +Liang. Break-it-fix-it: Unsupervised learning for program +repair. In International Conference on Machine Learning, +pages 11941–11952. PMLR, 2021. + +A +User noise operators +We implement the following noise operators: +1. Wrong Range: we replace the range operator :, with +one of the following symbols: {; , space "}, or we +delete the range operator. +2. Malformed Range: +A range consists of 4 el- +ements: +col1, +row1, +col2, +row2 +written +as +col1row1:col2row2. We randomly delete one of these +elements. For eg: col1:col2row2 +3. Space between Function and Arguments in a Call: +We introduce a space between the function name and +the opening parentheses for built-in functions. For exam- +ple: SUM(A1:A10) converts to SUM (A1:A10) +4. Change number of arguments: We change the num- +ber of arguments for functions with fixed function arity. +For example, IF has a minimum arity of 2 and maxi- +mum arity of 3. Specifically, if a function contains ar- +guments equal to its minimum function arity, then we +randomly delete one argument. Whereas, if the func- +tion’s max arity is equal to the number of arguments, +then we randomly copy one of the existing arguments +and pass it as an additional argument to the function. +For example, IF(A2>10, True, False) can become +IF(A2>10, True, False, False) +5. Swap arguments: If a function takes different types of +arguments, then we swap these arguments. For example: +IF(A1>10, 1, 2) can become IF(1, A1>10, 2). +6. Space between relational operators: +We add space +between relational operators, such as < =. +7. Swap relational operators: We swap relational opera- +tors, such as <= turns to =< +8. Inequality noise operator: In Excel <> is the inequality +operator. We replace this with the incorrect != or =!. +9. Invalid Equality: We also corrupt the equality operator. +The equality operator in Excel is =, we replace it with == +or ===. +10. Malformed Sheet Name: Multi-word sheet names in +Excel need to be enclosed within single quotes (’’). We randomly choose to either delete the single +quotes or replace them with double quotes. For example, +’Sheet 1’!A10 can become "Sheet 1"!A10. +11. Remove exclamation Mark: +In Excel, sheet names +are followed by an exclamation mark to denote sheet +reference. We delete this exclamation mark. +12. Malformed Strings: We corrupt strings by either delet- +ing the double quotes or replacing them with single +quotes. +13. Add Comma and Remove Parentheses: We randomly +choose to either insert a comma before a closing parenthe- +sis or insert a comma and delete the closing parentheses. +14. Add random operators: We define a set of operators +that we randomly insert into the formula at a random +position. These operators are: {+ - * / ^& < > = . +) +#} +15. Add operator at the end: +We randomly add one of +the operators mentioned previously at the end of the se- +quence. +16. Add Parentheses: We add opening and closing paren- +thesis at random places. +17. Corrupting Unreliable tokens: Following [Bavishi et +al., 2022], we randomly add, delete or replace unreliable +tokens. Unreliable tokens are tokens where users often +make mistakes, defined to be delimiters. +B +Codex Prompts +For all our codex experiments, we use the following prompts +for zeroshot and finetuning and use a temperature of 0.7 +B.1 +Repair - Zeroshot and Finetuning +##### Fix bugs in the below code +### Buggy Excel + +### Fixed Excel + +##### Fix bugs in the below code +### Buggy Excel +=SUMIFS( +Master!$P:$P, +Master!$F:$F,$A7, +Master856!$E:212Systems$B7 +) +### Fixed Excel +B.2 +Syntax Reconstruction - Zeroshot and +Finetuning +### Excel Tokens + +### Complete Excel Formula + +### Excel Tokens +INDEX Table1 SMALL IF +Table1 COMPANY_NAME = $E$1 +ROW Table1 COMPANY_NAME - 1 +ROW 2:2 3 +### Complete Excel Formula +B.3 +Autocomplete - Zeroshot +### Excel Formula + + +### Excel Formula +IF(FALSE,NA( + +Models +Exact Match +Sketch Match +0.25 +0.50 +0.75 +0.90 +0.99 +0.25 +0.50 +0.75 +0.90 +0.99 +FLAME (60M) +0.01 +0.06 +0.34 +0.70 +0.93 +0.10 +0.24 +0.55 +0.84 +0.94 +FLAME (16M) +0 +0.03 +0.24 +0.59 +0.89 +0.11 +0.25 +0.54 +0.76 +0.90 +Global Deduplication +0 +0.03 +0.15 +0.45 +0.64 +0.10 +0.25 +0.41 +0.59 +0.70 +T5 (Generic objectives and tokenizer) +0 +0.07 +0.07 +0.22 +0.21 +0.01 +0.09 +0.25 +0.37 +0.29 +Table 7: Design choice experiments for autocompletion task. We compare multiple pretraining design decisions: model size, pretraining data +curation, domain-specific pretraining objectives and tokenizer. We consider top-5 for Autocompletion (AC) with Exact Match (EM) and Sketch +Match (SM). We note that FLAME outperforms all the models. +Models +Exact Match +SketchMatch +0.25 +0.5 +0.75 +0.9 +Total +0.25 +0.5 +0.75 +0.9 +Total +Beam Search +0.00 +0.06 +0.33 +0.71 +0.92 +0.10 +0.25 +0.54 +0.82 +0.94 +Group Beam Search (groups = 2) +0.01 +0.06 +0.34 +0.70 +0.93 +0.10 +0.24 +0.55 +0.84 +0.94 +Nucleus Sampling +0.00 +0.04 +0.26 +0.59 +0.92 +0.14 +0.30 +0.53 +0.74 +0.92 +TopK Sampling +0.00 +0.04 +0.25 +0.62 +0.92 +0.15 +0.30 +0.55 +0.76 +0.92 +Table 8: Performance by decoder strategy for Autocompletion (top 5) with Exact Match and Sketch Match. Beam Search outperforms all the +strategies – Group Beam Search with a group size of 2, Nucleus Sampling, and Top K Sampling. +C +Training details +We use the following HuggingFace configuration to train +FLAME: +{ +"architectures": [ +"T5ForConditionalGeneration" +], +"d_ff": 1024, +"d_kv": 64, +"d_model": 512, +"decoder_start_token_id": 0, +"dropout_rate": 0.1, +"bos_token_id": 1, +"eos_token_id": 2, +"feed_forward_proj": "gated-gelu", +"initializer_factor": 1.0, +"is_encoder_decoder": true, +"layer_norm_epsilon": 1e-06, +"model_type": "t5", +"num_decoder_layers": 8, +"num_heads": 6, +"num_layers": 8, +"output_past": true, +"pad_token_id": 0, +"relative_attention_num_buckets": 32, +"tie_word_embeddings": false, +"vocab_size": 16479 +} +We use an AdaFactor optimizer, with 1e-4 learning rate, +clip factor of 1.0, with scale parameters and relative steps set +to false. For fine-tuning, we use a weight decay of 0.1 We use +linear learning rate schedule with 100 warm-up steps. +D +Design Decision (Autocompletion) +We detail our autocompletion evaluation where we evaluate +FLAME against different variations in Table 7. We observe that +FLAME beats all the different model variants. +E +Decoder Autocompletion +In Table 8, we detail autocompletion results for different de- +coding strategies. We find that Beam Search beats other decod- +ing methods in 7 out of 10 prefix lengths, and Top K Sampling +beats others in Sketch Match for smaller fractions of prefixes. + diff --git a/09FST4oBgHgl3EQfWDi_/content/tmp_files/load_file.txt b/09FST4oBgHgl3EQfWDi_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad5771f438b0537ae7a2840166dd7729fdcf3d5a --- /dev/null +++ b/09FST4oBgHgl3EQfWDi_/content/tmp_files/load_file.txt @@ -0,0 +1,962 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf,len=961 +page_content='FLAME: A small language model for spreadsheet formulas Harshit Joshi1 , Abishai Ebenezer1 , Jos´e Cambronero2∗ , Sumit Gulwani2∗ , Aditya Kanade3∗ , Vu Le2∗ , Ivan Radiˇcek4∗ , Gust Verbruggen5∗ 1Microsoft, India 2Microsoft, USA 3Microsoft Research, India 4Microsoft, Croatia 5Microsoft, Belgium {t-hjoshi, t-aebenezer, jcambronero, sumitg, kanadeaditya, levu, ivradice, gverbruggen}@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='com Abstract The widespread use of spreadsheet environments by billions of users presents a unique opportunity for formula-authoring assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Although large language models, such as Codex, can assist in general-purpose languages, they are expensive to train and challenging to deploy due to their large model sizes (up to billions of parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' More- over, they require hundreds of gigabytes of train- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We present FLAME, a T5-based model trained on Excel formulas that leverages domain insights to achieve competitive performance with a substantially smaller model (60M parameters) and two orders of magnitude less training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We cu- rate a training dataset using sketch deduplication, introduce an Excel-specific formula tokenizer for our model, and use domain-specific versions of masked span prediction and noisy auto-encoding as pretraining objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We evaluate FLAME on formula repair, formula auto-completion, and a novel task called syntax reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' FLAME (60M) can outperform much larger models, such as Codex-Davinci (175B), Codex-Cushman (12B), and CodeT5 (220M), in 6 out of 10 settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 1 Introduction Despite a much larger user base, spreadsheet environments do not have access to nearly the same range of productivity tools as available for general programming environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' The latter typically have code completion, refactoring, linting, and a wide range of extensions for additional functionality, like generating tests, inserting code snippets, and summarizing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Many of these advanced programming assistance tools are driven by advances in large language models trained on code (LLMCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Codex [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021a] is used for code completion [GitHub, 2021] and repair [Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022], AlphaCode [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022a] solves competitive programming problems, [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022b] built a code review system, and many other models show great performance in code related tasks [Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Fried et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Nijkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' ∗Listed in alphabetical order Formula Autocompletion Last Mile Repair Syntax Reconstruction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 FLAME (16M) FLAME (60M) CodeT5 (220M) Codex Cushman (12B) Codex Davinci (175B) Model Parameters (log-scale) Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='40 Z Figure 1: A summary of model comparisons in fine-tuned setting for different formula assistance tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We show the results under a top-5 cutoff on a public excel benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Note that all Codex-Davinci results are few-shot, and Autocompletion is zeroshot for all systems except CodeT5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For Autocompletion, results represent the fraction of benchmarks successfully (based on a sketch match metric) completed given 90% of the prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' To capture the complexity and variety of code and com- ments in different languages, these models need billions of parameters—the smallest variant of Codex, used by GitHub Copilot, has 12 billion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' As a result, these models are trained for long periods on corpora containing millions of programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, Incoder 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='7B used 159GB of code over a period of 24 days on 248 V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In addition to training costs, inference on large models is expensive due to extensive hardware requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, using Codex- Davinci to process 1000 tokens, including the prompt, costs $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='02 USD [OpenAI, 2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In a spreadsheet environment used by billions, these costs quickly add up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In this paper, we present FLAME, a Formula LAnguage Model for Excel trained exclusively on Excel formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' FLAME is based on T5-small [Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2020] and has only 60 mil- lions parameters, yet it can compete with much larger models (up to 175B parameters) on three formula authoring tasks: last-mile repair, formula auto-completion and syntax recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Syntax reconstruction is a novel task where all de- limiters are removed from a formula, resulting in a flat stream arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='13779v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='PL] 31 Jan 2023 =SUMIF(B1:B5, A1:A5, "Yes") =SUMIF(B1:B5, "Yes", A1:A5) Last Mile Repair =AVERAGEIFS(B4:M4 =AVERAGEIFS(B4:M4, B4:M4, ">0") Formula Autocompletion Syntax Reconstruction IFERROR VLOOKUP A2 Sheet2 $A$1:$E$22 5 0 "Not available" =IFERROR(VLOOKUP(A2, Sheet2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$A$1:$E$22, 5, 0), "Not available") Figure 2: We consider three downstream tasks: Last Mile Repair, Formula Autocompletion, and Syntax Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Red and green colors denote the input and the expected output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Yellow text denotes the buggy part of the formula in the repair task, where the user has swapped the correct order of arguments resulting in a type error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Each task shows a case that FLAME successfully solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' of tokens, and the model must recover the original formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Figure 1 shows a high-level summary of results as a function of model size on a public dataset, where FLAME can outper- form larger models in all three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Figure 2 provides real examples, solved by FLAME, for each of these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' There are three main challenges involved in training a model for Excel formulas: obtaining diverse training data, tokenizing their unique structure, and pretraining with objectives that teach the model about this distinctive structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Spreadsheets contain many duplicate formulas due to copying down for- mula cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We reduced our corpus from 927M formulas down to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1M by comparing formulas based on syntax, creating 540MB of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We combine formulas insights with byte pair encoding (BPE) to train an Excel-specific tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In addition to two generic objectives (tail-masking and de- noising auto-encoding), we introduce two new pretraining objectives designed for formulas: language-aware masked span prediction and user-inspired denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We extensively evaluate FLAME on three downstream tasks, showing that our proposed solutions to the modeling chal- lenges significantly improve the performance of FLAME over T5-based models and can compete with much larger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Specifically, we find that FLAME can outperform other models in 6 out of 10 settings in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We make the following contributions: We present FLAME, the first language model designed exclusively for Excel formulas (§3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' To this end, we introduce domain-specific dataset curation (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2), tok- enization (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='3), and pretraining objectives (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We extensively evaluate FLAME on three formula assis- tance tasks: last-mile repair, formula autocompletion, and syntax reconstruction (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We compare our performance to two variants of Codex (latest version of Cushman and Davinci) and CodeT5, and finetune Cushman for downstream tasks (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We show that FLAME can outperform larger models in 6 out of 10 settings (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We analyze the contribution of different design choices for FLAME (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2,§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='3) 2 Related Work Language models for code Multiple popular language model architectures have been successfully adapted to code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' CodeBERT [Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2020] trained BERT (encoder) on nat- ural language and code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' CodeT5 [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021] trained T5 (encoder-decoder) on a similar corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Codex [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021a], PolyCoder [Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022], or CodeGen [Ni- jkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] are all trained variants of GPT (decoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' These models are trained on multiple programming languages and use pretraining objectives to understand or generate code and natural language, but do not adapt them for specific lan- guages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In contrast, FLAME exploits a single domain to use domain-specific objectives, such as span masking that respects programming language tokens, to learn a better representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Evaluating code models Many tasks have been presented to evaluate code models, and CodeXGLUE [Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021] bundles most of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' These tasks are categorized by the modality (text/code) of their input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' FLAME is trained on formulas exclusively and is focused on formula tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We now describe related work for these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Formula repair A popular code authoring task is repairing small mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' DeepFix [Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2017], BIFI [Yasunaga and Liang, 2021], Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='Repair [Yasunaga and Liang, 2020], and TFix [Berabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021] use deep learning to perform syntax, compilation, or diagnostics repair in general-purpose program- ming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' LaMirage [Bavishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] generates repair engines for low-code languages and coins the term last- mile repair for these types of fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' RING [Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] uses Codex to fix last-mile errors across multiple languages, but it requires additional information, such as examples of repairs and compiler messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Formula autocompletion The generative nature of LLMCs makes them serve as code-completion engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' This feature has been shipped in commercial products, such as GitHub Copilot in Visual studio Code [GitHub, 2021] and IntelliCode in Visual Studio [Svyatkovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Spreadsheet- Coder [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021b] is a model designed for predicting simple formulas from context in the spreadsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Syntax reconstruction Syntax reconstruction, where all de- limiters in a formula are removed, resembles component-based program synthesis, where partial programs are combined into a program that satisfies a specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Components are pro- vided by a user [Jha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2010], generated by a model [Rah- mani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021], or defined by an API [Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 3 FLAME: Approach We now describe the FLAME architecture and how it overcomes the three key challenges (data, tokenization, and training) in pretraining a general language model for formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1 Architecture To facilitate both formula understanding and generation, FLAME follows an encoder-decoder architecture based on T5 [Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Encoder models like CodeBERT [Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2020] show remarkable code understanding capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Decoder models like CodeGen [Nijkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] and User-inspired Denoising INDEX(summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='N:N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' MATCH(A350;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$D:$D, 0, 0)) INDEX(summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='N:N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' MATCH(A350;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$D:$D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 0)) Change Function Arity Comma to Semi colon 17 user-inspired noise operators INDEX(summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='N:N, MATCH(A350, summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$D:$D, 0)) INDEX(summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='N:N, MAT Tail Masking INDEX(summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='N:N, (A350, summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$D:$D, 0) low mask rate, low average span length INDEX2(summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='N:N, yeMATCH(A350, summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$[D:$D, 0)) Random Noising INDEX(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='N:N, MATCHA350, summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=':$D, 0) high mask rate, low average span length Language-Aware Span Masking different combinations of high and low masking rate and average span lengths Figure 3: Four pretraining objectives used by FLAME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For each batch, we randomly (with weighted probability) choose one of the four objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Generic objectives (tail masking and random noise) are shown with a yellow header, while formula-specific variants (language-aware span masking and user-inspired noise) are shown with a green header.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We depict inserted tokens with red and deleted tokens with blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Codex [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021a] perform well on code generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Encoder-decoder models seek to blend these strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2 Training Data We start from a dataset of 927M formulas drawn from a corpus of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='8M publicly available Excel workbooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1 Each workbook contains one or more worksheets, and each worksheet contains zero or more formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Formulas in spreadsheets are often repeated with minor cell reference changes across rows or columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, a user can drag a formula to another cell to repeat a computation on neighboring cell values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We compute formula sketches to preserve a single instance of each unique formula per workbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In a formula sketch, numeric constants, string constants and cell references are replaced by their token type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, the sketch of =SUM(A1:A10) is =SUM(cell:cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' After applying sketch deduplication, we are left with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1M formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Note that ap- plying this globally to the corpus, rather than per workbook, results in only 591K formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We found this globally dedu- plicated corpus to be insufficient for training as it skews the distribution of formulas —see evaluation (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='3 Tokenizing Formulas Tokenization is an essential part of language models [Domingo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' A popular method for tokenization is byte pair encoding (BPE) [Sennrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' BPE iteratively joins consecutive tokens that appear together most frequently until a target vocabulary size is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' However, this procedure can have adverse effects on formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, SUM and ( are combined to get SUM(, which can reduce expressiveness and hurt performance for tasks like repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Our tokenizer considers punctuation, whitespace, built-in function names, and digits as individual tokens [Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] and applies BPE [Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2019] to the re- maining parts of formulas, like string constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Excel is case insensitive (with the exception of string contents) so we con- vert all input tokens to lowercase to map differently capitalized tokens to a single token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, without lowercasing, the same function SUM and sum will map to different tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' A formula =SUMIF(B1:B5, "Not available", A1:A5) 1These workbooks were collected as part of a large Excel corpus planned for public release by a separate group of authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' is tokenized as = sumif ( b 1 : b 5 , ␣ " not ␣ available " , ␣ a 1 : a 5 ) with space tokens denoted by ␣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='4 Pretraining Objectives for Training In this section, we describe the combination of generic and Excel-specific pretraining objectives, as summarized in Fig- ure 3, that we use to train FLAME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Masking objectives We use two forms of masking to pre-train FLAME, an Excel- specific variant of masked span prediction (MSP), and a generic tail masking objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Language-aware masked span prediction In contrast to traditional MSP, spans must respect Excel lexer bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, when an Excel cell reference BC18 is divided into four tokens B C 1 8, we ensure that either all or none of its constituent tokens is masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Consecutive masked tokens are represented with a single token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Inspired by Mixture- of-Denoisers [Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022], we mask spans of tokens using combinations of high (35%) and low (15%) masking rates, and big (6 tokens) and small (2 tokens) average span lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Generic tail masking We perform tail masking at the char- acter level and allow partial masks of complete tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We keep the leading {30%,40%,··· ,70%} tokens of the input sequence and append a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Noisy Auto-encoding Previous work in natural language processing has used denois- ing auto-encoding during pretraining [Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We incorporate two such objectives in FLAME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Random Noise We introduce generic noise by randomly inserting, deleting, or updating tokens in the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' The insertion and update operators randomly sample a token from the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Excel-specific user-inspired noise We introduce noise op- erators that mirror mistakes that real users might make when writing Excel formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, users often write formu- las with the incorrect function arity for in-built functions such as SUMIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We implement 17 noise operators (Appendix A) based on a combination of help forum and code analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We randomly choose one of these noise operators when introduc- ing noise into an input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Note that for all pretraining objectives, FLAME needs to generate a complete formula (rather than just mask values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Combining pretraining objectives Rather than applying all pretraining objectives on every batch and then combining losses, we pick a single objective for each batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We use the following probabilities {MSP: 50%, tail masking: 20%, user-inspired denoising: 20%, random denoising: 5%} for choosing the objective to be applied, and with a 5% probability, we leave the sequence intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 4 Experimental Setup We now describe our experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We start with the baseline models we compare against (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1), the training setup (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2), and then detail each downstream task in our evaluation, along with their corresponding datasets (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1 Baselines and Configurations We compare FLAME to the following much larger language models, summarized in Table 1: CodeT5: a 220 million parameter T5-based encoder- decoder model trained on both natural language and code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We present fine-tuned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Codex-Cushman: a 12 billion parameter autoregressive, decoder-only, GPT-3-based model trained on both natural language and code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We present both zeroshot and fine- tuned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Codex-Davinci: a 175 billion parameter autoregressive, decoder-only, GPT-3-based model trained on both natural language and code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We present zeroshot and few-shot results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We do not have resources to fine-tune Davinci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For Codex-based baselines, we use nucleus sampling [Holtz- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2019] (temperature=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='7) and sample 50 sequences per task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We sort these sequences based on their average token log probabilities following [Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We detail the prompts in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For CodeT5, we use beam search with a beam width of 50, and we consider the top 50 sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2 Training Details We pretrain FLAME for 10 epochs and finetune CodeT5 and FLAME on a cluster with 16 AMD MI200s, 96 cores and 900 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We finetune FLAME for 2 epochs for each down- stream task and finetune CodeT5 for 25 epochs with a patience of 5 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We carry out all Codex experiments on a cluster with 8 V100s, 40 cores, and 672 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For Codex fine- tuning we use low-rank adaptation (LoRA) [Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Refer to Appendix C for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='3 Downstream Tasks We consider three different downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' System Architecture Number of parameters Codex-Cushman Decoder 12 billion Codex-Davinci Decoder 175 billion CodeT5 (base) Encoder-Decoder 220 million FLAME (ours) Encoder-Decoder 60 million Table 1: Architecture and size comparison of baselines and FLAME Last-mile Repair Last-mile repair refers to repairs that require few edits and fix syntax and simple semantic errors, such as wrong function call arity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In this setting, FLAME is given the buggy formula as the input sequence, and the task is to generate the user’s intended (and syntactically correct) formula without any last-mile error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' The user has used the wrong call arity for ISERROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Red highlights the error in the buggy formula, and green denotes the required edit to match the groundtruth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Buggy Formula: =IF(ISERROR(G6 *1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2, "" ) ) Groundtruth Formula: =IF(ISERROR(G6 *1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2 ) , "") Fine Tuning We create a finetuning dataset for all systems by taking 200K well-formed formulas from Excel help forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We then randomly apply our user-inspired noise operators to generate broken versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Evaluation Metric We compute an exact match with re- spect to the ground truth repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We consider the top 1 and top 5 candidates produced by each system per formula and report the exact match fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Benchmarks We evaluate all systems on two benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We use the collection of 273 labeled Excel formulas used in recent last-mile repair literature [Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' The authors sourced these formulas from Excel help forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We refer to this benchmark set as Forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We also reserve a split of randomly sampled 500 formulas derived using the same procedure as our finetuning dataset to create a Test benchmark set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Autocompletion Code completion is a popular task for language models trained on code, both due to its autoregressive nature and the practical value of code completion as a feature in developers’ workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In this setting, FLAME is given a formula prefix, and the task is to generate the complete formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Formula Autocompletion Formula Prefix: =B2<=EDATE( Formula Completion: =B2<=EDATE(TODAY(),-33) Fine Tuning We curated a finetuning dataset for autocom- pletion by splitting 189k formulas and sampling a prefix length of {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2,··· ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='7,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='8} fraction of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Evaluation Metric When completing formulas, some parts can be hard to predict due to lack of context [Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021], such as cell references, sheet names, string literals, and numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Therefore, in addition to exatch match, we also consider sketch match for autocompletion with respect to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Precisely, for sketch match, we use the same sketch procedure described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' This uses the Excel lexer Model Last Mile Repair Syntax Reconstuction Forum Test Forum Test T@1 T@5 T@1 T@5 T@1 T@5 T@1 T@5 Cushman 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='91 Davinci (FS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='73 CodeT5 (220M) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 CodeT5 (60M) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 FLAME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 Table 2: Fine-tuned performance for Last Mile Repair and Syntax reconstruction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Codex-Davinci uses few-shots and is denoted by an FS suffix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' FLAME outperforms larger models at last-mile repair in the Forum benchmark at top-5, and comes in second at top-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In syntax reconstruction, FLAME outperforms all models at both cutoffs in the Forum benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Bold denotes best performing model and Underline represents second best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Models Exact Match Sketch Match 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='99 Cushman 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='86 Davinci 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='85 CodeT5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='22 FLAME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='94 Table 3: Zeroshot autcompletion performance of FLAME, Codex-Cushman and Codex-Davinci, and fine-tuned CodeT5 (as denoted by FT suffix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Given {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='50,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='99} fraction of formula prefix, we report the proportion of formulas completed in the top 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We observe that FLAME outperforms all the large language models in the exact match setting and most (3/5) of the sketch match settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Bold denotes best performing model and Underline represents second best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' to tokenize a formula and preserves built-in function names but replaces all other tokens with their token type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We then compare the sketches of the formulas for a match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For instance, in Example 3, predicting the numeric −33 is highly contextual, so in a sketch we match with its token type, Numeric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Benchmarks We evaluate autocompletion on a single bench- mark, consisting of the 273 ground truth formulas from the Forum last-mile repair benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For each formula, given exact match or sketch match metric, we predict completions at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='99 fractions of formula prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Syntax Reconstruction We introduce a new task that we term syntax reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' The input to this task consists of Excel formulas which we have processed to remove any delimiters, resulting in a flat stream of lexer tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Excel delimiters are defined to be the following set of tokens: {( ) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' { } [ ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' The model is then tasked with generating the original formula with appropriate delimiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Syntax Reconstruction given the excel tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Tokens: MAX 0 MOD C10 - B10 1 - D10 Reconstruction: MAX(0,MOD(C10-B10,1)-D10) Since, by definition, syntax reconstruction cannot introduce tokens into the output that are not delimiters or not in the orig- inal input token stream, FLAME employs constrained decoding to greedily remove invalid candidates from the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Our tokenizer design, particularly splitting on punctuation, makes this decoding strategy easier to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Fine Tuning We curate a finetuning dataset by sampling 200k formulas from the publicly available Excel corpus that we used for FLAME’s pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We keep the subset that con- tains at least one delimiter (139k) and remove all delimiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Evaluation Metric We compute an exact match with re- spect to the ground truth and consider the top 1 and top 5 candidates produced by each system per formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Benchmarks We derive a benchmark set from the last- mile repair benchmarks by removing the delimiters for every groundtruth formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We refer to this benchmark as Forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Finally, we also consider a Test split that reflects the same preparation as the fine tuning dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 5 Evaluation We explore the following research questions in our evaluation: RQ1: How does FLAME perform on formula intelligence tasks compared to substantially larger language models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' RQ2: How do pretraining design decisions such as data curation, model size, pretraining objectives, and tokenizer affect FLAME’s downstream performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' RQ3: How do various decoding strategies affect different downstream-task performances for FLAME?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1 RQ1: Larger Language Models We now compare FLAME to substantially larger language mod- els on our three formula intelligence tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Last Mile Repair and Syntax Reconstruction We finetune FLAME, CodeT5, and Codex-Cushman for last- mile repair and syntax reconstruction, and use few-shot prompts with three shots for Codex Davinci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Although one of Model Last Mile Repair Syntax Reconstuction Forum Test Forum Test T@1 T@5 T@1 T@5 T@1 T@5 T@1 T@5 Cushman 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='46 Davinci 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='45 FLAME 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='58 Table 4: Zeroshot last-mile repair and syntax reconstruction performance of FLAME and Codex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' FLAME outperforms all the larger models in Last Mile Repair task and solves more benchmarks than Codex-Cushman for the Syntax Reconstruction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Bold denotes best performing model and Underline represents second best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Model Zeroshot Finetuned LMR SR AC (EM) AC (SM) LMR SR Forum Test Forum Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 Forum Test Forum Test FLAME (60M) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 FLAME (16M) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='78 Global Deduplication 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='81 T5 (Generic objectives and tokenizer) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='74 Table 5: We compare multiple pretraining design decisions: model size, pretraining data curation, domain-specific pretraining objectives and tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We consider at top-1 for Last-Mile Repair (LMR) and Syntax Reconstruction (SR) and top-5 for Autocompletion (AC) with Exact Match (EM) and Sketch Match (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For details refer to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Smaller model performs worse across the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Curating data with global deduplication reduces performance by up to 30 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Removing domain-specific objectives and tokenizer impacts performance most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' our pretraining objectives closely resembles last-mile repair (noisy auto-encoding) we find that finetuning FLAME helps direct it towards a particular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We summarize the results in Table 2 and observe that on the Forum last-mile repair benchmark FLAME outperforms all models at top-5, and is second best to Codex-Cushman at top- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In the Test benchmark, we find that FLAME is second-best to Codex-Cushman at top-5 and is close to CodeT5’s second- best performance at top-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In the Test benchmark, Davinci’s performance is substantially worse than the fine-tuned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' On further analysis, we found that all models solve 73% of the Forum benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' FLAME solves 4% of the benchmarks that no other model solves and fails on 1% of the benchmarks that all other models fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' FLAME also generates syntactically correct formulas for 98% of the benchmarks in top 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In Figure 4, we show examples where FLAME gets the correct fix, and other models do not, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We note that in some cases, FLAME’s fixes appear to be more natural, but fail to match the user’s ground truth repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For syntax reconstruction Forum, we find that FLAME outper- forms other models across the top-1 and top-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Interestingly, CodeT5 also solves more syntax reconstruction tasks than both Codex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We hypothesize that since syntax recon- struction is a new task, as compared to the more traditional repair problem, after fine-tuning, encoder-decoder models per- form better than decoder-only models, as shown by [Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In Test, we find that FLAME performs similar to Codex-Cushman (same at top-1 and -2 points lower at top-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We find that 54% of the Forum syntax reconstruction bench- marks are solved by all the models, 1% is solved only by FLAME, and there are no benchmarks that all other models solve but FLAME doesn’t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=" We attribute this performance to our =IF('Jan 13'!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='B2="", \'Feb 13\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='B2="", \'Mar 13\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='B2="", \'Apr 13\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='B2="", yes, no) =IF(AND(\'Jan 13\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='B2="", \'Feb 13\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='B2="", \'Mar 13\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='B2="", \'Apr 13\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='B2=""), "yes", "no") Buggy Formula Ground Truth Fix FLAME Codex-Cushman Codex-Davinci CodeT5 X X X =VLOOKUP($Z25,$X$25:$Y:31,2,FALSE) =VLOOKUP($Z25,$X$25:$Y31,2,FALSE) Buggy Formula Ground Truth Fix FLAME Codex-Cushman Codex-Davinci CodeT5 X =VLOOKUP($Z25,$X$25:$Y$31,2,FALSE) FLAME Example 1 Example 2 Figure 4: Repair tasks with diverging performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In Example 1, the user did not use the AND function and missed double quotes around string literals yes and no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' FLAME fixes this (in top-5), while other models fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In Example 2 FLAME’s top candidate is syntactically valid but does not match the user’s fix, while other models’ predictions do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' pretraining design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' First, FLAME learns to generate syntactically correct code as a result of its noisy auto-encoding pretraining objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Second, FLAME learns the natural distri- bution of formulas by generating complete sequences during pretraining, rather than just mask values and sentinel tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Zeroshot Performance FLAME’s pretraining objectives al- low us to consider zeroshot performance for both last-mile repair and syntax reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In Table 4, we observe that FLAME outperforms Codex models for last-mile repair across all benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We attribute this to the closeness of our noisy auto-encoding pretraining objectives and the last-mile repair task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We find that in the syntax reconstruction task, FLAME out- performs Codex-Cushman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We believe this is because syntax reconstruction can be considered an extreme case of repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Formula Autocompletion The autoregressive nature of Codex models and FLAME’s pre- training objectives allows us to evaluate their zeroshot per- formance2 for formula auto-completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Note that we fine- tune CodeT5 for this task as it is pretrained on smaller span lengths (1 to 5 tokens) and generates special mask tokens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', ) in a zeroshot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We compute exact match and sketch match metrics with top-5 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In Table 3, we observe that FLAME performs better than all the larger models on the exact match metric and 3 out of 5 pre- fix lengths for sketch match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We note that Codex-Cushman and Codex-Davinci fail to complete 14% and 15% of the bench- marks with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='99 fraction of the prefix, respectively, whereas FLAME fails to complete 6% of the benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We observe significantly lower performance by CodeT5, likely due to the lack of longer masks spans during pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Surpris- ingly, Codex-Davinci performs slightly worse than the smaller Codex-Cushman for 3 out of 5 prefix lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Inspection of completions shows that Codex-Davinci tends to generate more tokens than required when completing these benchmark tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We also observe cases where models succeed with a shorter prefix but fail given a longer prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2 RQ2: Pretraining design decisions We investigate FLAME’s data curation, model size, the use of domain-specific pretraining objectives, and domain-specific tokenizer, and present results in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Training data curation Previous work [Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Kandpal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] have shown that deduplication can improve the performance of language models and reduce the memorization of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Therefore, we curate a pretraining dataset by performing workbook-level sketch-based formula deduplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Alterna- tively, one might consider performing global (pooled across all workbooks) sketch-based deduplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' This alternative results in a pretraining set of 591K formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Table 5 shows that training on this smaller corpus results in a lower perfor- mance model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We find that FLAME’s zeroshot performance falls by 14 points and finetuned performance falls by 18 points for last-mile repair in Forum benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Model size We trained two variants of FLAME with 16M and 60M parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Table 5 compares FLAME-16M and FLAME-60M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We find that performance declines slightly across tasks/benchmarks when we reduce the model size to 16M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' However, note that FLAME-16M can still outperform larger models such as Codex in 5 out of 10 zeroshot and finetuned settings, highlighting the efficacy of our design choices for FLAME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Pretraining objectives and Tokenizer To evaluate the effectiveness of our domain-specific pretrain- ing objectives and tokenizer, we pretrained a 60M parameters T5 model with generic pertaining objectives and tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Specifically, this model uses tail-masking, masked span pre- diction without accounting for lexer token boundaries, and 2We finetuned Codex-Cushman and FLAME but observed worse performance, possibly from over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' MAX C2 Sum C3:C4 SUM C5:C7 1 MAX(C2, Sum(C3:C4),SUM(C5:C7),1) MAX(C2,Sum!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='C3:C4,SUM(C5:C7),1) Tokens Formula T5 (Generic Pretraining and Tokenizer) Figure 5: Failing case of syntax reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Due to the different capitalization of Sum and SUM, the model treats them as different tokens, converting them to an identifier and a function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' random denoising objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Additionally, it uses the CodeT5 tokenizer trained on our pretraining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Table 5 shows that this variant performs worse across all tasks and benchmarks, both in a zeroshot and finetuned setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We attribute the huge drop, up to 62 points, in last-mile repair tasks in zeroshot to our user-inspired denoising pretraining objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Moreover, we hypothesize that FLAME’s good syntax reconstruction per- formance can be attributed to the domain-specific tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Figure 5 illustrates how the generic tokenizer treats tokens with different capitalizations, resulting in incorrect generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='3 RQ3: Decoding strategy In Table 6, we evaluate FLAME using four different decoding strategies, Beam Search, Group Beam Search [Vijayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2016], Nucleus Sampling [Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2019] and Top K Sampling [Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We find FLAME to perform bet- ter with group beam search decoding (group size of 2) for all the formula intelligence tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' However, for autocompletion with sketch match, nucleus sampling showed superior perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We believe this is because autocompletion requires more diverse results, particularly at shorter prefixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Refer to Appendix E for autocompletion table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Decoding Method LMR (Forum) SR (Forum) T@1 T@5 T@1 T@5 Beam Search 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 Group Beam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 Nucleus Sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 Top K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 Table 6: Performance by decoder strategy for last mile repair (LMR) and syntax reconstruction (SR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Beam and Grouped Beam Search have similar performance, and outperform Nucleus, Top K Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 6 Conclusions and Future Work We present FLAME, a small (60M parameter) language model for spreadsheet formulas, which captures domain-specific properties in its data curation, tokenization, and pretraining objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We implemented FLAME for Excel formulas and evaluate on three downstream tasks: last-mile repair, autocom- pletion, and a novel task that we term syntax reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We compare with the much larger models CodeT5, Codex- Cushman, and Codex-Davinci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' When fine-tuned, FLAME can achieve top performance in 6 of our 10 experimental settings, despite having two orders of magnitude fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Future work will explore downstream tasks that require additional spreadsheet context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' tables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' To tackle such tasks we will explore extending our pretraining objectives to incorporate context and the extent to which FLAME can integrate with existing table encoder models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Acknowledgments We thank Microsoft Research Cambridge for sharing the Ex- cel corpus used for pretraining FLAME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We thank OCTO at Microsoft (in particular Gopi Kumar and the AMD vTeam) for providing us with compute resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We also thank the Excel team for their feedback and encouragement in pursuing this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' References [Bavishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] Rohan Bavishi, Harshit Joshi, Jos´e Cambronero, Anna Fariha, Sumit Gulwani, Vu Le, Ivan Radiˇcek, and Ashish Tiwari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Neurosymbolic repair for low-code formula languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Proc.' metadata={'source': 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Alethea Power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Lukasz Kaiser,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Mohammad Bavar- ian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Clemens Winter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Philippe Tillet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Felipe Petroski Such,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Dave Cummings,' metadata={'source': 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William Saun- ders,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Christopher Hesse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Andrew N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa, Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish, Ilya Sutskever, and Wojciech Zaremba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Eval- uating large language models trained on code, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Intellicode compose: Code generation using transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In 28th ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineer- ing (ESEC/FSE ’20), May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' [Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] Yi Tay, Mostafa Dehghani, Vinh Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Tran, Xavier Garcia, Jason Wei, Xuezhi Wang, Hyung Won Chung, Dara Bahri, Tal Schuster, Huaixiu Steven Zheng, Denny Zhou, Neil Houlsby, and Donald Metzler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Ul2: Unifying language learning paradigms, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' [Vijayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2016] Ashwin K Vijayakumar, Michael Cogswell, Ramprasath R Selvaraju, Qing Sun, Stefan Lee, David Crandall, and Dhruv Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Diverse beam search: Decoding diverse solutions from neural sequence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' arXiv preprint arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='02424, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2021] Yue Wang, Weishi Wang, Shafiq Joty, and Steven CH Hoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Codet5: Identifier-aware unified pre- trained encoder-decoder models for code understanding and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='00859, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' [Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022] Frank F Xu, Uri Alon, Graham Neubig, and Vincent Josua Hellendoorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' A systematic evaluation of large language models of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In Proceedings of the 6th ACM SIGPLAN International Symposium on Machine Programming, pages 1–10, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' [Yasunaga and Liang, 2020] Michihiro Yasunaga and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Graph-based, self-supervised program repair from diagnostic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In International Conference on Ma- chine Learning, pages 10799–10808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' [Yasunaga and Liang, 2021] Michihiro Yasunaga and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Break-it-fix-it: Unsupervised learning for program repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 11941–11952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' A User noise operators We implement the following noise operators: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Wrong Range: we replace the range operator :, with one of the following symbols: {;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' , space "}, or we delete the range operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Malformed Range: A range consists of 4 el- ements: col1, row1, col2, row2 written as col1row1:col2row2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We randomly delete one of these elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For eg: col1:col2row2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Space between Function and Arguments in a Call: We introduce a space between the function name and the opening parentheses for built-in functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For exam- ple: SUM(A1:A10) converts to SUM (A1:A10) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Change number of arguments: We change the num- ber of arguments for functions with fixed function arity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, IF has a minimum arity of 2 and maxi- mum arity of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Specifically, if a function contains ar- guments equal to its minimum function arity, then we randomly delete one argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Whereas, if the func- tion’s max arity is equal to the number of arguments, then we randomly copy one of the existing arguments and pass it as an additional argument to the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, IF(A2>10, True, False) can become IF(A2>10, True, False, False) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Swap arguments: If a function takes different types of arguments, then we swap these arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example: IF(A1>10, 1, 2) can become IF(1, A1>10, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Space between relational operators: We add space between relational operators, such as < =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Swap relational operators: We swap relational opera- tors, such as <= turns to =< 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Inequality noise operator: In Excel <> is the inequality operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We replace this with the incorrect !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='= or =!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='. 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Invalid Equality: We also corrupt the equality operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' The equality operator in Excel is =, we replace it with == or ===.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Malformed Sheet Name: Multi-word sheet names in Excel need to be enclosed within single quotes (’’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We randomly choose to either delete the single quotes or replace them with double quotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For example, ’Sheet 1’!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='A10 can become "Sheet 1"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='A10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Remove exclamation Mark: In Excel, sheet names are followed by an exclamation mark to denote sheet reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We delete this exclamation mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Malformed Strings: We corrupt strings by either delet- ing the double quotes or replacing them with single quotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Add Comma and Remove Parentheses: We randomly choose to either insert a comma before a closing parenthe- sis or insert a comma and delete the closing parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Add random operators: We define a set of operators that we randomly insert into the formula at a random position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' These operators are: {+ - * / ^& < > = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' ) #} 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Add operator at the end: We randomly add one of the operators mentioned previously at the end of the se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Add Parentheses: We add opening and closing paren- thesis at random places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Corrupting Unreliable tokens: Following [Bavishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=', 2022], we randomly add, delete or replace unreliable tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Unreliable tokens are tokens where users often make mistakes, defined to be delimiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' B Codex Prompts For all our codex experiments, we use the following prompts for zeroshot and finetuning and use a temperature of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1 Repair - Zeroshot and Finetuning ##### Fix bugs in the below code ### Buggy Excel ### Fixed Excel ##### Fix bugs in the below code ### Buggy Excel =SUMIFS( Master!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$P:$P, Master!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$F:$F,$A7, Master856!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='$E:212Systems$B7 ) ### Fixed Excel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='2 Syntax Reconstruction - Zeroshot and Finetuning ### Excel Tokens ### Complete Excel Formula ### Excel Tokens INDEX Table1 SMALL IF Table1 COMPANY_NAME = $E$1 ROW Table1 COMPANY_NAME - 1 ROW 2:2 3 ### Complete Excel Formula B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='3 Autocomplete - Zeroshot ### Excel Formula ### Excel Formula IF(FALSE,NA( Models Exact Match Sketch Match 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='99 FLAME (60M) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='94 FLAME (16M) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='90 Global Deduplication 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='70 T5 (Generic objectives and tokenizer) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='29 Table 7: Design choice experiments for autocompletion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We compare multiple pretraining design decisions: model size, pretraining data curation, domain-specific pretraining objectives and tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We consider top-5 for Autocompletion (AC) with Exact Match (EM) and Sketch Match (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We note that FLAME outperforms all the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Models Exact Match SketchMatch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='9 Total 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='9 Total Beam Search 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='94 Group Beam Search (groups = 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='94 Nucleus Sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='92 TopK Sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='92 Table 8: Performance by decoder strategy for Autocompletion (top 5) with Exact Match and Sketch Match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' Beam Search outperforms all the strategies – Group Beam Search with a group size of 2, Nucleus Sampling, and Top K Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' C Training details We use the following HuggingFace configuration to train FLAME: { "architectures": [ "T5ForConditionalGeneration" ], "d_ff": 1024, "d_kv": 64, "d_model": 512, "decoder_start_token_id": 0, "dropout_rate": 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1, "bos_token_id": 1, "eos_token_id": 2, "feed_forward_proj": "gated-gelu", "initializer_factor": 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='0, "is_encoder_decoder": true, "layer_norm_epsilon": 1e-06, "model_type": "t5", "num_decoder_layers": 8, "num_heads": 6, "num_layers": 8, "output_past": true, "pad_token_id": 0, "relative_attention_num_buckets": 32, "tie_word_embeddings": false, "vocab_size": 16479 } We use an AdaFactor optimizer, with 1e-4 learning rate, clip factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='0, with scale parameters and relative steps set to false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' For fine-tuning, we use a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content='1 We use linear learning rate schedule with 100 warm-up steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' D Design Decision (Autocompletion) We detail our autocompletion evaluation where we evaluate FLAME against different variations in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We observe that FLAME beats all the different model variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' E Decoder Autocompletion In Table 8, we detail autocompletion results for different de- coding strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} +page_content=' We find that Beam Search beats other decod- ing methods in 7 out of 10 prefix lengths, and Top K Sampling beats others in Sketch Match for smaller fractions of prefixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09FST4oBgHgl3EQfWDi_/content/2301.13779v1.pdf'} diff --git a/1tAzT4oBgHgl3EQft_25/content/tmp_files/2301.01685v1.pdf.txt b/1tAzT4oBgHgl3EQft_25/content/tmp_files/2301.01685v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..863af1e90155d01d41d5290d959275d36135082c --- /dev/null +++ b/1tAzT4oBgHgl3EQft_25/content/tmp_files/2301.01685v1.pdf.txt @@ -0,0 +1,2226 @@ +arXiv:2301.01685v1 [math.AP] 4 Jan 2023 +Global existence and decay of small solutions for +quasi-linear second-order uniformly dissipative +hyperbolic-hyperbolic systems +Matthias Sroczinski∗ +January 5, 2023 +Abstract +This paper is concerned with quasilinear systems of partial differential equations +consisting of two hyperbolic operators interacting dissipatively. Its main theorem es- +tablishes global-in-time existence and asymptotic stability of strong solutions to the +Cauchy problem close to homogeneous reference states. Notably, the operators are not +required to be symmetric hyperbolic, instead merely the existence of symbolic sym- +metrizers is assumed. The dissipation is characterized by conditions equivalent to the +uniform decay of all Fourier modes at the reference state. On a technical level, the +theory developed herein uses para-differential operators as its main tool. Apparently +being the first to apply such operators in the context of global-in-time existence for +quasi-linear hyperbolic systems, the present work contains new results in the field of +para-differential calculus. In the context of theoretical physics, the theorem applies +to recent formulations for the relativistic dynamics of viscous, heat-conductive fluids +notably such as that of Bemfica, Disconzi and Noronha [1] (Phys. Rev. D, 98:104064, +2018.). +Keywords. hyperbolic systems, initial value problem, global existence, asymptotic +stability, para-differential operators, fluid mechanics +AMS subject classifications. +Primary 35A01, 35B35, 35L72, 35L15, 35S50, +35Q35, 35Q75 +∗Department +of +Mathematics, +University +of +Konstanz, +78457 +Konstanz, +Germany. +matthias.sroczinski@uni-konstanz.de, https://orcid.org/0000-0002-5472-2741 +1 + +1 +Introduction and main result +In this paper, we study systems of partial differential equations that are given by the su- +perposition of two hyperbolic operators and show that homogeneous states are nonlinearly +stable in the sense that small perturbations thereof lead to global-in-time decaying solutions. +Concretely, we consider the Cauchy problem for quasi-linear systems of the form +d +� +j=0 +Aj(u(t, x))uxj(t, x) = +d +� +j,k=0 +(Bjk(u(t, x))uxj(t, x))xk, +x0 = t ≥ 0, x = (x1, . . . , xd) ∈ Rd, +(1.1) +u(0, x) = u0(x), +ut(0, x) = u1(x), +x ∈ Rd, +(1.2) +where both the operator on the right hand side and the operator on the left hand side are +hyperbolic and each of them acts dissipatively on the trajectories generated by the other one. +Such systems occur in theroretical physics as recent formulations for the (special-)relativistic +dynamics of viscous, heat conductive fluids [15, 16, 17, 1, 12, 2]. Our results apply to these +formulations. The main theorem is the following. +1.1 Theorem. Consider d ≥ 3, s > d/2 + 1, ¯u ∈ Rn and let (1.1) satisfy conditions +(HA), (HB) and (D) from Section 3. Then there exist constants δ > 0 and C = C(δ) > 0 +such that the following holds: For all u0, u1 with u0 − ¯u ∈ Hs+1(Rd, Rn) ∩ L1(Rd, Rn), u1 ∈ +Hs(Rd, Rn)∩L1(Rd, Rn) as well as ∥u0 − ¯u∥Hs+1, ∥u1∥Hs, ∥u− ¯u∥L1, ∥u1∥L1 < δ there exists a +unique global solution u of (1.1), (1.2) satisfying u − ¯u ∈ C([0, ∞), Hs+1) ∩ C1([0, ∞), Hs), +l = 0, . . . , s + 1 and, for all t ∈ [0, ∞), +∥u(t) − ¯u∥Hs + ∥ut(t)∥Hs−1 ≤ C(1 + t)− d +4(∥u0 − ¯u∥Hs + ∥u1∥Hs−1 + ∥u0 − ¯u∥L1 + ∥u1∥L1), +(1.3) +∥u(t) − ¯u∥2 +Hs+1 + ∥ut(t)∥2 +Hs + +� t +0 +∥u(τ) − ¯u∥2 +Hs+1 + ∥ut(τ)∥2 +Hs dτ +(1.4) +≤ C(∥u0 − ¯u∥2 +Hs+1 + ∥u1∥2 +Hs + ∥u0 − ¯u∥2 +L1 + ∥u1∥2 +L1) +While conditions (HA) and (HB) specify the assumed hyperbolicity, condition (D), essentially +obtained in [14], characterizes the needed decay behaviour for the Fourier modes of the +associated linearized system. +Based on the famous Kawashima-Shizuta condition [24, 34], analogous results are well-known +for symmetric hyperbolic-parabolic systems and first-order hyperbolic systems with relax- +ation, cf. [9, 33, 41, 19, 26, 42, 5] among others.1 Regarding dissipative second-order hyper- +bolic systems there are fewer results available, cf. notably [11, 27, 32] and references therein, +all of those treat systems whose structure is different form the one we consider here. The +1Note that the often available reformulations of (1.1) as first-order hyperbolic systems do typically not +satisfy the assumptions of these works. +2 + +most prominent example for equations satisfying condition (D) are probably damped wave +equations with a non-linear convection term, which alternatively can be viewed as conserva- +tion laws with hyperbolic artificial viscosity. In this case, (D) reduces to Whitham’s famous +sub-characteristic condition [40] and various in-depth results on the asymptotic behaviour +of solutions have been achieved in this context, cf. [31, 25, 39, 20, 10, 23]. Closest related +to the present work are [35, 36, 37], there a predecessor of Theorem 1.1 was shown for the +systems proposed in [16, 17]. +The theory developed in the present work requires novel techniques in the use of para- +differential operators. Developed by Bony [6] and Meyer [30, 29], such operators have been +used in the context of hyperbolic equations by G´erard and Rauch [18], Taylor [38] and +M´etivier [28]. +However, quite different from these works, we will in particular need to +precisely understand how the norms of para-differential operators depending on the functions +inducing their symbols. +The paper is organized as follows. In the crucial Section 2 general results on para-differential +operators needed for the argumentation in Section 3 and 4 will be established. The present +work apparently being the first that uses such operators to treat global-in-time solutions to +quasi-linear hyperbolic systems, we prove corresponding new results on that dependence. The +technical highlight in this regard will be a modified version of the strong G˚arding inequality. +In Section 3 we construct a para-differential operator which is specifically associated with the +system’s dissipativity. Section 4 is dedicated to the proof of Theorem 1.1. The challenging +part is the treatment of the highest derivatives. +Here we have to use the sophisticated +estimates of Section 2 and the construction of Section 3. Finally, Section 5 shows that models +of equations of dissipative relativistic fluid dynamics satisfy the assumptions of Theorem 1.1. +2 +Results on para-differential operators +A tour through the theory of para-differential operators from scratch to fine properties, this +section relies on Appendix C of Benzoni-Gavage and Serre [4] and Section 9 of H¨ormander +[22], however with strong attention to symbols induced by what later will be the solution to +the PDE system considered. In its initial part interpolating between brevity and legibility, +the section culminates in the aforementioned novel version of the strong G˚arding inequality. +2.1 +Notation, definitions and basics +For topological vector-spaces V, W we write L(V, W) for the space of continuous linear oper- +ators form V to W (or L(V ) if W = V ). Throughout this section consider fixed dimensions +n, d ∈ N and let m denote some real number. For x, ξ ∈ Rd we just write xξ for their +Euclidian scalar product. +3 + +Let E be a finite-dimensional C-Banach space. We denote the E-valued Schwartz space by +S(Rd, E), and by S′(Rd, E) := L(S(Rd), E) the space of continuous linear mappings from +S(Rd) to E, i.e. the space of E-valued temperate distributions, both equipped with the +standard locally convex topologies. For f ∈ S(Rd, E) the Fourier transform is +(Ff)(ξ) = ˆf(ξ) = (2π)−d/2 +� +Rd f(x)e−ixξdx +with inverse +(F −1 ˆf)(x) = (2π)−d/2 +� +Rd +ˆf(ξ)eixξdξ. +We write F1 and F2 for the Fourier transform with respect to the first and the second variable +for functions f ∈ S(Rd × Rd, E), i.e. +(F1f)(η, y) = F(f(·, y))(η) = (2π)−d/2 +� +Rd f(x, y)e−ixηdx, +(F2f)(x, ξ) = F(f(x, ·))(ξ) = (2π)−d/2 +� +Rd f(x, y)e−iyξdy. +As usual we extend F, F1, F2 to continuous operators on S′(Rd, E), S′(Rd × Rd, E) and +unitary operators on L2(Rd, E), L2(Rd × Rd, E) also denoted by F, F1, F2. +We will use ⟨ξ⟩ := (1 + |ξ|2) +1 +2, ξ ∈ Rd, Λm := F −1⟨·⟩mF. As usual +Hm(Rd, E) := {u ∈ L2(Rd, E) : Λmu ∈ L2(Rd, E)}, +are the L2-based E-valued Sobolev spaces with norm +∥u∥Hm(Rd,E) := ∥Λmu∥L2(Rd,E). +If E is a Hilbert space we consider the scalar product on Hm(Rd, E) +⟨u, v⟩Hm(Rd,E) := ⟨Λmu, Λmv⟩L2(Rd,E). +We also use L∞-based Sobolev spaces +W k,∞(Rd, E) := {u ∈ L∞(Rd, E) : ∂α +x u ∈ L∞(Rd, E), |α| ≤ k} +with norm +∥u∥W k,∞(Rd,E) = max +|α|≤k ∥∂α +x u∥L∞(Rd,E). +We often just write Hm, ∥u∥m, ⟨u, v⟩m, W k,∞ instead of Hm(Rd, E), ∥u∥Hm(Rd,E), ⟨u, v⟩Hm(Rd,E), +W k,∞(Rd, E) if there is no concern for confusion, and ∥u∥ for ∥u∥0. +For A ∈ Cn×n we denote the adjoint matrix by A∗ = ¯At and for T ∈ L(S(Rd, Cn)) we write +T ∗ for the adjoint operator with respect to the L2(Rd, Cn) inner product. As usual we call T +4 + +self-adjoint if T = T ∗ and positive (strictly positive) if ⟨Tf, f⟩0 ≥ 0 (⟨Tf, f⟩ > 0), in which +case we also write T ≥ 0 (T > 0). +Next, we turn to the basic definitions concerning pseudo-differential operators which will be +used in the present paper. We consider the following symbol classes. +2.1 Definition. +(i) Sm := Sm(Rd, Cn×n) is the set of all functions a ∈ C∞(Rd×Rd, Cn×n) +for which for any α, β ∈ N0 there exists Cαβ > 0 such that +|∂β +x∂α +ξ a(x, ξ)| ≤ Cαβ⟨ξ⟩m−|α|. +(2.1) +With semi-norms being the optimal constants in (2.1), Sm is a Fr´echet space. +(ii) Sm +1,1 := Sm +1,1(Rd, Cn×n) is the set of functions a ∈ C∞(Rd × Rd) for which for any +α, β ∈ Nd +0 there exist Cαβ > 0 such that +|∂β +x∂α +ξ a(x, ξ)| ≤ Cαβ⟨ξ⟩m−|α|+|β +(2.2) +for all (x, ξ) ∈ Rd × Rd. With semi-norms being the optimal constants in (2.2), Sm +1,1 is +a Fr´echet space. +(iii) For a ∈ Sm +1,1 the mapping op[a] ∈ L(S(Rd, Cn)) defined by +(op[a]f)(x) := (2π)− d +2 +� +eixξa(x, ξ)Ff(ξ) dξ. +(2.3) +is called the pseudo-differential operator with symbol a. We also write a := Sym[op[a]]. +As first shown in [7, 8] for a ∈ Sm +1,1 the operator op[a] extends to a bounded operator from +Hl+m to Hl only if op[a]∗ also has a symbol in S1,1 +m . But the operator norm of op[a] can +in general not be controlled by semi-norms of a uniformly over this subspace. As for our +applications to dissipative hyperbolic systems it is essential that the norm of op[a] is small +if the semi-norms of a are small we have to make sure that the symbols occurring in the +present work belong to the following smaller subspaces. +2.2 Definition. For L ∈ (0, 1], Sm,L +1,1 +is the subspace of all a ∈ Sm +1,1 such that F1a vanishes +on NL := {(η, ξ) ∈ Rd × Rd : |η + ξ| + 1 < L|ξ|} in the sense of distributions, i.e. +a(F1φ) = 0 for all φ ∈ S(Rd × Rd) with supp φ ⊂ NL. +(2.4) +2.3 Proposition. Let L ∈ (0, 1]. For all l ∈ R and a ∈ Sm,L +1,1 +op[a] extends to a continuous +operator form Hl+m to Hl and op is itself continuous from Sm,L +1,1 +to L(Hl+m, Hl). +Proof. Cf. [22], Proposition 9.3.1. +The symbols in Sections 2 and 3 will be induced by functions (x, ξ) �→ F(u(x), ξ) where +F ∈ C∞(Rn × Rd), u ∈ W k,∞(Rd, Rn) for some k ∈ R, i.e. they belong to the following +symbol class. +5 + +2.4 Definition. For any k ∈ N0 the set Γm +k of symbols of order m with regularity k is the +set of functions A : Rd × Rd �→ Cn×n such that, +(i) for almost all x ∈ Rd the mapping ξ �→ A(x, ξ) is in C∞(Rd, Cn×n) +(ii) for any α ∈ Nd +0 and ξ ∈ Rd the mapping x �→ ∂α +ξ A(x, ξ) belongs to W k,∞(Rd, Cn×n) +and there exists Cα > 0 not depending on ξ such that +∥∂α +ξ A(·, ξ)∥W k,∞ ≤ Cα⟨ξ⟩m−|α|. +(2.5) +With the semi-norms being the optimal constants in (2.5), Γm +k is a Fr´echet space. +Para-differential operators associated with symbols in Γm +k are defined as follows. +2.5 Definition. For ǫ = (ǫ1, ǫ2) with 0 < ǫ1 < ǫ2 < 1 we call a function χ ∈ C∞(Rd × Rd) +an admissible ǫ-cut-off if χ is even with respect to each variable, χ(Rd × Rd) ⊂ [0, 1], +χ(η, ξ) = +� +1, +|η| ≤ ǫ1|ξ| and |ξ| ≥ 1 +0, +|η| ≥ ǫ2⟨ξ⟩ or |ξ| ≤ ǫ2 +(2.6) +for all η, ξ ∈ Rd and for all α, β ∈ Nd there exists Cα,β > 0 such that for all ξ, η ∈ Rd +|∂β +η ∂α +ξ χ(η, ξ)| ≤ Cα,β⟨ξ⟩−|α|−|β|. +2.6 Proposition. Let χ be an admissible ǫ-cut-off. Set Kχ := F −1 +1 (χ) and consider the +function Rχ : Γm +k → C∞(Rd × Rd) given by +Rχ(A) := Kχ ∗1 A, +A ∈ Γm +k . +Then Rχ defines a bounded linear operator from Γm +k to Sm,1−ǫ2 +1,1 +∩ Γm +k . Here +(Kχ ∗1 A)(x, ξ) = +� +Rd Kχ(x − y, ξ)A(y, ξ) dy. +Proof. Apart from the aspect that a is not only in Sm +1,1 but even in Sm,1−ǫ2 +1,1 +the proof can +be found in [4], Proposition C.16. But that aspect follows in a straightforward manner as +|η + ξ| + 1 ≤ (1 − ǫ2)|ξ| implies |ξ| − |η| + 1 ≤ (1 − ǫ2)|ξ| and thus |η| ≥ ǫ2⟨ξ⟩ and χ vanishes +for such η, ξ. +2.7 Definition. Let χ be an admissible ǫ-cut-off. +For A ∈ Γm +k the (χ-)para-differential +operator with symbol A is defined by +Opχ[A] := op[Rχ(A)]. +As Rχ ∈ L(Γm +k , Sm,1−ǫ2 +1,1 +), Opχ = op ◦Rχ defines a continuous linear operator from Γm +k to +L(Hl+m, Hl) (l ∈ R). In particular the L(Hl+m, Hl)-norm of Opχ[A] can be estimated by a +constant depending on l, χ and a finite sum of Γk +m-semi-norms of A. +6 + +The following shows that, regarding its dependence on χ, opχ[A] is determined by A up to +a lower order operator, if k ≥ 1. +2.8 Lemma. Let χ be an admissible ǫ-cut-off and k ≥ 1. Then the following holds: +(i) The mapping Rχ − Id is a continuous operator from Γm +k to Γm−1 +k−1 . +(ii) If ˜χ is an admissible ˜ǫ-cut-off, then Rχ − R˜χ is a continuous operator from Γm +k to +Sm−1,1−τ +1,1 +∩ Γm−1 +k−1 with τ = max{ǫ2, ˜ǫ2}. +Proof. Cf. [4], Proposition C.13, Corollary C.5. +We end this subsection by stating two additional results on para-differential operators for +later usage. The proofs are contained in [4], Appendix C. To simplify notation we fix an +admissible ǫ-cut-off χ and suppress the dependence of Rχ and Opχ on χ in the following. +We call an operator K infinitely smoothing if K ∈ L(Hs, Hl) for all s, l ∈ R. +2.9 Lemma. Let b ∈ Sm be constant with respect to the first variable. Then the following +holds: +(i) op(b) − Op[b] is infinitely smoothing. +(ii) Op[b] = Op[b∗] +(iii) Op[Ab] = Op[A]F −1bF for any A ∈ Γµ +k. +2.10 Lemma. For each k > 0 there exists C > 0 such that for all f ∈ L∞ ∩ Hk, A ∈ +W 1,∞ ∩ Hk +∥A − Op[A]f∥k ≤ C(∥A∥Hk∥f∥L∞ + ∥A∥W 1,∞∥f∥Hk−1). +2.2 +Adjoints and products +For the argumentation in Section 3 it will be essential to control the norms of operators +Op[A∗] − Op[A]∗, Op[BA] − Op[B] Op[A], A ∈ Γm +1 , B ∈ Γµ +1, µ ∈ R, in terms of the semi- +norms of A and B. While for a ∈ Sm,L +1,1 , b ∈ Sm,L +1,1 +there exist symbols g ∈ Sm +1,1, h ∈ Sm+µ +1,1 +such that op[a]∗ = op[g], op[b] op[a] = op[h] and that, provided ∂xja ∈ Sm +1,1(j = 1, . . . , d), +op[a∗] − op[a]∗ ∈ L(Hl+m−1, Hl), op[b] op[a] − op[ba] ∈ L(Hl+m+µ−1, Hl), l ∈ R, it is not +true in general that g, h are again in some class Sm,L +1,1 , Sm+µ,L +1,1 +which would allow to control +their operator norms. However, for our purposes it is sufficient to consider symbols of the +particular form a = R(A), b = R(B) for A ∈ Γm +1 , B ∈ Γµ +1 and we will show that in this case +the symbols of op[a]∗ = Op[A]∗, op[b] op[a] = Op[B] Op[A] are in fact again in Sm,L +1,1 , Sm+µ,L +1,1 +for some L ∈ (0, 1]. +As a first step note that for symbols in S(Rd × Rd, Cn×n) there exist neat formulas for the +symbols of adjoint and product operators, which also can be found in [22]. +7 + +2.11 Lemma. If a ∈ S(Rd × Rd), then op[a]∗ = op[g] with F1g(η, ξ) = (F1a(−η, η + ξ))∗. +2.12 Lemma. If a, b ∈ S(Rd × Rd, Cn×n), then op[b] op[a] = op[h] with +F1h(η, ξ) = +� +Rd F1b(η − θ + ξ, θ)F1a(θ − ξ, ξ)dθ. +The significance of this result lies in the following observation. +2.13 Lemma. Let A ∈ Γm +0 . +Then there exists a sequence (aν)ν≥1 ⊂ S(Rd × Rd, Cn×n) +such that op[aν]u → Op[A]u, ν → ∞ in S(Rd, Cn) for all u ∈ S(Rd, Cn). Furthermore +for all δ ∈ (ǫ2, 1), ǫ2 being the constant of the ǫ-cut-off, there exists ν0 > 0 such that +supp F1aν ⊂ {(η, ξ) ∈ Rd × Rd : |η| ≤ δ⟨ξ⟩} for all ν ≥ ν0. +Proof. The first part of the statement is shown as in [21], proof of Theorem 18.1.8. However, +we have to slightly modify the construction to also obtain the second part. Set a := R(A). +Choose ˆφ ∈ S(Rd) with supp ˆφ ⊂ B0(1), F −1 ˆφ(0) = 1 and define φ := F −1 ˆφ, +aν(x, ξ) := φ(x/ν)φ(ξ/ν)a(x, ξ), +x, ξ ∈ Rd. +The asserted convergence then follows as ibid. +It remains to show the statement concerning the supports. Set ψν(x, ξ) := φ(x/ν)φ(ξ/ν) +(ξ, η ∈ Rd, ν ≥ 1). As F1(aν) = (2π)−d/2(F1ψν) ∗1 (F1a) and F1a = χF1A, it is sufficient +to show that for given δ ∈ (ǫ2, 1) and ν sufficiently large χ(η − θ, ξ)F1ψν(θ, ξ) = 0 for all +θ, η, ξ ∈ Rd |η| ≥ δ⟨ξ⟩. Clearly +F1ψν(θ, ξ) = νd ˆφ(θν)φ(ξ/ν). +As by construction ˆφ(θν) = 0 for |θ| ≥ ν−1 we can assume |θ| ≤ ν−1. Then |η| ≥ δ⟨ξ⟩ yields +|η − θ| ≥ |η| − |θ| ≥ δ⟨ξ⟩ − ν−1. +Hence choosing ν so large that ν−1 ≤ δ − ǫ2 gives (note ⟨ξ⟩ ≥ 1) +|η − θ| ≥ δ⟨ξ⟩ − (δ − ǫ2)⟨ξ⟩ = ǫ2⟨ξ⟩. +But this implies χ(η − θ, ξ) = 0, which finishes the proof. +2.14 Proposition. Let A ∈ Γm +0 . Then there exists b = b(A) ∈ Sm,1−ǫ2 +1,1 +such that Op[A]∗ = +op[b(A)]. Furthermore if A ∈ Γ1 the operator +T : Γm +1 → Sm−1,1−ǫ2 +1,1 +, a �→ b(A) − R(A)∗ +is continuous. In particular the mapping +A �→ Op[A∗] − Op[A]∗ = op[R(A)∗ − b(A)] +is continuous from Γm +1 to L(Hl+m−1, Hl) for any l ∈ R. +8 + +Proof. Set a := R(A). As A ∈ Sm,1−ǫ2 +1,1 +, the existence of b := b(A) ∈ Sm +1,1 with op[b] = Op[A]∗ +follows by [22], Lemma 9.4.1. +Next we prove that F1b vanishes on N1−ǫ2. If A ∈ S(Rd×Rd, Cn×n) also a ∈ S(Rd×Rd, Cn×n) +and Lemma 2.11 gives +F1b(η, ξ) = (F1a(−η, η + ξ))∗. +If now |η + ξ| + 1 ≤ (1 − ǫ2)|ξ| then ǫ2|ξ| ≤ |η| and thus +ǫ2⟨η + ξ⟩ ≤ ǫ2(1 + |η + ξ|) ≤ (1 − ǫ2)ǫ2|ξ| ≤ (1 − ǫ2)|η| ≤ |η|, +which implies F1a(−η, η + ξ) = (χF1A)(−η, η + ξ) = 0. +For general A choose a sequence (aν)ν≥1 ⊂ S(Rd × Rd, Cn×n) with op[aν]u → Op[A]u in +S(Rd, Cn) for all u ∈ S(Rd, Cn). This implies op[aν]∗ → Op[a]∗ = op[b] in S′(Rd × Rd, Cn×n) +and it is straightforward to show that this yields bν → b ∈ S′(Rd × Rd, Cn×n), where +F1bν(η, ξ) = F1aν(−η, η+ξ). By Lemma 2.13 F1aν(η, ξ) vanishes for |η| ≥ δ⟨ξ⟩, if δ ∈ (ǫ2, 1). +As seen above this yields bν ∈ Sm,1−δ +1,1 +. In conclusion b = limν→∞ bν ∈ Sm,1−δ +1,1 +for all δ > ǫ2, +i.e. b ∈ Sm,1−ǫ2 +1,1 +. +Lastly A ∈ Γm +1 directly gives ∂δ +xA ∈ Γm +0 and hence ∂δ +xR(A) = R(∂δ +xA) ∈ Sm +1,1 (|δ| = 1) . By +[22], Lemma 9.6.1 (applied to N = 1, mN = m − 1) we now obtain b − R(A) ∈ Sm−1 +1,1 +and its +Sm +1,1-semi-norms are bounded by a constant times a sum of finitely many Sm +1,1-semi-norms of +∂δ +xR(A) (|δ| = 1). As also b − R(A) ∈ Sm−1,1−ǫ2, the assertion follows by the continuity of R +and op. +Concerning the analysis of product operators we first consider the difference Rχ(AB) − +Rχ(A)Rχ(B). +2.15 Lemma. For A ∈ Γm +1 , B ∈ Γµ +1 and an ǫ-cut-off χ with ǫ2 < 1/2 we have Rχ(B)Rχ(A) ∈ +Sm+µ,1−2ǫ2 +1,1 +. Furthermore the bilinear operator +T : Γm +1 × Γm +1 → Sm+µ−1,1−2ǫ2 +1,1 +, +(A, B) �→ Rχ(AB) − Rχ(A)Rχ(B) +is continuous. +Proof. We suppress the superscript χ in the following. As R(A) ∈ Sm +1,1, R(B) ∈ Sµ +1,1, it +is clear that R(B)R(A) ∈ Sm+µ +1,1 . Thus regarding the first assertion we need to show that +R(B)R(A) vanishes on N1−2ǫ2. +Since F1(R(B)R(A)) = (2π)−d/2F1R(B) ∗1 F1R(B) and +F1R(A) = χF1A, F1R(B) = χF1B, it is sufficient to prove that χ(η − θ, ξ)χ(θ, ξ) vanish for +all θ, η, ξ ∈ Rd with |η +ξ|+1 ≤ (1−2ǫ2)|ξ|. Take such θ, η, ξ. If χ(θ, ξ) ̸= 0 then |θ| ≤ ǫ2⟨ξ⟩ +and |η + ξ| + 1 ≤ (1 − 2ǫ2)|ξ| implies |η| ≥ 2ǫ2|ξ| + 1. Together this yields +|η − θ| ≥ |η| − |θ| ≥ 2ǫ2|ξ| + 1 − ǫ2ξ ≥ ǫ2⟨ξ⟩. +Now χ(η − θ, ξ) vanishes for such η, θ, ξ, wich completes the argument. +9 + +In regard to the continuity we write +R(BA) − R(B)R(A) = R(BA) − BA + B(A − R(A)) − (R(B) − B)R(A). +Hence it follows from Lemma 2.8 (i) and the continuity of R that T is continuous as an +operator to Γm+µ−1 +0 +. Thus the proof is finished if we show that each Sm+µ−1 +1,1 +-semi-norm can +be bounded by a constant times a finite sum of Γm+µ−1 +0 +-semi-norms of T(A, B). We show +that even the following holds: For all α, β ∈ N0 there exists Cβ > 0 such that +|∂β +x∂α +ξ T(A, B)(x, ξ)| ≤ Cαβ|∂α +ξ T(A, B)(x, ξ)|⟨ξ⟩|β|, +x, ξ ∈ Rd. +(2.7) +By Bernstein’s Lemma applied to ∂α +ξ T(a, b)(·, ξ) (cf. +e.g. +[4], Lemma C.3) this can be +deduced from the fact that for all ξ ∈ Rd +supp +� +(FT(A, B))(·, ξ) +� +⊂ B(0, 2ǫ2⟨ξ⟩). +In fact, +supp(F(R(ba)(·, ξ)) ⊂ supp(χ(·, ξ)) ⊂ B(0, 2ǫ2⟨ξ⟩) +holds by definition of χ and that F1(R(b)R(a)) vanishes for all η, ξ with |η| > 2ǫ2⟨ξ⟩ follows +by the same argumentation as in the first part of the proof. +We can now prove our main proposition concerning products of para-differential operators. +2.16 Proposition. Let A ∈ Γm +0 , B ∈ Γµ +0. Then for L := (1 − ǫ2)2 there exists h(B, A) ∈ +Sµ+m,L +1,1 +such that Op[B] Op[A] = op[h(B, A)]. Furthermore the operator +Γm +1 × Γm +1 → L(Hl+µ+m−1, Hl), +(B, A) �→ Opχ[B] Op[A] − Op[BA] +is continuous for all l ∈ R. +Proof. The existence of a h = h(B, A) ∈ Sm+µ +1,1 +such that +op[h(B, A)] = op[R(B)] op[R(A)] = Op[B] Op[A] +follows direclty from [22], Lemma 9.5.1 as R(A) ∈ Sm,1−ǫ2 +1,1 +. We now prove that h satisfies +(2.4) for L = (1 − ǫ2)2. First assume a := R(A), b := R(B) ∈ S(Rd × Rd). By Lemma 2.12 +F1h(η, ξ) = +� +Rd F1b(η − θ + ξ, θ)F1a(θ − ξ, ξ)dθ. +(2.8) +Let η, ξ ∈ Rd with |η + ξ| + 1 ≤ (1 − ǫ2)2|ξ|. If F1a(θ − ξ, ξ) = F1R(A)(θ − ξ, ξ) ̸= 0 we have +|θ − ξ| ≤ ǫ2⟨ξ⟩ ≤ ǫ2 + ǫ2|ξ|, which gives (1 − ǫ2)|ξ| ≤ |θ| + ǫ2. We arrive at +|η + ξ − θ| ≥ |θ| − |η + ξ| ≥ |θ| − (1 − ǫ2)2|ξ| + 1 ≥ |θ| − (1 − ǫ2)|θ| − (1 − ǫ2)ǫ2 + 1 += ǫ2θ + ǫ2 + (1 − ǫ2)2 ≥ ǫ2⟨θ⟩. +10 + +But this implies F1b(η + ξ − θ, θ) = F1R(B)(η + ξ − θ, θ) = 0, which finishes the argument. +For general A, B choose sequences (aν)ν≥1, (bν)ν≥1, ⊂ S(Rd × Rd) with op[aν]u → Op[A]u, +op[bν]u → Op[B]u in S(Rd × Rd, Cn) for all u ∈ S(Rd × Rd, Cn) as constructed im Lemma +2.13. Then clearly +op[hν]u = op[bν] op[aν]u → Op[B] Op[A]u = op[h]u +in S(Rd, Cn), where hν is defined by (2.8) with a, b replaced by aν, bν. This implies hν → h +in S′(Rd × Rd, Cn×n). As for all 1 > δ > ǫ2 supp F1aν, supp F1bν ⊂ {(η, ξ) ∈ Rd × Rd : |η| ≤ +δ⟨ξ⟩} for ν sufficiently large we get by the same reasoning as above that for all 1 > δ > ǫ2 +hν vanishes on N(1−δ)2 for ν sufficiently large, which proves that h vanishes on N(1−ǫ2)2. +To prove the second assertion note that by Lemma 2.8 (ii), the mapping G �→ Opχ[G] − +Op˜χ[G] is continuous from Γk +1 to L(Hl+k−1, Hl), k, l ∈ R, for any admissible cut-offs χ, ˜χ. +Hence we can assume w.l.o.g ǫ2 < 1 +2. By Lemma 2.15 and the continuity of op +(B, A) �→ Op[BA] − op[R(B)R(A)] = op[R(BA) − R(B)R(A)] +is also continuous as mapping from Γm +1 × Γm +1 to L(Hl+µ+m−1, Hl). What is left to show ist +the continuity of +(B, A) �→ Op[B] Op[A] − op[R(B)R(A)] = op[h(B, A) − R(B)R(A)]. +As R(A) ∈ Sm,1−ǫ2 +1,1 +and +∂xjR˜χ(A) = R(∂xjA) ∈ Sm +1,1, +∂xjR˜χ(B) = R(∂xjB) ∈ Sm +1,1, +j = 1, . . . , d. +all semi-norms of h(B, A) − R(B)R(A) can be estimated by a constant times a finite sum +of products of semi norms of ∂xjR(A), ∂xkR(B). Thus as h(B, A) − R(B)R(A) ∈ Sm−1,L +1,1 +for +l = min{1 − 2ǫ2, (1 − ǫ2)2} the assertion follows from the continuity of op and R. +2.3 +Estimates for operators with symbols induced by Sobolev func- +tions +In Section 3 the results of Sections 2.1, 2.2 are applied to symbols of the form (x, ξ) �→ +F(u(x), ξ), where F ∈ C∞(U × Rd, Cn×n) (U ⊂ Rn some 0-neighbourhood) and u ∈ +Hs(Rd, Rn) for s sufficiently large. For this purpose we prove the results below. +In the following let U ⊂ RN be a 0-neighbourhood. +2.17 Definition. We denote by Sm(U) := Sm(U, Cn×n) the set of all functions F ∈ C∞(U × +Rd, Cn×n) for which for any α, β ∈ Nd +0 there exists Cαβ > 0 such that for all (u, ξ) ∈ U × Rd +|∂β +x∂α +ξ F(u, ξ)| ≤ Cαβ⟨ξ⟩m−|α|. +(2.9) +11 + +For functions F : U × Rd → Cn×n and u : Rd → U we consider the composition +Fu : Rd × Rd → Cn×n, (x, ξ) �→ F(u(x), ξ). +2.18 Lemma. Let F ∈ Sm(U) and u ∈ Hs with s > d/2. Then Fu ∈ Γm +k for k = [s − d/2] +and for all α ∈ Nd +0 and each Γm +k -semi-norm pα(Fu) it holds +pα(Fu) ≤ Cα(∥u∥s, F), +and if additionally F(0, ξ) = 0, then +pα(Fu) ≤ ˜Cα(∥u∥s, F)∥u∥s, +where Cα, ˜Cα depend on α, F and continuously on ∥u∥s. +Proof. By Sobolev embedding Hs ֒→ W k,∞. Thus we have Fu(·, ξ) ∈ W k,∞ and +∥∂α +ξ Fu(·, ξ)∥W k,∞ ≤ C(∥u∥W k,∞)∥∂α +ξ F(·, ξ)∥W k,∞(U) ≤ C(∥u∥s)Cα(F)⟨ξ⟩m−|α. +all ξ ∈ Rd. If F(0, ξ) = 0, we even get, for all ξ ∈ Rd, +∥∂α +ξ Fu(·, ξ)∥W k,∞ ≤ C(∥u∥W k,∞)∥u∥W k,∞∥∂α +ξ Fu(·, ξ)∥W k,∞(U) ≤ C(∥u∥s, F)∥u∥s⟨ξ⟩m−|α|. +The following proposition will be central for the energy estimates in Section 3. It follows +directly by the continuity of Op : Γm +k → L(Hl+m, Hl) and Lemma 2.18 as well as Propositions +2.14, 2.16 and the facts that Op[F0]∗ = op[F ∗ +0 ] and op[G0F0] − op[G0] op[F0] is infinitely +smoothing by Lemma 2.9. +2.19 Proposition. Let F ∈ Sm(U), l ∈ R. Then for all u ∈ Hs with s > d/2 there exists +Cl = Cl(F, ∥u∥) > 0 depending on l, F and monotonically increasingly on ∥u∥s such that: +(i) ∥ Op[Fu]∥L(Hl+m,Hl) ≤ Cl(∥u∥s) and for F(0, ·) = 0 ∥ Op[Fu]∥L(Hl+m,Hl) ≤ Cl∥u∥s, +(ii) for s > d/2 + 1, Op[Fu]∗ − Op[F ∗ +u] ∈ L(Hl−1+m, Hm) and +∥ Op[Fu]∗ − Op[F ∗ +u]∥L(Hl−1+m,Hm) ≤ Cl∥u∥s +(iii) for G ∈ Sµ(U) and s > d/2 + 1 there exist Cl,2 = Cl,2(G, ∥u∥s) depending on G and +monotonically increasingly on ∥u∥s such that +∥ Op[Gu] Op[Fu] − Op[GuFu]∥L(Hl+µ−1+m,Hm) ≤ Cl,2Cl∥u∥s +up to an infinitely smoothing operator, which is determined by F(0, ·), G(0, ·). +12 + +2.20 Proposition. Let F ∈ Sm(U) and u ∈ C1([0, T], Hs) (T > 0) for s > d/2. Then for +each l ∈ R the mapping +[0, T] → L(Hl+m, Hl), +t �→ Op[Fu(t)] +is continuously differentiable and there exists Cl depending on l and F but not on u such +that for all t ∈ [0, T] +∥ d +dt Op[Fu(t)]∥L(Hl+m,Hl) ≤ Cl∥∂tu(t)∥s0 +(2.10) +Proof. If +d +dtFu(t) ∈ Γm +0 we get by continuity and linearity of Op +d +dt Op[Fu(t)] = Op[∂tFu(t)] +To prove this and (2.10) it is sufficient to show that for any α ∈ Nd +0 there exists Cα = Cα(F) +auch that for all ξ ∈ Rd +∥∂α +ξ ∂tFu(t)(·, ξ)∥L∞ ≤ Cα∥∂tu(t)∥s⟨ξ⟩m−|α|. +Let α ∈ Nd +0 and set F α +u(t) := ∂α +ξ Fu(t). We have for all x, ξ ∈ Rd +∂tF α +u(t)(x, ξ) = +n +� +j=1 +∂tuj∂ujF α(u(t, x), ξ) +Due to F ∈ Sm(U) this yields +∥∂tF α +u(t)(x, ξ)∥L∞ ≤ ∥∂tu(t)∥L∞ +� +|β|=1 +∥∂β +uF α(·, ξ)∥L∞ ≤ Cα(F)∥∂tu∥s⟨ξ⟩m−|α|. +Lastly we prove a version of the strict G˚arding inequality for F ∈ Sm(U). First consider the +following lemma which is a modification of a construction in [21], proof of Thm. 18.1.6. +2.21 Lemma. There exists an even function ψ ∈ S(Rd × Rd) with unit integral, Op[ψ] = +Op[ψ]∗, ⟨op[ψ]v, v⟩ ≥ 0 (v ∈ S(Rd)) and F1ψ compactly supported. +Proof. Choose an even function ˆφ ∈ C∞ +0 (Rd × Rd) with L2-norm one and set φ = F −1 +1 +ˆφ. By +definition F1φ is compactly supported and clearly φ is even and has L2-norm one. Next, let +ψ ∈ S(Rd) be the symbol of op[ψ]∗ op[ψ]. As ibid. it follows that ψ is even and has unit +integral. op[ψ] = op[ψ]∗, ⟨op[ψ]v, v⟩L2 ≥ 0 (u ∈ S(Rd)) holds by definition. Now, let ρ be +the symbol of op[φ]∗. By Lemma 2.11 we get +F1ρ(η, ξ) = (F1φ)∗(−η, η + ξ), +η, ξ ∈ Rd +13 + +and thus by Lemma 2.12 +F1ψ(η, ξ) = +� +Rd F1ρ(η − θ, θ + ξ)F1φ(θ, ξ)dθ = +� +Rd F1φ(θ − η, η + ξ)F1φ(θ, ξ)dθ. +As F1φ is compactly supported, we can choose C > 0 such that F1φ(θ, ξ) = 0 if |θ| ≥ C +or |ξ| ≥ C. Then by definition F1ψ(η, ξ) = 0 if |ξ| ≥ C. Given |η| ≥ 2C and |θ| ≤ C we +conclude |θ − η| ≥ |η| − |θ| ≥ C, i.e. F1φ(θ − η, η + ξ) = 0. In conclusion we have proven +that F1ψ is in fact compactly supported. In particular ψ ∈ S(Rd × Rd). +Next we introduce a method to decompose symbols in Sm +1,1 into an infinite sum of infinitely +smoothing symbols; cf. [22]. +First, choose a function ρ ∈ D(Rd) even and monotonically decaying along rays such that +ρ(Rd) ⊂ [0, 1] and +ρ(ξ) = +� +1, +|ξ| ≤ 1 +2 +0, +|ξ| ≥ 1 . +For ν ∈ N0 define ρν, ζν ∈ D(Rd) by +ρν(ξ) := ρ(ξ/2ν), +ζν(ξ) = ρν+1(ξ) − ρν(ξ), ξ ∈ Rd +Additionally set ζ−1 := ρ. +2.22 Definition. For a function a : Rd × Rd → Cn×n and ν ≥ −1 define +aν(x, ξ) := a(x, ξ)ζν(ξ). +Note that a = � +ν≥−1 aν. +It is straightforward to show the following. +2.23 Lemma. Let a ∈ Sm +1,1. Then aν ∈ S−r for all r ∈ R and for any α, β ∈ N0 x, ξ ∈ Rd +|∂β +x∂α +ξ aν(x, ξ)|⟨ξ⟩r ≤ C2ν(r+m−|α|+|β|) � +γ≤α +Cγβ(a), +where Cγβ(a) are semi-norms of a. +2.24 Proposition. Let s > d/2, u ∈ Hs+2 and F ∈ Sm(U) such that there exists an R > 0 +with F(y, ξ) + F(y, ξ)∗ ≥ 0 for all y ∈ U and ξ ∈ Rd with |ξ| > R. Then there exists +C = C(∥u∥s+2, F) > 0 and for all q ∈ R there exists c = c(∥u∥s+2, F, q) > 0, both increasing +functions of ∥u∥s+2, such that for all v ∈ S(Rd, Cn) +⟨(Op[Fu] + Op[Fu]∗)v, v⟩L2 ≥ −C∥u∥ +1 +2 +s+2∥v∥2 +(m−1)/2 − c∥v∥2 +−q. +14 + +Proof. In the following it is straightforward to see that all constants can be chosen to be +increasing functions of ∥u∥s+2. First note that by Proposition 2.19 for all l ∈ R +∥ Op[Fu] + Op[Fu]∗ − Op[Fu + F ∗ +u]∥L(Hl+m−1,Hl) ≤ Cl∥u∥s+1. +Thus +⟨(Op[Fu] + Op[Fu]∗)v, v⟩L2 ≥ ⟨Op[Fu + F ∗ +u]v, v⟩L2 − C∥u∥s+1∥v∥2 +(m−1)/2, +v ∈ S(Rd). +Hence it is sufficent to prove the result for Op[Fu] + Op[Fu]∗ replaced by Op[Fu + F ∗ +u], i.e. +we can assume w.l.o.g F(u, ξ) = F(u, ξ)∗ ≥ 0. +It holds R(Fu) = R(F ∗ +u) = R(Fu)∗. By assumption this gives pointwise in Rd × {|ξ| ≥ R} +for all v ∈ Cn +⟨(R(Fu))v, v⟩Cn ≥ ⟨(R(Fu) − Fu)v, v⟩Cn ≥ −|R(Fu) − Fu||v|2 +≥ −(|R(Fu − F0) − (Fu − F0)| + |R(F0) − F0|)|v|2. +By Lemma 2.18 Fu−F0 ∈ Γm +2 with all semi-norms bounded by a positive constant depending +on F times ∥u∥s+2. By Lemma 2.8 (i) this yields R(Fu − F0) − (Fu − F0) ∈ Γm−1 +1 +with semi- +norms bounded in the same way. Thus +|R(Fu − F0) − (Fu − F0)| ≤ C0∥u∥s+2⟨ξ⟩m−1. +Using also that R(F0) − F0 has compact support we conclude that for all q ∈ R +|R(Fu − F0) − (Fu − F0)| + |R(F0) − F0| ≤ C0∥u∥s+2⟨ξ⟩m−1 + c0 +q⟨ξ⟩−q. +Therefore on Rd × {|ξ| ≥ R} +a := R(Fu) + C0∥u∥s+2⟨ξ⟩m−1 + c0⟨ξ⟩−r ≥ 0 +and a = a∗, a ∈ Sm,1−ǫ2 +1,1 +. As +Op[Fu] = op[R(Fu)] = op[a] − C0∥u∥s+2 op[⟨ξ⟩m−1] − c0 op[⟨ξ⟩−r] +it is now sufficient to show +⟨op[a]v, v⟩L2 ≥ −C∥u∥1/2 +s+2∥v∥2 +(m−1)/2 − c∥v∥−q +for all q ∈ R. +To this end we proceed similarly as in the proof of Theorem 9.7.1 in [22] but with a crucial +modification. First, decompose a = � +ν≥−1 aν according to Definition 2.22. As for all ν0 +¯aν0 := �ν0 +ν=−1 aν ∈ S−q for any q ∈ R with norm depending on µ, ν0 according to Lemma 2.23, +i.e. ∥ op[¯aν0]v∥ ≤ cν0,µ∥v∥−r, we only need to consider � +ν≥ν0 aν for some ν0 ∈ N. Naturally, +in a first step we choose ν0 large enough to obtain 2ν0−2 > R and thus by assumption +15 + +aν(x, ξ) ≥ 0 for all x, ξ ∈ Rd, ν ≥ ν0. But we will later see that we may have to choose ν0 +even larger. +W.l.o.g. assume u ̸= 0. Otherwise the result readily follows as F0 ≥ 0 is constant with +respect to x and Op[F0] − op[F0] is infinitely smoothing. +Choose an even function ψ ∈ S(Rd × Rd) with unit integral such that op[ψ] = op[ψ]∗, +⟨op[ψ]v, v⟩ ≥ 0 (v ∈ S(Rd)) and F1ψ compactly supported as constructed in Lemma 2.21. +For ν ∈ N0 set qν := 2ν/2 and write aν = bν + hν with +bν(x, ξ) := +� +Rd +� +Rd ψ((x − y)qνµ, (ξ − θ)/(qνµ))aν(y, θ) dy dθ +(2.11) += +� +Rd +� +Rd ψ(y, θ)aν(x − y/(qνµ), ξ − θqνµ) dy dθ, +(2.12) +where µ := ∥u∥s+2. As aν ≥ 0 and op[ψ] is a positive operator it is straightforward to obtain +the positivity of bν. Hence the theorem is proven provided +⟨op[h]v, v⟩L2 ≥ −Cµ∥u∥ +1 +2 +k+1∥v∥(m−1)/2, +v ∈ S(Rd). +(2.13) +To this end we show h ∈ Sm−1,L +1,1 +for some L ∈ (0, 1) and that all semi-norms of h are bounded +by a constant times ∥u∥ +1 +2 +s+2. Then (2.13) follows from Proposition 2.3. +First we verify h ∈ Sm−1 +1,1 +and the estimate on the semi-norms, i.e. +| +� +ν≥ν0 +∂β +x∂α +ξ hν(x, ξ)| ≤ Cαβ∥u∥ +1 +2 +s+2⟨ξ⟩m−1−|α|+|β|. +(2.14) +Let α = β = 0. Fix ξ ∈ Rd and consider ν ∈ N0 with |ξ| < 2ν−2 or |ξ| > 2ν+2. As aν(y, θ) = 0 +for 2ν−1 ≤ |θ| ≤ 2ν+1 we then have hν(x, ξ) = −bν(x, ξ) and it follows by basic estimates (cf. +[22]) that in the support of the first integrand in (2.11) +|ξ − θ| ≥ 1 +5(2ν + |ξ|) +and thus +|ξ − θ|/qν = 2−ν/2|ξ − θ| ≥ 1 +5(2ν + |ξ|) +1 +2. +(2.15) +As a ∈ Sm +1,1 and supp aν ⊂ {(x, θ) ∈ Rd × Rd : 2ν−1 ≤ |θ| ≤ 2ν} +|aν(y, θ)| ≤ C⟨θ⟩m ≤ C(1 + 2ν)m. +Hence ψ ∈ S(Rd × Rd) and (2.15) yield +|hν| ≤ Cm(1 + 2ν)m +� � +(|ξ − θ|/(qνµ))−2(|m|+1)(1 + |ξ − θ|/(qνµ))−n−1(1 + |(x − y)|qνµ)−n−1dy dθ +≤ Cm,nµ2(|m|+1)(1 + 2ν)m(2ν + |ξ|)−2|m|−2 +≤ Cm,nµ2(|m|+1)(1 + |ξ|)m−12−ν. +(2.16) +16 + +Thus +� +{ν:|ξ|<2ν−2 or |ξ|>2ν+2} +|hν| ≤ C∥u∥ +1 +2 +s+2⟨ξ⟩m−1. +(2.17) +Now consider ν ∈ N0 with 2ν−2 ≤ |ξ| ≤ 2ν+2. As ψ is an even function with unit integral we +get from (2.12) +hν = aν − bν = +� � +ψ(y, θ) +� +aν(x, ξ) − aν(x − y/(qνµ), ξ − θqνµ) +� +dy dθ += +� � +ψ(y, θ) +� +� +|α+β|<2 +∂β +x∂α +ξ aν(x, ξ)(−y)β(−θ)α − aν(x − y/(qνµ), ξ − θqνµ) +� +dy dθ. +(2.18) +By Taylor’s fomula we can estimate (w.l.o.g. assume |θ| ≤ |ξ|) +�� +� +|α+|β|<2 +∂β +x∂α +ξ aν(x, ξ)(−y)β(−θ)α − aν(x − y/(qνµ), ξ − θqνµ) +�� +≤ C +� +|α|+|β|=2 +sup +x,ξ∈Rd |∂β +x∂α +ξ aν(x, ξ)||yβθα|(qνµ)|α|−|β|. +(2.19) +Note that a = R(Fu). By Lemma 2.18 Fu ∈ Γm +2 and thus ∂β +xFu ∈ Γm +2−|β| for |β| ≤ 2. Hence +for each γ ∈ Nd +0, ξ ∈ Rd +∥∂γ +ξ ∂β +xFu(·, ξ)∥W 2−|β|,∞ ≤ Cγ⟨ξ⟩m−|γ| +and for |β| ≥ 1 we also have ∂β +xFu|u=0 = 0. Thus again by Lemma 2.18 +∥∂γ +ξ ∂β +xFu(·, ξ)∥W 2−|β|,∞ ≤ Cγ∥u∥s+2⟨ξ⟩m−|γ|. +Clearly +∂β +xa = ∂β +xR(Fu) = R +� +∂β +xFu). +and we conclude from Proposition 2.6 that ∂β +xa ∈ Sm +1,1 and for all x, ξ ∈ Rd +|∂β +x∂γ +ξ a(x, ξ)| ≤ Cγ⟨ξ⟩m−|γ| +� +1, +|β| = 0, +∥u∥s+2, +1 ≤ |β| ≤ 2 . +Then by Lemma 2.23 +sup +x,ξ∈Rd |∂β +x∂γ +ξ aν| ≤ Cγ2ν(m−|α|) ≤ Cγ +� +1, +|β| = 0, +∥u∥s+2, +1 ≤ |β| ≤ 2 +(2.20) +From (2.18), (2.19), (2.20) and µ = ∥u∥ +1 +4 +s+2, qν = 2ν/2 we now get for 2ν−2 ≤ |ξ| ≤ 2ν+2 +|hν| ≤ C +� +ψ(y, θ)(|θ|2 + |θ||y| + |y|2) dy dθ +� +2ν(m−2)2ν∥u∥ +1 +2 +s+2 + 2ν(m−1)∥u∥s+2 + 2νm∥u∥s+22−ν∥u∥ +− 1 +2 +s+2 +� +≤ Cµ2ν(m−1)∥u∥ +1 +2 +s+2 ≤ C∥u∥ +1 +2 +s+2⟨ξ⟩m−1, +17 + +where we used ψ ∈ S(Rd × Rd) and 2ν−2 ≤ |ξ| ≤ 2ν+2 in the last line. Together with (2.17) +this shows (2.14) for α = β = 0. +Now note that ∂β +x∂α +ξ bν is given by (2.12) with aν replaced by ∂β +x∂α +ξ aν. Hence we obtain +(2.14) for α, β ̸= 0 by applying the argumentation above with aν replaced by ∂β +x∂α +ξ aν and m +replaced by m − |α| + |β|. +To finish the proof we show that F1h vanishes on NL = {(η, ξ) ∈ Rd × Rd : |η + ξ| < L|ξ|} +with L := min{1 − ǫ2, 1 +2}. Then the estimate on the operator norm follows by the continuity +of op. As a = R(Fu) ∈ Sm,1−ǫ2 +1,1 +, it suffices to prove that F1b vanishes on N 1 +2. +By standard arguments on convolution and Fourier transform we have for all g ∈ S(Rd ×Rd) +bν(F1g) = (µqν)−d/2 +� +Rd +� +Rd aν(y, θ)F1f(y, θ) dθ dy, +where +f(η, θ) = +� +Rd F1ψ(η/(qνµ), (ξ − θ)/qνµ)g(η, ξ)dη. +(2.21) +Let supp g ⊂ N1/2. By construction we have supp F1ψ ⊂ {(ξ, η) ∈ Rd × Rd : |η|, |ξ| ≤ D, } +for some D > 0. Next choose ν0 ∈ N so large that 3Dµ ≤ qν0/2. Then for ν ≥ ν0 on the +support of the integrand of (2.21) we have |η|, |ξ − θ| ≤ Dqνµ and |ξ + η| + 1 < 1 +2|ξ|. The +first and third inequality yield +|ξ| < 2|η| ≤ 2Dqνµ +and thus the second one gives +|θ| ≤ Dqνµ + |ξ| < 3Dqνµ ≤ qνqν0/2 ≤ 2ν/22ν0/2−1 ≤ 2ν−1. +But this implies bν(y, θ) = 0 for all y ∈ Rd. Therefore we have proven bν(F1g) = 0 for +all ν ≥ ν0 and supp g ⊂ {(ξ, η) ∈ Rd × Rd : |ξ + η| < +1 +2|ξ|}. Hence this also holds for +b = � +ν≥ν0 bν. +3 +Dissipativity +Throughout this section we consider (1.1), (1.2) with smooth matrix families Aj, Bjk : U → +Rn×n, u0, u1 : Rd → U and u : [0, T] × Rd → U for some domain U ⊂ Rn. Carrying out the +differentiation with respect to xk on the right-hand side and distinguishing between space +and time derivatives we write (1.1) as +−B00(u)utt = +d +� +j,k=1 +Bjk(u)uxjxk+ +d +� +j=1 +(B0j(u)+Bj0(u)ut)xj−A0(u)ut− +d +� +j=1 +Aj(u)uxj+Q(u, Dt,xu), +18 + +where Q is of the form +Q(u, Dt,xu) = +n +� +l=1 +d +� +j,k=0 +Qljk(u)ul +xkuxj. +We will see in the proofs that the specific form of the matrices Qljk(u) does not play any role. +Hence multiplying (1.1) by (−B00)−1, we can assume −B00 = In without loss of generality, +which we will always do in the following. +Next, denote by +B(u, ξ) := +d +� +j,k=1 +Bjk(u)ξjξk, +C(u, ξ) := +d +� +j=1 +(B0j(u) + Bj0(u))ξj, +A(u, ξ) := +d +� +j=1 +Aj(u)ξj, +ξ = (ξ1, . . . , ξn) ∈ Rd. +the symbols of the second and first order parts, respectively. Then the hyperbolicity of both +sides of (1.1) is expressed by the following conditions: +(HA) (a) there exists a smooth bounded family of hermitian uniformly positive definite ma- +trices Σ : U → Rn×n such that Σ(u)A0(u) is symmetric and uniformly positive on U, +(b) the matrix family A0(u)−1A(u, ξ) permits a symbolic symmetrizer H(u, ξ), +(HB) with +B(u, ξ) = +� +0 +|ξ|In +−|ξ|−1B(u, ξ) +iC(u, ξ) +� +, +ξ = (ξ1, ..., ξd) ∈ Rd, +the matrix family iB(u, ξ) permits a symbolic symmetrizer H(u, ξ). +Above we use the following notion of a symbolic symmetrizer (cf. e.g. [38]). +3.1 Definition. Let K ∈ C∞(U × Rd \ {0}, Cn×n). A symbolic symmetrizer for K is a +smooth mapping S ∈ C∞(U × Rd \ {0}, Cn×n) positive homogeneous of degree 0 with respect +to the second argument, bounded as well as all its derivatives on U ×Sd−1 such that for some +c > 0 and all (u, ξ) ∈ U × Rd \ {0} +S(u, ξ) = S(u, ξ)∗ ≥ cIn, +and S(u, ξ)K(u, ξ) = (S(u, ξ)K(u, ξ))∗. +3.2 Remark. K admits a symbolic symmetrizer if K is positive homogeneous of degree 1, +for all (u, ω) ∈ U ×Sd−1 all eigenvalues of K(u, ξ) are real, semi-simple (i.e. their geometric +and algebraic multiplicities coincide) and their multiplicities do not depend on (u, ω) (cf. +[38], Proposition 5.2 C). If this holds for A0(u)−1A(u, ξ) or B(u, ξ) the respective operator +is often called constantly hyperbolic. +19 + +We now fix a homogeneous state ¯u ∈ U and assume the following dissipativity conditions on +the coefficient matrices. +Condition (D). Matrices Aj(¯u), Bjk(¯u) have three properties: +(D1) For every ω ∈ Sd−1, all restrictions, as a quadratic form, of +W1 = H(¯u, ω)(A0(¯u))−1� +− B(¯u, ω) + (A0(¯u))−1(A(¯u, ω))(A0(¯u))−1A(¯u, ω) ++ C(¯u, ω)(A0(¯u))−1A(¯u, ω) +� +, +on the eigenspaces E = J−1 +E (Cn) of +W0 = (A0(¯u))−1A(¯u, ω) +are uniformly negative in the sense that +J∗ +E (W1 + W ∗ +1 ) JE ≤ −¯c J∗ +EJE +with one ¯c > 0. +(D2) For every ω ∈ Sd−1, all restrictions, as a quadratic form, of +W1 = H(¯u, ω)A(¯u, ω), +A(¯u, ω) = +� +0 +0 +−iA(¯u, ω) +−A0(¯u) +� +(3.1) +on the eigenspaces E = J −1 +E (C2n) of +W0 = B(¯u, ω) +(3.2) +are uniformly negative in the sense that +J ∗ +E (W1 + W∗ +1) JE ≤ −¯c IE +with one ¯c > 0.. +(D3) All solutions (λ, ξ) ∈ C × (Rd \ {0}) of the dispersion relation of (1.1) at ¯u = 0 have +Re(λ) < 0. +3.3 Remark. Note that as (D) is an open condition there exists a neighbourhood of ¯u such +that Bjk(u), Aj(u) satisfy (D) with ¯u replaced by u for all u ∈ U0 with ¯c independent of u. +The following remark is useful in the proofs below. +3.4 Remark. It is straightforward to show that (D1) and (D2) are equivalent to the same +conditions with W0, W1 replaced by +¯W0 := H(¯u, ω) +1 +2A(¯u)−1A(0, ω)H(¯u, ω)− 1 +2, +¯W1 := H(¯u, ω)− 1 +2W1H(¯u, ω)− 1 +2 +and W0, W1 replaced by +¯ +W0 := H(¯u, ω) +1 +2B(¯u, ω)H(¯u, ω)− 1 +2, +¯ +W1 := H(¯u, ω) +1 +2A(¯u, ω)H(¯u, ω)− 1 +2. +20 + +From now on we always assume (HA), (HB) and (D). As we could also consider (1.1), (1.2) +in the variable u − ¯u, we can w.l.o.g. restrict our argumentation to the case ¯u = 0. +We write (1.1) as the first-order in time system +ut = v +vt = +d +� +j=1 +(Bj0 + B0j)(u)vxj + +d +� +j,k=1 +Bjk(u)uxjxk − A0(u)v − +d +� +j=1 +Aj(u)uxj + Q(u, Dt,xu) (3.3) +and denote by +¯ +M(u, ξ) := +� +0 +In +M(u, ξ) +N(u, ξ) +� +, +(3.4) +with +M(u, ξ) = −iA(u, ξ) − B(u, ξ), +N(u, ξ) = iC(u, ξ) − A0(u), +the Fourier symbol of (3.3). We also define +M(u, ξ) := Z(ξ) ˜ +M(u, ξ)Z(ξ)−1 +Z(ξ) = +�⟨ξ⟩In +0 +0 +In +� +. +First we treat the linearization of (1.1) at the reference state u = 0, i.e. +d +� +j=0 +Aj(0)uxj = +d +� +j,k=0 +Bij(0)uxixj. +(3.5) +Such linear systems were studied in [14], however under the stronger assumptions, that the +coefficient matrices are symmetric and A0 is positive definite. Then (HA) is clearly satisfied +with FA = In and H = A0. Also, condition (HB) (b) ibid. +requires the existence of a +matrix family S : Sd−1 → Cn×n such that iS(ω)B(0, ω)S(ω)−1 is real symmetric. But one +can easily check that this can be relaxed to the assumption that iS(ω)B(0, ω)S(ω)−1 is +hermitian, which is satisfied in the present context for S(ω) := H(0, ω) +1 +2. Lastly, we want +to point out that (D1), (D2) ibid. were stated in the equivalent form mentioned in Remark +3.4. +We will make plausible below that the weaker conditions in the present work are still sufficient +to retrieve the main result of [14], namely: +3.5 Proposition. There exist a c > 0 and a family ξ �→ T (ξ), Rd → C2n×2n of linear +transformations of C2n which, together with their inverses T (ξ)−1, are uniformly bounded, +such that +T (ξ)M(0, ξ)T −1(ξ) + (T (ξ)M(0, ξ)T −1(ξ))∗ ≤ −cρ(ξ)I2n, +ξ ∈ Rd, +(3.6) +where ρ(ξ) = |ξ|2/(1 + |ξ|2). +21 + +As outlined in [14] this brings about the pointwise decay of solutions in Fourier space and +thus the following decay estimate for the inhomogeneous linear Cauchy problem. +3.6 Corollary. For any s ∈ N0 there exists C > 0 such that the following holds: For all +u0 ∈ Hs+1 ∩ L1, u1 ∈ Hs ∩ L1 and f ∈ C([0, T], Hs ∩ L1) the solution u of +f + +d +� +j=0 +Aj(0)uxj = +d +� +j,k=0 +Bij(0)uxixj +with u(0) = u0, ut(0) = u1 satisfies +∥u(t)∥s+1 + ∥ut(t)∥s ≤ C(1 + t)− d +4(∥u0∥s+1 + ∥u0∥L1 + ∥u1∥s + ∥u1∥L1) +C +� t +0 +(1 + t − τ)− d +4(∥f(τ)∥s + ∥f(τ)∥L1) dτ +for all t ∈ [0, T]. +Proof of Proposition 3.5. As stated above the proof can be found essentially in [14]. We just +illustrate at which points it has to be slightly modified. +The existence of a bounded family T (ξ) ⊂ Gl2n2 satisfying (3.6) is proven separately for the +three different regimes |ξ| ≤ r0, r0 ≤ |ξ| ≤ r∞ and |ξ| ≥ r∞ for suitable r0, r∞ > 0. In +the latter two cases only ((H)B) and conditions (D2), (D3) are used. The symmetry of the +matrices plays no role whatsoever. +For small values of |ξ| writting ξ = ξω for ξ > 0, ω ∈ Sd−1 one finds a bounded family of +invertible R(ξ, ω) with R(ξ, ω)−1 also bounded and (supressing the argument u = 0) +R(ξ, ω) ¯ +M(ξω)R(ξ, ω)−1 = +� +X(ξ, ω) +0 +0 +Y (ξ, ω) +� +, +where +X(ξ, ω) = iξ(A0)−1A(ω) ++ ξ2(A0)−1� +− B(ω) + (A0)−1(A(ω))(A0)−1A(ω) + C(ω)(A0)−1A(ω) +� ++ O(ξ3) +Y (ξ, ω) = −A0 + O(ξ3). +This is due to the fact that A0(0) is invertible and again makes no use of the symmetry. +Hence for +ˇR(ξ, ω) = +� +H(ω) +1 +2 +0 +0 +F +1 +2 +A +� +ˇR(ξ, ω) +we get +ˇR(ξ, ω)M(ξω) ˇR(ξ, ω)−1 = +� +iξ ¯W0 + ξ2 ¯W1 + O(ξ3) +0 +−F +1 +2 +A A0F +− 1 +2 +A . +� +. +2For m ∈ N Glm denotes the space of invertible m × m-matrices. +22 + +with ¯W0, ¯W1 as in Remark 3.4. Since F +1 +2 +A A0F +− 1 +2 +A +is positive definite the existence of the +family T (ξ) now follows for sufficiently small ξ by condition (D1) and [14], Lemma 5.3 +In Section 4 we will see that, given d ≥ 3, s > d/2 + 1, Corollary 3.6 directly implies the +decay of a solution to the quasi-linear problem (1.1) in Hs−1 but only provided that its +Hs-norm is a-priori known to be small. To close this gap we need to show that the Hs-norm +of a small solution can be bounded by the initial conditions and L2-norms of lower order +derivatives. The rest of this section is devoted to a construction preparing such a result. +In the following for ξ ∈ Rd we write ξ = ξω with ξ = |ξ| ∈ [0, ∞), ω = ξ/|ξ| ∈ Sd−1. +For r > 0, u ∈ Rn, ξ ∈ Rd and ω ∈ Sd−1 by Bn(u, r), Bd(ξ, r), BS(ω, r) we denote the balls +with radius r and center u, ξ, ω with respect to the metrices on Rn, Rd, Sd−1. +For some +ω∗ ∈ Sd−1 and δ > 0 we use +P(ω∗, δ) = Bn(0, δ) × [0, δ) × BS(ω∗, δ) +. +3.7 Proposition. There exist r > 0, c∞ > 0 and a mapping D∞ ∈ C∞(Ω∞, C2n×2n), Ω∞ := +¯U0 × {ξ ∈ Rd : |ξ| ≥ r−1}, ¯U0 := Bn(0, r) ⊂ U, such that: +(i) For all (u, ξ) ∈ Ω∞ +D∞(u, ξ) = D∞(u, ξ)∗ ≥ c∞In, +and +D∞(u, ξ)M(u, ξ) + (D∞(u, ξ)M(u, ξ))∗ ≤ −c∞I2n. +(ii) For any α, β ∈ Nd +0 there exist Cαβ > 0 with +|∂β +u∂α +ξ D∞(u, ξ)| ≤ Cαβ⟨ξ⟩−|α|, +(u, ξ) ∈ Ω∞. +(3.7) +Proof. Consider the mapping K : U × (0, ∞) × Sd−1 → C2n×2n defined by +K(u, η, ω) = +� +0 +In +−iηA(u, ω) − B(u, ω) +−iC(u, ω) − ηA0(u) +� +, +ω ∈ Sd−1. +(3.8) +and H(u, ω) denote the symmetrizer of B(u, ω) as in condition (HB) (b). Set +W(u, η, ω) := H(u, ω) +1 +2K(u, η, ω)H(u, ω)− 1 +2. +Since +K(0, 0, ω) = +� +0 +In +−B(0, ω) +iC(0, ω) +� += B(0, ω) +3Note that in said Lemma it is sufficient to assume that iM(0, ω) is selfadjoint instead of requiring +iM(0, ω) to be real symmetric. +23 + +and +∂K +∂η (0, 0, ω) = +� +0 +0 +−iA(0, ω) +−A0(0) +� += A(0, ω) +W satisfies +W(0, 0, ω) = ¯ +W0, +∂W(0, 0, ω) +∂η += ¯ +W1, +with ¯ +W0, ¯ +W1 as in Remark 3.4. Now fix ω0 ∈ Sd−1. By virtue of condition (D2) it follows +from Lemma 5 in [14] that there exists δ0 > 0, c0 > 0 and T0 ∈ C∞(P(ω∗, δ0), Gl2n) with +T −1 +0 +also bounded such that pointwise on P(ω0, δ0) +T0WT −1 +0 ++ (T0WT −1 +0 +)∗ ≤ −˜cηI2n +for some ˜c > 0. Hence ˜D0 := H +1 +2T ∗ +0 T0H +1 +2 ∈ C∞(P(δ0, ω0), C2n×2n) satisfies +˜D0(u, ξ, ω) = ˜D0(u, ξ, ω)∗ ≥ cI2n, +(u, ξ, ω) ∈ P(δ0, ω0) +for some c > 0 and thus +˜D0K + ( ˜D0K)∗ ≤ −c˜cηI. +In conclusion we have shown the following: For each ω ∈ Sd−1 there exist δω > 0, cω > 0 +and Dω ∈ C∞(P(ω, δω), C2n×2n) such that for all (u, ξ, ¯ω) ∈ P(ω, δω) +Dω(u, η, ¯ω) = Dω(u, η, ¯ω)∗ ≥ cωI +Dω(u, η, ¯ω)K(u, η, ¯ω) + (Dω(u, η, ¯ω)K(u, η, ¯ω))∗ ≤ −cωξ2I. +(3.9) +As Sd−1 is compact we may choose ω1, . . . , ωr such that +¯l� +l=1 +BS(ωl, δl/2) = Sd−1 (δl := δωl). +Set r0 = min{δ1, . . . , δr}, c0 = min{cω1, . . . , cωr}. Then for l = 1, . . . , ¯l and Pl := Bn(0, r0) × +[0, r0) × BS(ωl, δl) choose functions φl ∈ C∞(Sd−1, [0, 1]) with supp φl ⊂ BS(ωj, δl), φl = 1 +on BS(ωj, δj/2) and extend Dl := Dωl trivially by 0 to a function defined on Bn(0, r0) × +[0, r0) × Sd−1 =: Ω0. Define +D0 : Ω0 → C2n×2n : (u, η, ω) �→ +¯l +� +l=1 +φl(ω)Dl(u, η, ω). +Then D0 ∈ C∞(Ω0, C2n×2n), and D0(u, η, ω) is hermitian for all (u, η, ω) ∈ Ω0. Furthermore +for (u, η, ω) ∈ Ω0 we have ω ∈ BS(ωk, δ/2) for some k ∈ {1, . . . , ¯l} and thus as Dl(u, η, ω) ≥ 0 +D0(u, η, ω) = +¯l +� +l=1 +φl(ω)Dl(u, η, ω) ≥ Dk(u, η, ω) ≥ c0I. +with the same reasoning we see +D0(u, η, ω)K(u, η, ω) + (D0(u, η, ω)K(u, η, ω))∗ ≤ −c0ηI2n, +(u, η, ω) ∈ Ω0. +24 + +Now note that for all u, ξ, ω +ξK(u, 1/ξ, ω) = +� +0 +ξIn +−iA(u, ω) − ξB(u, ω) +−iξC(u, ω) − A0(U) +� += ˜Z(ξ)M(u, ξω) ˜Z(ξ)−1, +where +˜Z(ξ) = +� ⟨ξ⟩ +ξ In +0 +0 +In +� +. +As clearly ˜Z, ˜Z−1 ∈ C∞((r−1 +0 , ∞), C2n×2n) are symmetric and positive definite on (r−1 +0 , ∞), +for r := r0/2, Ω∞ := Bn(0, r) × {ξ ∈ Rd : |ξ| ≥ r−1} the mapping +D∞ : Ω∞ → C2n×2n, (u, ξ) �→ ˜Z(|ξ|)D0(u, 1/|ξ|, ξ/|ξ|) ˜Z(|ξ|) +is in C∞(Ω∞, C2n×2n) and for all (u, ξ) ∈ Ω∞ +D∞(u, ξ) = D∞(u, ξ)∗ ≥ c∞I2n +for some c∞ > 0. Since for ξ = ξω ∈ U0 +D∞(u, ξ)M(u, ξ) = ξ ˜Z(ξ)D0(u, 1/ξ, ω) ˜Z(ξ)K(u, 1ξ, ω) = ξ ˜Z(ξ) ˜D0(u, 1/ξ, ω)K(u, 1/ξ) ˜Z(ξ), +we also have +D∞(u, ξ)M(u, ξ) + (D∞(u, ξ)M(u, ξ))∗ ≤ −c∞I2n +for some c∞ > 0. +It remains to verify (3.7). First note that the functions ξ �→ ⟨|ξ|⟩/|ξ|, ξ �→ ξk/|ξ|, k = 1, . . . , d +and ξ �→ 1/|ξ| are positive homogeneous of degree 0 and −1, respectively. Thus for any +α ∈ Nd +0 there exists Cα > 0 such that for all ξ ∈ Rd with |ξ| > 2r−1 +0 +|DαZ(ξ)| + |Dα(ξk/|ξ|)| + |Dα(1/|ξ|)| ≤ Cα⟨ξ⟩−|α|. +Since D0 as well as all of its derivatives are bounded on Bn(0, r0/2) × [0, r0/2] × Sd−1 the +estimate (3.7) follows by product and chain rule. +25 + +4 +Proof of Theorem 1.1 +To begin with, we remark that local well-posednes sof (1.1), (1.2) follows from the existing +theory for hyperbolic systems of any order [38].4 Our task thus consists in showing that +under an a priori smallness assumption the solution satisfies the decay and energy estimates +(1.3) and (1.4), for, w.l.o.g., ¯u = 0. Then we can extend them globally by standard methods +(cf. e.g. [24], proof of Theorem 3.6). We show the following. +4.1 Proposition. Consider d ≥ 3, s > d/2 + 1 and assume (HB), (HA) and (D). Then +there exist constants µ > 0, δ = δ(µ) > 0, and C = C(µ, δ) > 0 (all independent of T) +such that the following holds: For all u0 ∈ Hs+1, u1 ∈ Hs with ∥u0∥s+1 + ∥u1∥s < δ and all +u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs) satisfying (1.1), (1.2) and +sup +t∈[0,T] +∥u(t)∥2 +s+1 + ∥ut(t)∥2 +s + +� T +0 +∥u(τ)∥2 +s+1 + ∥ut(τ)∥2 +s dτ ≤ µ +we have for all t ∈ [0, T] +∥u(t)∥s + ∥ut(t)∥s−1 ≤ C(1 + t)− d +4(∥u0∥s + ∥u0∥L1 + ∥u1∥s−1 + ∥u1∥L1), +(4.1) +∥u(t)∥2 +s+1 + ∥ut(t)∥2 +s + +� t +0 +∥u(τ)∥2 +s+1 + ∥ut(τ)∥2 +s ≤ C(∥u0∥2 +s+1 + ∥u0∥2 +L1 + ∥u1∥2 +s1 + ∥u1∥2 +L1) +(4.2) +We split the proof into two parts corresponding to the following two assertions. +4.2 Proposition. In the situation of Proposition 4.1 there exist µ > 0, δ > 0, and C > +0 such that the following holds: For all u0 ∈ Hs+1 ∩ L1, u1 ∈ Hs ∩ L1 with ∥u0∥s+1 + +∥u1∥s, ∥u0∥L1 + ∥u1∥L1 < δ and all u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs) satisfying (1.1), +(1.2) and +sup +t∈[0,T] +∥u(t)∥2 +s+1 + ∥ut(t)∥2 +s+1 + +� T +0 +∥u(τ)∥2 +s+1 + ∥ut(τ)∥2 +s dτ ≤ µ +(1.3) holds for all t ∈ [0, T]. +4.3 Proposition. In the situation of Proposition 4.1 there exist µ > 0, and C > 0 such that +the following holds: For all u0 ∈ Hs+1, u1 ∈ Hs and all u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs) +satisfying (1.1), (1.2) and +sup +t∈[0,T] +∥u(t)∥2 +s+1 + ∥ut(t)∥2 +s+1 + +� T +0 +∥u(τ)∥2 +s+1 + ∥ut(τ)∥2 +s dτ ≤ µ +we have for all t ∈ [0, T] +∥u(t)∥2 +s+1 + ∥ut(t)∥2 +s + +� t +0 +∥u(τ)∥2 +s + ∥ut(τ)∥2 +s−1dτ +≤ C(∥u0∥2 +s+1 + ∥u1∥2 +s) + C +� t +0 +∥u(τ)∥2 +s + ∥ut(τ)∥2 +s−1 dτ. +(4.3) +4For example, the recent result in [3], which applies to the class we study in Section 5, is of this type. +26 + +From there Proposition 4.1 clearly follows by multiplying (4.3) with a sufficiently small factor +integrating, (1.3) with respect to t, and adding the resulting inequalities. +For notational reasons we write the first order representation (3.3) of (1.1) in the compact +form +Ut = L(u)U + (0, Q(u, Dx,tu))t +(4.4) +with U = (u, ut), +L(u) = +� +0 +In +�d +j,k=1 Bjk(u)∂xj∂xk − �d +j=1 Aj(u)∂xj +�d +j=1( ¯Bj0 + ¯B0j)(u)∂xj − A0. +� +Proof of Proposition 4.2. As s > d/2+1 we find by Moser type inequalities (cf. [4] Appendix +C and the references therein) +∥(L2(u) − L2(0))U∥s−1 + ∥(L2(u) − L2(0))U∥L1 ≤ Cµ∥u∥s−1(∥u∥s+1 + ∥ut∥s), +where L(u)U = (U2, L2(u)U). Furthermore +∥Q(u, Dx,tu))∥s−1 + ∥Q(u, Dx,tu)∥L1 ≤ Cµ∥u∥s∥ut∥s−1. +Now writing system (4.4) as L(0)U = (0, L2(0)−L2(u)+Q(u, Dx,tu)) and applying Corollary +3.6 to f = (L2(0) − L2(u)) + Q(u, Dx,tu) with s replaced by s − 1 yields +∥u(t)∥s + ∥ut(t)∥s−1 ≤ C(1 + t)− d +4(∥u0∥s + ∥u0∥L1 + ∥u1∥s−1 + ∥u1∥L1) ++ Cµ sup +τ∈[0,t] +(∥u(τ)∥s+1 + ∥ut(τ)∥s) +� t +0 +(1 + t − τ)− d +4(∥u(τ)∥s + ∥ut∥s−1)dτ. +(4.5) +As t → (1 + t)− d +4 is square-integrable over [0, ∞) for d ≥ 3 this gives (1.3) as in e.g. [24], +proof of Proposition 3.3. +Proof of Proposition 4.3. From now Cµ always denotes some constant depending monoton- +ically increasing on µ, whose concrete value may change at every instance. +For 0 < ǫ < 1 let Jǫ be the Friedrichs mollifier and set V = (Λu, ut), W := Wǫ := ΛsJǫ(Λu, ut) +and +Mu(x, ξ) = Mu(t)(x, ξ) = M(u(t, x), ξ) += +� +0 +⟨ξ⟩In +� +− B(u, ξ) − A(u, ξ) +� +⟨ξ⟩−1 +C(u, ξ) − A0(u) +� +. +We start with the following observation. +27 + +4.4 Lemma. W satisfies the differential equation +Wt = Op[Mu]W + R1, +(4.6) +for some R1 ∈ L2 satisfying +∥R1∥ ≤ Cµ∥V ∥2 +s + C∥V ∥s−1 +(4.7) +Proof. Set +˜L(u) := +�ΛIn +0 +0 +In +� +L(u) +�Λ−1In +0 +0 +In +� +. +Then +Vt = Op[Mu]V + ˜R1 +(4.8) +where +˜R1 = (˜L(u) − Op[Mu])V + (0, Q(u, Dx,tu)). +As we have already seen in the proof of Proposition 4.2 (now with s − 1 replaced by s) +∥Q(u, Dx,tu)∥s ≤ Cµ∥V ∥2 +s. +By Lemma 2.9 +∥(˜L(0) − Op[M0])V ∥s ≤ C∥V ∥s−1 +and due to Lemma 2.9 (iii) all terms appearing in +(˜L(u) − ˜L(0) − Op[Mu − M0])V +are of the form (a(u) − Op[au])f, where a is a smooth function with a(0) = 0 and f ∈ +{∂l +t∂β +xu| l ≤ 1, l + |β| ≤ 2} ⊂ Hs−1 ֒→ L∞. Hence Lemma 2.10 yields +∥(˜L(u) − ˜L(0) − Op[Mu − M0])V ∥s ≤ Cµ(∥u∥s∥V ∥s + ∥V ∥s−1). +In conclusion we have shown +∥ ˜R1∥s ≤ Cµ(∥V ∥2 +s + ∥V ∥s−1). +(4.9) +Now apply ΛsJǫ to (4.8) and obtain +Wt = Op[Mu]W + R1, +(4.10) +where +R1 = [ΛsJǫ, Op[Mu]]V + ΛsJǫ ˜R1 +Note that (Jǫ)ǫ∈(0,1) is a family of pseudo-differential operators, constant with respect to x, +with symbols uniformly bounded in S0. Thus we get from (4.9) +∥ΛsJǫR1∥ ≤ Cµ∥V ∥2 +s + C∥V ∥s−1 +and from Proposition 2.19 (iii) +∥[ΛsJǫ, Op[Mu]]V ∥ ≤ Cµ∥u∥s∥V ∥s + C∥V ∥s−1, +which proves the assertion. +28 + +Next, let D∞ ∈ C∞(Bnr (0) × {ξ ∈ Rd : |ξ| ≥ r}, C2n×2n) be the mapping constructed in +Proposition 3.7 and extend it trivially by zero to a function defined on Bn +r (0)×Rd := U0×Rd. +Choose a function φ ∈ C∞(Rd), with 0 ≤ φ ≤ 1, φ(ξ) = 0 for |ξ| ≤ 2r and φ(ξ) = 1 for +|ξ| ≥ 3r. Set +D(v, ξ) := φ(ξ)D∞(v, ξ), +(v, ξ) ∈ U0 × Rd +Let µ be sufficiently small such that u(t, x) ∈ Bnr (0) for all (t, x) ∈ [0, T] × Rd and define +Du(x, ξ) := Du(t)(x, ξ) = D(u(t, x), ξ), +(t, x, ξ) ∈ [0, T] × Rd × Rd. +Choose another function ψ ∈ C∞(Rd), with 0 ≤ ψ ≤ 1, ψ(ξ) = 0 for |ξ| ≥ 5r, ψ(ξ) = 1 for +|ξ| ≤ 4r and define +˜Du(x, ξ) = Du(x, ξ) + ψ(ξ)I2n. +4.5 Lemma. The family of operators (Gu(t))t∈[0,T] defined by +Gu(t) := 1 +2(Op[ ˜Du(t)] + Op[ ˜Du(t)]∗) + op[ ˜D0] − Op[ ˜D0] +is self-adjoint and uniformly positive definite in L(L2) for µ sufficiently small. Furthermore +1 +2 +d +dt +� +GuW, W⟩ = Re⟨Gu Op[Mu]W, W⟩ + R2, +for some R2 ∈ R with +|R2| ≤ Cµ∥W∥(∥V ∥2 +s + ∥V ∥s∥W∥ + ∥V ∥s−1). +Proof. By Proposition 3.7 ˜D, D ∈ S0(U) and ˜Du = ˜D∗ +u is uniformly positive definite. In +particular, op[ ˜D0] = op[ ˜D0]∗ is a self-adjoint and uniformly positive definite operator on +L(L2) (cf. Lemma 2.9). Due to ibid. also Op[ ˜D0]∗ = Op[ ˜D0], i.e. +Gu = op[ ˜D0] + 1 +2(Op[ ˜Du − ˜D0] + Op[ ˜Du − ˜D0]∗). +Proposition 2.19 (i) gives +∥ Op[ ˜Du − ˜D0]∥L(L2) ≤ Cµ∥u∥s, +which yields the first assertion. +Now apply Gu to (4.6), take the L2 scalar product with W and consider the real part to find +Re⟨GuWt, W⟩ = Re⟨Gu Op[Mu]W, W⟩ + Re⟨GuR1, W⟩ := Re⟨Gu Op[Mu]W, W⟩ + R21. +(4.11) +Due to (4.7) and ∥Gu∥L(L2) ≤ Cµ, +∥R21∥ ≤ Cµ∥W∥(∥V ∥2 +s + ∥V ∥s−1). +(4.12) +29 + +As Gu is self-adjoint we get +Re⟨GuWt, W⟩ = 1 +2 +d +dt +� +GuW, W⟩ − Re +�� d +dtGu +� +W, W⟩ +(4.13) +and 2.20 (iv) yields +2∥ d +dtGu∥L(L2) ≤ +�� d +dt Op[ ˜Du] +�� +L(L2) ≤ Cµ∥ut∥s. +(4.14) +The second statement then clearly follows from (4.11)-(4.14). +The last step consists in showing the following. +4.6 Lemma. It holds +Re⟨Gu Op[Mu]W, W⟩ ≤ −c∥W∥2 + Cµ∥W∥2(∥u∥ +1 +2 +s+1 + ∥u∥s) + Cµ∥W∥2 +−1). +From Lemmas 4.5, 4.6 we obtain +1 +2 +d +dt⟨GuW, W⟩+c∥W∥2 ≤ Cµ∥W∥(∥V ∥2 +s+∥V ∥s∥W∥+∥V ∥ +1 +2)+Cµ(∥V ∥2 +s−1+∥W∥2 +−1). (4.15) +As Λ−kW = Λ−kWǫ → V as ǫ → 0 uniformly with respect to t for 0 ≤ k ≤ s and Gu is +uniformly postive definite, we find by integrating (4.15) +∥V (t)∥2 +s + +� t +0 +∥V ∥2 +s dτ ≤ Cµ(∥V (0)∥ + +� t +0 +∥V (τ)∥3 +s + ∥V (τ)∥ +5 +2s + ∥V (τ)∥s−1)dτ, +t ∈ [0, T], +which yields the assertion since ∥V ∥2 +s = ∥u∥2 +s+1 + ∥ut∥2. +Proof of Lemma 4.6. Set κ := op[ ˜D0] − Op[ ˜D0], which is infinitely smoothing. Then +Gu = 1 +2(Op[ ˜Du] + Op[ ˜Du]∗) + κ. +As ∥Mu∥L(Hl,Hl−1) ≤ Cµ, l ∈ R, due to Proposition 2.19 (i) we find +Re⟨κ Op[Mu]W, W⟩ ≤ Cµ∥W∥2 +−1 +By construction ˜Du = ˜D∗ +u and thus 2.19 (ii) yields +Re +� +(1 +2(Op[ ˜Du]∗ − Op[Du]) Op[Mu]W, W +� +≤ Cµ∥u∥s∥W∥2. +Next note that ˜Du(x, ·) − Du(x, ·) is compactly supported with support not depending on t. +Therefore Op[ ˜Du − Du] is infinitely smoothing and +Re⟨Op[ ˜Du − Du] Op[Mu]W, W⟩ ≤ Cµ∥W∥2 +−1. +30 + +In conclusion +Re⟨Gu Op[Mu]W, W⟩ = Re⟨Op[Du] Op[Mu]W, W⟩ + Re⟨κ Op[Mu]W, W⟩ ++ 1 +2 Re⟨(Op[ ˜Du]∗ − Op[ ˜Du]) Op[Mu])W, W⟩ + Re⟨Op[ ˜Du − Du]MuW, W⟩ +≤ Re⟨Op[Du] Op[Mu]W, W⟩ + Cµ(∥u∥s∥W∥2 + ∥W∥2 +−1) +(4.16) +By Proposition 2.19 (iii) +∥(Op[Du] Op[Mu] − Op[DuMu])W∥ ≤ Cµ∥u∥s∥W∥ + C∥W∥−1. +Hence +Re⟨Op[Du] Op[Mu]W, W⟩ ≤ Re⟨Op[DuMu]W, W⟩ + Cµ∥u∥s∥W∥2 + C∥W∥2 +−1. +(4.17) +Set Xu := DuMu + c∞/2I2n with c∞ as in Lemma 3.7. Note that c∞ does not depend on µ. +Since Op[I2n] − IdL2 is infinitely smoothing we conclude +Re⟨Op[DuMu]W, W⟩ ≤ Re⟨Op[Xu]W, W⟩ − c∞ +2 ∥W∥2 + C∥W∥2 +−1. +(4.18) +By Proposition 3.7 +Xu(x, ξ) + X ∗ +u(x, ξ) = DuMu(x, ξ) + (DuMu)(x, ξ)∗ + c∞ ≤ 0, +for x ∈ Rd und ξ ∈ Rd with |ξ| ≥ 3r. Since u ∈ Hs+1 and s + 1 ≥ d/2 + 2, Proposition 2.24 +applied to −Xu gives +Re⟨Op[Xu]W, W⟩ ≤ Cµ(∥u∥ +1 +2 +s+1∥W∥2 + ∥W∥−1). +(4.19) +(4.18) and (4.19) lead to +Re⟨Op[DuMu]W, W⟩ ≤ −c∥W∥2 + Cµ(∥u∥ +1 +2 +s+1∥W∥2 + ∥W∥2 +−1) +(4.20) +for c independent of u. Clearly the assertion follows from (4.16), (4.17), (4.18) and (4.20). +5 +A class of examples from dissipative relativistic fluid +dynamics +We consider the Euler-augmented Navier-Stokes formulation of dissipative relativistic fluid +dynamics on flat Minkowski space-time derived in [12] as a generalization of a model proposed +in [1]. For barotropic fluids it consists of a system of four equations which, using Einstein’s +summation convention, read +Aαβγ(ψǫ)∂ψγ +∂xδ = +∂ +∂xβ +� +Bαβγδ(ψǫ)∂ψγ +∂xδ +� +, +α = 0, 1, 2, 3, +(5.1) +31 + +where all Greek indices run from 0 to 3, Aαβγ, Bαβγδ are contravariant tensors and the +unknown function ψǫ = (ψ0, ψ1, ψ2, ψ3)t determining the state of the fluid is a 4-vector with +respect to the Minkowski-metric of flat space-time. More specifically ψǫ = uǫ/θ with uǫ being +the 4-velocity, θ the temperature of the fluid. We show that the results of the present work +imply non-linear stability of the homogeneous reference state ¯ψ = ¯uǫ/¯θ, where ¯uǫ = (1, 0, 0, 0) +represents the fluid’s rest frame and ¯θ > 0 is a constant temperature. +For a fluid with equation of state p = ρ/r, 1 ≤ r < ∞, p being the pressure, ρ the specific +internal energy, the coefficent matrices evaluated at ¯ψ are given by [12] (w.lo.g. assume +¯θ = 1)5 +A0( ¯ψ) = +� +r +0 +0 +I3 +� +, +Aj( ¯ψ) = +� +0 +(ej)t +ej +0 +� +, +B00( ¯ψ) = +� +−r2µ +0 +0 +−νI3 +� +, +B0j( ¯ψ) = Bj0( ¯ψ) = 1 +2 +� +0 +−(µr + ν)(ej)t +−(µr + ν)ej +0 +� +, +Bij( ¯ψ) = +� +−νδij +0 +0 +ηδij + 1 +2(−µ + 1 +3η + ζ)(ei ⊗ ej + ej ⊗ ei) +� +, +i, j = 1, 2, 3, +where η, ζ > 0 quantify the fluid’s viscosity, ν, µ > 0 with µ > ˜η := 4 +3η + ζ reflect a frame +change and Aβ(ψǫ) := (Aαβγ(ψǫ))0≤α,γ≤3, Bβγ(ψǫ) := (Bαβγδ(ψǫ))0≤α,γ≤3, β, δ = 0, . . . , 3. +We do not give the detailed non-linear formulation at this point and just refer to [12]. The +only information we need for the argumentation below is the fact that for all β, δ = 0, . . . , 3 +and all states ψǫ the coefficient matrices Aβ(ψǫ), Bβδ(ψǫ), β, δ = 0, . . . , 3 are symmetric (cf. +ibid.). +We show (HA), (HB), (D) for the matrices (−B00)−1Bβδ, (−B00)−1Aβ. +(HA) is straightforward: As −B00(ψǫ), A0(ψǫ) are positive definite at ψǫ = ¯ψ and symmetric +for all states they are symmetric positive definite also in a neighbourhood of ¯ψ. Thus (HA) +(a) is satisfied with FA(u) = −B00(u) and (HA) (b) with H(u) = A0(u). +Regarding (HB) Freist¨uhler proved ibid. that at the reference state ψǫ = ¯ψ for each ω ∈ S2 +the matrix +˜B(ψǫ, ω) = +� +0 +I4 +−(−B00)− 1 +2B(ψǫ, ω)(−B00)− 1 +2 +i(−B00)− 1 +2C(ψǫ, ω)(−B00)− 1 +2 +� +, +where +B(ψǫ, ω) = +d +� +ij=0 +Bij(ψǫ)ωiωj, +C(ψǫ, ω) = 2 +d +� +j=0 +B0j(ψǫ)ωj, +ω = (ω1, . . . , ω2) ∈ S2, +5Here e1, e2, e3 and δij denote the conanical basis of R3 and the Kronecker symbol, respectively. +32 + +has four simple and two semi-simple purely imaginary eigenvalues. This is then also true for +B(ψǫ, ω) := +� +0 +I4 +(−B00)−1B(ψǫ, ω) +i(−B00)−1C(ψǫ, ω), +� += T −1 ˜B(ψǫ, ω)T +with T = diag((−B00) +1 +2, (−B00) +1 +2). Now in the present context the geometric multiplicities +of purely imaginary eigenvalues of B(ψǫ, ω) are state invariant properties. Therefore there +exists a symbolic symmetrizer of B due to Remark 3.2. +To see this invariance note that (even in the general setting in Section 3) the eigenvectors +v = v(u, ω) ∈ C2n \ {0} to an eigenvalue λ = λ(u, ω) ∈ C of B(u, ω) are exactly of the form +v = (v1, λv1) with v1 ∈ Cn such that eλt+iξv1 is a plane wave solution to the linearization +of (1.1) at u. As (5.1) is a covariant expression, eλt+iξv being a plane wave solution with +λ ∈ iR is also a covariant property (cf. e.g. [13]). +It remains to show (D1), (D2), (D3). In the following we only consider matrices evaluated +at ¯ψ. The Fourier-symbols correpsonding to the differential operators in (5.1) are given by +A(ω) = +d +� +j=1 +Ajωj = +�0 +ωt +ω +0 +� +, +B(ω) = +d +� +j,k=1 +Bjkωjωk = +�−r2µ +0 +0 +η + (−µ + 1 +3η + ζ)ω ⊗ ω +� +, +C(ω) = 2 +d +� +j=1 +B0jξj +� +0 +−(µr + ν)ωt +−(µr + ν)ω +0 +� +, +ω = (ω1, ω2, ω3) ∈ Sd−1. +It is straightforward to see that for any ω ∈ Sd−1 the matrices A0, Aj(ω), B00, Bjk(ω), C(ω) +all decompose in sense of linear operators as A0 = A0 +l ⊕ A0 +t, A(ω) = Al ⊕ At, B00 = +B00 +l +⊕ B00 +t , B(ω) = Bl ⊗ Bt, C(ω) = Cl ⊕ Ct with respect to the orthogonal decomposition +C4 = (C×ωC)⊕({0}×{ω}⊥). Thus we can verify the conditions for A0 +l , Al, B00 +l , Bl, Cl and +A0 +t, At, B00 +t , Bt, Ct separately. We have +A0 +t = I2, +At = 0, +B00 = −νI2, +Bt = ηI2, +Ct = 0. +As η > 0, these matrices correspond to coefficients of damped wave equations and it is +well-known that such equations satisfy (D). One can also check this easily by virtue of [14], +Theorem 4 and Lemma 5. +Next +A0 +l = +�r +0 +0 +1 +� +, +Al = +�0 +1 +1 +0 +� +, +B00 +l += +�−r2µ +0 +0 +−ν +� +, +Bl = +�−ν +0 +0 +˜η − µ +� +, +Ct = +� +0 +−(µr + ν) +−(µr + ν) +0 +� +. +It was shown in [14] that +˜Aj = (−B00 +l )− 1 +2Aj +l (−B00 +l )− 1 +2, +˜Bjk +l += (−B00 +l )− 1 +2Bjk +l (−B00 +l )− 1 +2 +satisfy (D). But then also ˇAj := (−B00 +l )−1Aj +l , ˇBjk := (−B00 +l )−1Bjk +l +satisfy (D). +33 + +To see this note that ˇAj = S ˜AjS−1, ˇBjk = S ˜BjkS−1 with S = (−B00) +1 +2. Hence we have +ˇW0 = S ˜W0S−1 for ˇW0 and ˜W0 as in (D1) for the matrices ˇAj, ˇBjk and ˜Aj, ˜Bjk, respectively. +Further the symbolic symmetrizer ˜H = ˜A0 of ( ˜A0)−1 ˜A(ω) the matrix ˇH = S−1 ˜A0S−1 is a +symbolic symmetrizer for ( ˇA0)−1 ˇA(ω). This yields ˇW1 = S−1 ˜W1S−1 with ˇW1, ˜W1 as in (D1) +for the respective matrices. If now v is an eigenvector of ˇW0, S−1v is an eigenvector of ˜W0 +and as ˜Aj, ˜Bjk satisfy (D1) we get +⟨( ˜W1 + ˜W ∗ +1 )S−1v, S−1v⟩ ≤ −c|S−1v|2 ≤ −ˇc|v|2, +i.e. ⟨( ˇW1 + ˇW ∗ +1 )v, v⟩ ≤ −ˇc|v|2, which proves (D1) for ˇAj, ˇBjk. +(D2) follows analogously since with S = diag((−B00) +1 +2, (−B00) +1 +2) the matrix S−1 ˜H(ω)S−1 +is a symbolic symmetrizer for ˇB(ω) if ˜H(ω) is a symbolic symmetrizer for ˜B(ω). +Lastly, (D3) is satisfied trivially, as the matrices introduce equivalent systems of PDEs and +thus solutions to the dispersion relation are identical for the two systems. +Statements and declarations +Funding. This work was supported by DFG Grants No. FR 822/10-1, 10-1/2) +Competing interests. The author has no competing interests to declare that are relevant +to the content of this article. +Acknowledgement. The author would like to sincerely thank Heinrich Freist¨uhler for his +highly helpful suggestions and comments as well as many fruitful discussions. +References +[1] F. S. 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Anal., 172(2):247–266, 2004. +37 + diff --git a/1tAzT4oBgHgl3EQft_25/content/tmp_files/load_file.txt b/1tAzT4oBgHgl3EQft_25/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..538dac96696e693e7ed4fef8ac3b47b0b9049f60 --- /dev/null +++ b/1tAzT4oBgHgl3EQft_25/content/tmp_files/load_file.txt @@ -0,0 +1,1241 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf,len=1240 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='01685v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='AP] 4 Jan 2023 Global existence and decay of small solutions for quasi-linear second-order uniformly dissipative hyperbolic-hyperbolic systems Matthias Sroczinski∗ January 5, 2023 Abstract This paper is concerned with quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Its main theorem es- tablishes global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Notably, the operators are not required to be symmetric hyperbolic, instead merely the existence of symbolic sym- metrizers is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The dissipation is characterized by conditions equivalent to the uniform decay of all Fourier modes at the reference state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' On a technical level, the theory developed herein uses para-differential operators as its main tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Apparently being the first to apply such operators in the context of global-in-time existence for quasi-linear hyperbolic systems, the present work contains new results in the field of para-differential calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the context of theoretical physics, the theorem applies to recent formulations for the relativistic dynamics of viscous, heat-conductive fluids notably such as that of Bemfica, Disconzi and Noronha [1] (Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' D, 98:104064, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' hyperbolic systems, initial value problem, global existence, asymptotic stability, para-differential operators, fluid mechanics AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Primary 35A01, 35B35, 35L72, 35L15, 35S50, 35Q35, 35Q75 ∗Department of Mathematics, University of Konstanz, 78457 Konstanz, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='sroczinski@uni-konstanz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='de, https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='org/0000-0002-5472-2741 1 1 Introduction and main result In this paper, we study systems of partial differential equations that are given by the su- perposition of two hyperbolic operators and show that homogeneous states are nonlinearly stable in the sense that small perturbations thereof lead to global-in-time decaying solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Concretely, we consider the Cauchy problem for quasi-linear systems of the form d � j=0 Aj(u(t, x))uxj(t, x) = d � j,k=0 (Bjk(u(t, x))uxj(t, x))xk, x0 = t ≥ 0, x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , xd) ∈ Rd, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) u(0, x) = u0(x), ut(0, x) = u1(x), x ∈ Rd, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) where both the operator on the right hand side and the operator on the left hand side are hyperbolic and each of them acts dissipatively on the trajectories generated by the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Such systems occur in theroretical physics as recent formulations for the (special-)relativistic dynamics of viscous, heat conductive fluids [15, 16, 17, 1, 12, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Our results apply to these formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The main theorem is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Consider d ≥ 3, s > d/2 + 1, ¯u ∈ Rn and let (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) satisfy conditions (HA), (HB) and (D) from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then there exist constants δ > 0 and C = C(δ) > 0 such that the following holds: For all u0, u1 with u0 − ¯u ∈ Hs+1(Rd, Rn) ∩ L1(Rd, Rn), u1 ∈ Hs(Rd, Rn)∩L1(Rd, Rn) as well as ∥u0 − ¯u∥Hs+1, ∥u1∥Hs, ∥u− ¯u∥L1, ∥u1∥L1 < δ there exists a unique global solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) satisfying u − ¯u ∈ C([0, ∞), Hs+1) ∩ C1([0, ∞), Hs), l = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , s + 1 and, for all t ∈ [0, ∞), ∥u(t) − ¯u∥Hs + ∥ut(t)∥Hs−1 ≤ C(1 + t)− d 4(∥u0 − ¯u∥Hs + ∥u1∥Hs−1 + ∥u0 − ¯u∥L1 + ∥u1∥L1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) ∥u(t) − ¯u∥2 Hs+1 + ∥ut(t)∥2 Hs + � t 0 ∥u(τ) − ¯u∥2 Hs+1 + ∥ut(τ)∥2 Hs dτ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4) ≤ C(∥u0 − ¯u∥2 Hs+1 + ∥u1∥2 Hs + ∥u0 − ¯u∥2 L1 + ∥u1∥2 L1) While conditions (HA) and (HB) specify the assumed hyperbolicity, condition (D), essentially obtained in [14], characterizes the needed decay behaviour for the Fourier modes of the associated linearized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Based on the famous Kawashima-Shizuta condition [24, 34], analogous results are well-known for symmetric hyperbolic-parabolic systems and first-order hyperbolic systems with relax- ation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [9, 33, 41, 19, 26, 42, 5] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 Regarding dissipative second-order hyper- bolic systems there are fewer results available, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' notably [11, 27, 32] and references therein, all of those treat systems whose structure is different form the one we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The 1Note that the often available reformulations of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) as first-order hyperbolic systems do typically not satisfy the assumptions of these works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2 most prominent example for equations satisfying condition (D) are probably damped wave equations with a non-linear convection term, which alternatively can be viewed as conserva- tion laws with hyperbolic artificial viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In this case, (D) reduces to Whitham’s famous sub-characteristic condition [40] and various in-depth results on the asymptotic behaviour of solutions have been achieved in this context, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [31, 25, 39, 20, 10, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Closest related to the present work are [35, 36, 37], there a predecessor of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 was shown for the systems proposed in [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The theory developed in the present work requires novel techniques in the use of para- differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Developed by Bony [6] and Meyer [30, 29], such operators have been used in the context of hyperbolic equations by G´erard and Rauch [18], Taylor [38] and M´etivier [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' However, quite different from these works, we will in particular need to precisely understand how the norms of para-differential operators depending on the functions inducing their symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the crucial Section 2 general results on para-differential operators needed for the argumentation in Section 3 and 4 will be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The present work apparently being the first that uses such operators to treat global-in-time solutions to quasi-linear hyperbolic systems, we prove corresponding new results on that dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The technical highlight in this regard will be a modified version of the strong G˚arding inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In Section 3 we construct a para-differential operator which is specifically associated with the system’s dissipativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Section 4 is dedicated to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The challenging part is the treatment of the highest derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Here we have to use the sophisticated estimates of Section 2 and the construction of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Finally, Section 5 shows that models of equations of dissipative relativistic fluid dynamics satisfy the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2 Results on para-differential operators A tour through the theory of para-differential operators from scratch to fine properties, this section relies on Appendix C of Benzoni-Gavage and Serre [4] and Section 9 of H¨ormander [22], however with strong attention to symbols induced by what later will be the solution to the PDE system considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In its initial part interpolating between brevity and legibility, the section culminates in the aforementioned novel version of the strong G˚arding inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 Notation, definitions and basics For topological vector-spaces V, W we write L(V, W) for the space of continuous linear oper- ators form V to W (or L(V ) if W = V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Throughout this section consider fixed dimensions n, d ∈ N and let m denote some real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For x, ξ ∈ Rd we just write xξ for their Euclidian scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 3 Let E be a finite-dimensional C-Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We denote the E-valued Schwartz space by S(Rd, E), and by S′(Rd, E) := L(S(Rd), E) the space of continuous linear mappings from S(Rd) to E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' the space of E-valued temperate distributions, both equipped with the standard locally convex topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For f ∈ S(Rd, E) the Fourier transform is (Ff)(ξ) = ˆf(ξ) = (2π)−d/2 � Rd f(x)e−ixξdx with inverse (F −1 ˆf)(x) = (2π)−d/2 � Rd ˆf(ξ)eixξdξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We write F1 and F2 for the Fourier transform with respect to the first and the second variable for functions f ∈ S(Rd × Rd, E), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (F1f)(η, y) = F(f(·, y))(η) = (2π)−d/2 � Rd f(x, y)e−ixηdx, (F2f)(x, ξ) = F(f(x, ·))(ξ) = (2π)−d/2 � Rd f(x, y)e−iyξdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As usual we extend F, F1, F2 to continuous operators on S′(Rd, E), S′(Rd × Rd, E) and unitary operators on L2(Rd, E), L2(Rd × Rd, E) also denoted by F, F1, F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We will use ⟨ξ⟩ := (1 + |ξ|2) 1 2, ξ ∈ Rd, Λm := F −1⟨·⟩mF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As usual Hm(Rd, E) := {u ∈ L2(Rd, E) : Λmu ∈ L2(Rd, E)}, are the L2-based E-valued Sobolev spaces with norm ∥u∥Hm(Rd,E) := ∥Λmu∥L2(Rd,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If E is a Hilbert space we consider the scalar product on Hm(Rd, E) ⟨u, v⟩Hm(Rd,E) := ⟨Λmu, Λmv⟩L2(Rd,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We also use L∞-based Sobolev spaces W k,∞(Rd, E) := {u ∈ L∞(Rd, E) : ∂α x u ∈ L∞(Rd, E), |α| ≤ k} with norm ∥u∥W k,∞(Rd,E) = max |α|≤k ∥∂α x u∥L∞(Rd,E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We often just write Hm, ∥u∥m, ⟨u, v⟩m, W k,∞ instead of Hm(Rd, E), ∥u∥Hm(Rd,E), ⟨u, v⟩Hm(Rd,E), W k,∞(Rd, E) if there is no concern for confusion, and ∥u∥ for ∥u∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For A ∈ Cn×n we denote the adjoint matrix by A∗ = ¯At and for T ∈ L(S(Rd, Cn)) we write T ∗ for the adjoint operator with respect to the L2(Rd, Cn) inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As usual we call T 4 self-adjoint if T = T ∗ and positive (strictly positive) if ⟨Tf, f⟩0 ≥ 0 (⟨Tf, f⟩ > 0), in which case we also write T ≥ 0 (T > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Next, we turn to the basic definitions concerning pseudo-differential operators which will be used in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We consider the following symbol classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (i) Sm := Sm(Rd, Cn×n) is the set of all functions a ∈ C∞(Rd×Rd, Cn×n) for which for any α, β ∈ N0 there exists Cαβ > 0 such that |∂β x∂α ξ a(x, ξ)| ≤ Cαβ⟨ξ⟩m−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) With semi-norms being the optimal constants in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1), Sm is a Fr´echet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (ii) Sm 1,1 := Sm 1,1(Rd, Cn×n) is the set of functions a ∈ C∞(Rd × Rd) for which for any α, β ∈ Nd 0 there exist Cαβ > 0 such that |∂β x∂α ξ a(x, ξ)| ≤ Cαβ⟨ξ⟩m−|α|+|β (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) for all (x, ξ) ∈ Rd × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' With semi-norms being the optimal constants in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2), Sm 1,1 is a Fr´echet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (iii) For a ∈ Sm 1,1 the mapping op[a] ∈ L(S(Rd, Cn)) defined by (op[a]f)(x) := (2π)− d 2 � eixξa(x, ξ)Ff(ξ) dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) is called the pseudo-differential operator with symbol a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We also write a := Sym[op[a]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As first shown in [7, 8] for a ∈ Sm 1,1 the operator op[a] extends to a bounded operator from Hl+m to Hl only if op[a]∗ also has a symbol in S1,1 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' But the operator norm of op[a] can in general not be controlled by semi-norms of a uniformly over this subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As for our applications to dissipative hyperbolic systems it is essential that the norm of op[a] is small if the semi-norms of a are small we have to make sure that the symbols occurring in the present work belong to the following smaller subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For L ∈ (0, 1], Sm,L 1,1 is the subspace of all a ∈ Sm 1,1 such that F1a vanishes on NL := {(η, ξ) ∈ Rd × Rd : |η + ξ| + 1 < L|ξ|} in the sense of distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' a(F1φ) = 0 for all φ ∈ S(Rd × Rd) with supp φ ⊂ NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let L ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For all l ∈ R and a ∈ Sm,L 1,1 op[a] extends to a continuous operator form Hl+m to Hl and op is itself continuous from Sm,L 1,1 to L(Hl+m, Hl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [22], Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The symbols in Sections 2 and 3 will be induced by functions (x, ξ) �→ F(u(x), ξ) where F ∈ C∞(Rn × Rd), u ∈ W k,∞(Rd, Rn) for some k ∈ R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' they belong to the following symbol class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For any k ∈ N0 the set Γm k of symbols of order m with regularity k is the set of functions A : Rd × Rd �→ Cn×n such that, (i) for almost all x ∈ Rd the mapping ξ �→ A(x, ξ) is in C∞(Rd, Cn×n) (ii) for any α ∈ Nd 0 and ξ ∈ Rd the mapping x �→ ∂α ξ A(x, ξ) belongs to W k,∞(Rd, Cn×n) and there exists Cα > 0 not depending on ξ such that ∥∂α ξ A(·, ξ)∥W k,∞ ≤ Cα⟨ξ⟩m−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5) With the semi-norms being the optimal constants in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5), Γm k is a Fr´echet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Para-differential operators associated with symbols in Γm k are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For ǫ = (ǫ1, ǫ2) with 0 < ǫ1 < ǫ2 < 1 we call a function χ ∈ C∞(Rd × Rd) an admissible ǫ-cut-off if χ is even with respect to each variable, χ(Rd × Rd) ⊂ [0, 1], χ(η, ξ) = � 1, |η| ≤ ǫ1|ξ| and |ξ| ≥ 1 0, |η| ≥ ǫ2⟨ξ⟩ or |ξ| ≤ ǫ2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6) for all η, ξ ∈ Rd and for all α, β ∈ Nd there exists Cα,β > 0 such that for all ξ, η ∈ Rd |∂β η ∂α ξ χ(η, ξ)| ≤ Cα,β⟨ξ⟩−|α|−|β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let χ be an admissible ǫ-cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set Kχ := F −1 1 (χ) and consider the function Rχ : Γm k → C∞(Rd × Rd) given by Rχ(A) := Kχ ∗1 A, A ∈ Γm k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then Rχ defines a bounded linear operator from Γm k to Sm,1−ǫ2 1,1 ∩ Γm k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Here (Kχ ∗1 A)(x, ξ) = � Rd Kχ(x − y, ξ)A(y, ξ) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Apart from the aspect that a is not only in Sm 1,1 but even in Sm,1−ǫ2 1,1 the proof can be found in [4], Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' But that aspect follows in a straightforward manner as |η + ξ| + 1 ≤ (1 − ǫ2)|ξ| implies |ξ| − |η| + 1 ≤ (1 − ǫ2)|ξ| and thus |η| ≥ ǫ2⟨ξ⟩ and χ vanishes for such η, ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let χ be an admissible ǫ-cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For A ∈ Γm k the (χ-)para-differential operator with symbol A is defined by Opχ[A] := op[Rχ(A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As Rχ ∈ L(Γm k , Sm,1−ǫ2 1,1 ), Opχ = op ◦Rχ defines a continuous linear operator from Γm k to L(Hl+m, Hl) (l ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In particular the L(Hl+m, Hl)-norm of Opχ[A] can be estimated by a constant depending on l, χ and a finite sum of Γk m-semi-norms of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 6 The following shows that, regarding its dependence on χ, opχ[A] is determined by A up to a lower order operator, if k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let χ be an admissible ǫ-cut-off and k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then the following holds: (i) The mapping Rχ − Id is a continuous operator from Γm k to Γm−1 k−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (ii) If ˜χ is an admissible ˜ǫ-cut-off, then Rχ − R˜χ is a continuous operator from Γm k to Sm−1,1−τ 1,1 ∩ Γm−1 k−1 with τ = max{ǫ2, ˜ǫ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [4], Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='13, Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We end this subsection by stating two additional results on para-differential operators for later usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The proofs are contained in [4], Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' To simplify notation we fix an admissible ǫ-cut-off χ and suppress the dependence of Rχ and Opχ on χ in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We call an operator K infinitely smoothing if K ∈ L(Hs, Hl) for all s, l ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let b ∈ Sm be constant with respect to the first variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then the following holds: (i) op(b) − Op[b] is infinitely smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (ii) Op[b] = Op[b∗] (iii) Op[Ab] = Op[A]F −1bF for any A ∈ Γµ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='10 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For each k > 0 there exists C > 0 such that for all f ∈ L∞ ∩ Hk, A ∈ W 1,∞ ∩ Hk ∥A − Op[A]f∥k ≤ C(∥A∥Hk∥f∥L∞ + ∥A∥W 1,∞∥f∥Hk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2 Adjoints and products For the argumentation in Section 3 it will be essential to control the norms of operators Op[A∗] − Op[A]∗, Op[BA] − Op[B] Op[A], A ∈ Γm 1 , B ∈ Γµ 1, µ ∈ R, in terms of the semi- norms of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' While for a ∈ Sm,L 1,1 , b ∈ Sm,L 1,1 there exist symbols g ∈ Sm 1,1, h ∈ Sm+µ 1,1 such that op[a]∗ = op[g], op[b] op[a] = op[h] and that, provided ∂xja ∈ Sm 1,1(j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , d), op[a∗] − op[a]∗ ∈ L(Hl+m−1, Hl), op[b] op[a] − op[ba] ∈ L(Hl+m+µ−1, Hl), l ∈ R, it is not true in general that g, h are again in some class Sm,L 1,1 , Sm+µ,L 1,1 which would allow to control their operator norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' However, for our purposes it is sufficient to consider symbols of the particular form a = R(A), b = R(B) for A ∈ Γm 1 , B ∈ Γµ 1 and we will show that in this case the symbols of op[a]∗ = Op[A]∗, op[b] op[a] = Op[B] Op[A] are in fact again in Sm,L 1,1 , Sm+µ,L 1,1 for some L ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As a first step note that for symbols in S(Rd × Rd, Cn×n) there exist neat formulas for the symbols of adjoint and product operators, which also can be found in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='11 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If a ∈ S(Rd × Rd), then op[a]∗ = op[g] with F1g(η, ξ) = (F1a(−η, η + ξ))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='12 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If a, b ∈ S(Rd × Rd, Cn×n), then op[b] op[a] = op[h] with F1h(η, ξ) = � Rd F1b(η − θ + ξ, θ)F1a(θ − ξ, ξ)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The significance of this result lies in the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='13 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let A ∈ Γm 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then there exists a sequence (aν)ν≥1 ⊂ S(Rd × Rd, Cn×n) such that op[aν]u → Op[A]u, ν → ∞ in S(Rd, Cn) for all u ∈ S(Rd, Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Furthermore for all δ ∈ (ǫ2, 1), ǫ2 being the constant of the ǫ-cut-off, there exists ν0 > 0 such that supp F1aν ⊂ {(η, ξ) ∈ Rd × Rd : |η| ≤ δ⟨ξ⟩} for all ν ≥ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The first part of the statement is shown as in [21], proof of Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' However, we have to slightly modify the construction to also obtain the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set a := R(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Choose ˆφ ∈ S(Rd) with supp ˆφ ⊂ B0(1), F −1 ˆφ(0) = 1 and define φ := F −1 ˆφ, aν(x, ξ) := φ(x/ν)φ(ξ/ν)a(x, ξ), x, ξ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The asserted convergence then follows as ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It remains to show the statement concerning the supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set ψν(x, ξ) := φ(x/ν)φ(ξ/ν) (ξ, η ∈ Rd, ν ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As F1(aν) = (2π)−d/2(F1ψν) ∗1 (F1a) and F1a = χF1A, it is sufficient to show that for given δ ∈ (ǫ2, 1) and ν sufficiently large χ(η − θ, ξ)F1ψν(θ, ξ) = 0 for all θ, η, ξ ∈ Rd |η| ≥ δ⟨ξ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Clearly F1ψν(θ, ξ) = νd ˆφ(θν)φ(ξ/ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As by construction ˆφ(θν) = 0 for |θ| ≥ ν−1 we can assume |θ| ≤ ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then |η| ≥ δ⟨ξ⟩ yields |η − θ| ≥ |η| − |θ| ≥ δ⟨ξ⟩ − ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence choosing ν so large that ν−1 ≤ δ − ǫ2 gives (note ⟨ξ⟩ ≥ 1) |η − θ| ≥ δ⟨ξ⟩ − (δ − ǫ2)⟨ξ⟩ = ǫ2⟨ξ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' But this implies χ(η − θ, ξ) = 0, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='14 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let A ∈ Γm 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then there exists b = b(A) ∈ Sm,1−ǫ2 1,1 such that Op[A]∗ = op[b(A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Furthermore if A ∈ Γ1 the operator T : Γm 1 → Sm−1,1−ǫ2 1,1 , a �→ b(A) − R(A)∗ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In particular the mapping A �→ Op[A∗] − Op[A]∗ = op[R(A)∗ − b(A)] is continuous from Γm 1 to L(Hl+m−1, Hl) for any l ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set a := R(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As A ∈ Sm,1−ǫ2 1,1 , the existence of b := b(A) ∈ Sm 1,1 with op[b] = Op[A]∗ follows by [22], Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Next we prove that F1b vanishes on N1−ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If A ∈ S(Rd×Rd, Cn×n) also a ∈ S(Rd×Rd, Cn×n) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='11 gives F1b(η, ξ) = (F1a(−η, η + ξ))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If now |η + ξ| + 1 ≤ (1 − ǫ2)|ξ| then ǫ2|ξ| ≤ |η| and thus ǫ2⟨η + ξ⟩ ≤ ǫ2(1 + |η + ξ|) ≤ (1 − ǫ2)ǫ2|ξ| ≤ (1 − ǫ2)|η| ≤ |η|, which implies F1a(−η, η + ξ) = (χF1A)(−η, η + ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For general A choose a sequence (aν)ν≥1 ⊂ S(Rd × Rd, Cn×n) with op[aν]u → Op[A]u in S(Rd, Cn) for all u ∈ S(Rd, Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' This implies op[aν]∗ → Op[a]∗ = op[b] in S′(Rd × Rd, Cn×n) and it is straightforward to show that this yields bν → b ∈ S′(Rd × Rd, Cn×n), where F1bν(η, ξ) = F1aν(−η, η+ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='13 F1aν(η, ξ) vanishes for |η| ≥ δ⟨ξ⟩, if δ ∈ (ǫ2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As seen above this yields bν ∈ Sm,1−δ 1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In conclusion b = limν→∞ bν ∈ Sm,1−δ 1,1 for all δ > ǫ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' b ∈ Sm,1−ǫ2 1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Lastly A ∈ Γm 1 directly gives ∂δ xA ∈ Γm 0 and hence ∂δ xR(A) = R(∂δ xA) ∈ Sm 1,1 (|δ| = 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By [22], Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 (applied to N = 1, mN = m − 1) we now obtain b − R(A) ∈ Sm−1 1,1 and its Sm 1,1-semi-norms are bounded by a constant times a sum of finitely many Sm 1,1-semi-norms of ∂δ xR(A) (|δ| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As also b − R(A) ∈ Sm−1,1−ǫ2, the assertion follows by the continuity of R and op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Concerning the analysis of product operators we first consider the difference Rχ(AB) − Rχ(A)Rχ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='15 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For A ∈ Γm 1 , B ∈ Γµ 1 and an ǫ-cut-off χ with ǫ2 < 1/2 we have Rχ(B)Rχ(A) ∈ Sm+µ,1−2ǫ2 1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Furthermore the bilinear operator T : Γm 1 × Γm 1 → Sm+µ−1,1−2ǫ2 1,1 , (A, B) �→ Rχ(AB) − Rχ(A)Rχ(B) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We suppress the superscript χ in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As R(A) ∈ Sm 1,1, R(B) ∈ Sµ 1,1, it is clear that R(B)R(A) ∈ Sm+µ 1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus regarding the first assertion we need to show that R(B)R(A) vanishes on N1−2ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Since F1(R(B)R(A)) = (2π)−d/2F1R(B) ∗1 F1R(B) and F1R(A) = χF1A, F1R(B) = χF1B, it is sufficient to prove that χ(η − θ, ξ)χ(θ, ξ) vanish for all θ, η, ξ ∈ Rd with |η +ξ|+1 ≤ (1−2ǫ2)|ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Take such θ, η, ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If χ(θ, ξ) ̸= 0 then |θ| ≤ ǫ2⟨ξ⟩ and |η + ξ| + 1 ≤ (1 − 2ǫ2)|ξ| implies |η| ≥ 2ǫ2|ξ| + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Together this yields |η − θ| ≥ |η| − |θ| ≥ 2ǫ2|ξ| + 1 − ǫ2ξ ≥ ǫ2⟨ξ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Now χ(η − θ, ξ) vanishes for such η, θ, ξ, wich completes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 9 In regard to the continuity we write R(BA) − R(B)R(A) = R(BA) − BA + B(A − R(A)) − (R(B) − B)R(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8 (i) and the continuity of R that T is continuous as an operator to Γm+µ−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus the proof is finished if we show that each Sm+µ−1 1,1 semi-norm can be bounded by a constant times a finite sum of Γm+µ−1 0 semi-norms of T(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We show that even the following holds: For all α, β ∈ N0 there exists Cβ > 0 such that |∂β x∂α ξ T(A, B)(x, ξ)| ≤ Cαβ|∂α ξ T(A, B)(x, ξ)|⟨ξ⟩|β|, x, ξ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7) By Bernstein’s Lemma applied to ∂α ξ T(a, b)(·, ξ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [4], Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) this can be deduced from the fact that for all ξ ∈ Rd supp � (FT(A, B))(·, ξ) � ⊂ B(0, 2ǫ2⟨ξ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In fact, supp(F(R(ba)(·, ξ)) ⊂ supp(χ(·, ξ)) ⊂ B(0, 2ǫ2⟨ξ⟩) holds by definition of χ and that F1(R(b)R(a)) vanishes for all η, ξ with |η| > 2ǫ2⟨ξ⟩ follows by the same argumentation as in the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We can now prove our main proposition concerning products of para-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='16 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let A ∈ Γm 0 , B ∈ Γµ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then for L := (1 − ǫ2)2 there exists h(B, A) ∈ Sµ+m,L 1,1 such that Op[B] Op[A] = op[h(B, A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Furthermore the operator Γm 1 × Γm 1 → L(Hl+µ+m−1, Hl), (B, A) �→ Opχ[B] Op[A] − Op[BA] is continuous for all l ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The existence of a h = h(B, A) ∈ Sm+µ 1,1 such that op[h(B, A)] = op[R(B)] op[R(A)] = Op[B] Op[A] follows direclty from [22], Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 as R(A) ∈ Sm,1−ǫ2 1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We now prove that h satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4) for L = (1 − ǫ2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' First assume a := R(A), b := R(B) ∈ S(Rd × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='12 F1h(η, ξ) = � Rd F1b(η − θ + ξ, θ)F1a(θ − ξ, ξ)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8) Let η, ξ ∈ Rd with |η + ξ| + 1 ≤ (1 − ǫ2)2|ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If F1a(θ − ξ, ξ) = F1R(A)(θ − ξ, ξ) ̸= 0 we have |θ − ξ| ≤ ǫ2⟨ξ⟩ ≤ ǫ2 + ǫ2|ξ|, which gives (1 − ǫ2)|ξ| ≤ |θ| + ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We arrive at |η + ξ − θ| ≥ |θ| − |η + ξ| ≥ |θ| − (1 − ǫ2)2|ξ| + 1 ≥ |θ| − (1 − ǫ2)|θ| − (1 − ǫ2)ǫ2 + 1 = ǫ2θ + ǫ2 + (1 − ǫ2)2 ≥ ǫ2⟨θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 10 But this implies F1b(η + ξ − θ, θ) = F1R(B)(η + ξ − θ, θ) = 0, which finishes the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For general A, B choose sequences (aν)ν≥1, (bν)ν≥1, ⊂ S(Rd × Rd) with op[aν]u → Op[A]u, op[bν]u → Op[B]u in S(Rd × Rd, Cn) for all u ∈ S(Rd × Rd, Cn) as constructed im Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then clearly op[hν]u = op[bν] op[aν]u → Op[B] Op[A]u = op[h]u in S(Rd, Cn), where hν is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8) with a, b replaced by aν, bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' This implies hν → h in S′(Rd × Rd, Cn×n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As for all 1 > δ > ǫ2 supp F1aν, supp F1bν ⊂ {(η, ξ) ∈ Rd × Rd : |η| ≤ δ⟨ξ⟩} for ν sufficiently large we get by the same reasoning as above that for all 1 > δ > ǫ2 hν vanishes on N(1−δ)2 for ν sufficiently large, which proves that h vanishes on N(1−ǫ2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' To prove the second assertion note that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8 (ii), the mapping G �→ Opχ[G] − Op˜χ[G] is continuous from Γk 1 to L(Hl+k−1, Hl), k, l ∈ R, for any admissible cut-offs χ, ˜χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence we can assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g ǫ2 < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='15 and the continuity of op (B, A) �→ Op[BA] − op[R(B)R(A)] = op[R(BA) − R(B)R(A)] is also continuous as mapping from Γm 1 × Γm 1 to L(Hl+µ+m−1, Hl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' What is left to show ist the continuity of (B, A) �→ Op[B] Op[A] − op[R(B)R(A)] = op[h(B, A) − R(B)R(A)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As R(A) ∈ Sm,1−ǫ2 1,1 and ∂xjR˜χ(A) = R(∂xjA) ∈ Sm 1,1, ∂xjR˜χ(B) = R(∂xjB) ∈ Sm 1,1, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' all semi-norms of h(B, A) − R(B)R(A) can be estimated by a constant times a finite sum of products of semi norms of ∂xjR(A), ∂xkR(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus as h(B, A) − R(B)R(A) ∈ Sm−1,L 1,1 for l = min{1 − 2ǫ2, (1 − ǫ2)2} the assertion follows from the continuity of op and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3 Estimates for operators with symbols induced by Sobolev func- tions In Section 3 the results of Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2 are applied to symbols of the form (x, ξ) �→ F(u(x), ξ), where F ∈ C∞(U × Rd, Cn×n) (U ⊂ Rn some 0-neighbourhood) and u ∈ Hs(Rd, Rn) for s sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For this purpose we prove the results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the following let U ⊂ RN be a 0-neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='17 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We denote by Sm(U) := Sm(U, Cn×n) the set of all functions F ∈ C∞(U × Rd, Cn×n) for which for any α, β ∈ Nd 0 there exists Cαβ > 0 such that for all (u, ξ) ∈ U × Rd |∂β x∂α ξ F(u, ξ)| ≤ Cαβ⟨ξ⟩m−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9) 11 For functions F : U × Rd → Cn×n and u : Rd → U we consider the composition Fu : Rd × Rd → Cn×n, (x, ξ) �→ F(u(x), ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let F ∈ Sm(U) and u ∈ Hs with s > d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then Fu ∈ Γm k for k = [s − d/2] and for all α ∈ Nd 0 and each Γm k -semi-norm pα(Fu) it holds pα(Fu) ≤ Cα(∥u∥s, F), and if additionally F(0, ξ) = 0, then pα(Fu) ≤ ˜Cα(∥u∥s, F)∥u∥s, where Cα, ˜Cα depend on α, F and continuously on ∥u∥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Sobolev embedding Hs ֒→ W k,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus we have Fu(·, ξ) ∈ W k,∞ and ∥∂α ξ Fu(·, ξ)∥W k,∞ ≤ C(∥u∥W k,∞)∥∂α ξ F(·, ξ)∥W k,∞(U) ≤ C(∥u∥s)Cα(F)⟨ξ⟩m−|α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' all ξ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If F(0, ξ) = 0, we even get, for all ξ ∈ Rd, ∥∂α ξ Fu(·, ξ)∥W k,∞ ≤ C(∥u∥W k,∞)∥u∥W k,∞∥∂α ξ Fu(·, ξ)∥W k,∞(U) ≤ C(∥u∥s, F)∥u∥s⟨ξ⟩m−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The following proposition will be central for the energy estimates in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It follows directly by the continuity of Op : Γm k → L(Hl+m, Hl) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18 as well as Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='14, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='16 and the facts that Op[F0]∗ = op[F ∗ 0 ] and op[G0F0] − op[G0] op[F0] is infinitely smoothing by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let F ∈ Sm(U), l ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then for all u ∈ Hs with s > d/2 there exists Cl = Cl(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ∥u∥) > 0 depending on l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' F and monotonically increasingly on ∥u∥s such that: (i) ∥ Op[Fu]∥L(Hl+m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='Hl) ≤ Cl(∥u∥s) and for F(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ·) = 0 ∥ Op[Fu]∥L(Hl+m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='Hl) ≤ Cl∥u∥s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (ii) for s > d/2 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Op[Fu]∗ − Op[F ∗ u] ∈ L(Hl−1+m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hm) and ∥ Op[Fu]∗ − Op[F ∗ u]∥L(Hl−1+m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='Hm) ≤ Cl∥u∥s (iii) for G ∈ Sµ(U) and s > d/2 + 1 there exist Cl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2 = Cl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ∥u∥s) depending on G and monotonically increasingly on ∥u∥s such that ∥ Op[Gu] Op[Fu] − Op[GuFu]∥L(Hl+µ−1+m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='Hm) ≤ Cl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2Cl∥u∥s up to an infinitely smoothing operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' which is determined by F(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' G(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='20 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let F ∈ Sm(U) and u ∈ C1([0, T], Hs) (T > 0) for s > d/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then for each l ∈ R the mapping [0, T] → L(Hl+m, Hl), t �→ Op[Fu(t)] is continuously differentiable and there exists Cl depending on l and F but not on u such that for all t ∈ [0, T] ∥ d dt Op[Fu(t)]∥L(Hl+m,Hl) ≤ Cl∥∂tu(t)∥s0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If d dtFu(t) ∈ Γm 0 we get by continuity and linearity of Op d dt Op[Fu(t)] = Op[∂tFu(t)] To prove this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='10) it is sufficient to show that for any α ∈ Nd 0 there exists Cα = Cα(F) auch that for all ξ ∈ Rd ∥∂α ξ ∂tFu(t)(·, ξ)∥L∞ ≤ Cα∥∂tu(t)∥s⟨ξ⟩m−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let α ∈ Nd 0 and set F α u(t) := ∂α ξ Fu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We have for all x, ξ ∈ Rd ∂tF α u(t)(x, ξ) = n � j=1 ∂tuj∂ujF α(u(t, x), ξ) Due to F ∈ Sm(U) this yields ∥∂tF α u(t)(x, ξ)∥L∞ ≤ ∥∂tu(t)∥L∞ � |β|=1 ∥∂β uF α(·, ξ)∥L∞ ≤ Cα(F)∥∂tu∥s⟨ξ⟩m−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Lastly we prove a version of the strict G˚arding inequality for F ∈ Sm(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' First consider the following lemma which is a modification of a construction in [21], proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='21 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' There exists an even function ψ ∈ S(Rd × Rd) with unit integral, Op[ψ] = Op[ψ]∗, ⟨op[ψ]v, v⟩ ≥ 0 (v ∈ S(Rd)) and F1ψ compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Choose an even function ˆφ ∈ C∞ 0 (Rd × Rd) with L2-norm one and set φ = F −1 1 ˆφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By definition F1φ is compactly supported and clearly φ is even and has L2-norm one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Next, let ψ ∈ S(Rd) be the symbol of op[ψ]∗ op[ψ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' it follows that ψ is even and has unit integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' op[ψ] = op[ψ]∗, ⟨op[ψ]v, v⟩L2 ≥ 0 (u ∈ S(Rd)) holds by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Now, let ρ be the symbol of op[φ]∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='11 we get F1ρ(η, ξ) = (F1φ)∗(−η, η + ξ), η, ξ ∈ Rd 13 and thus by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='12 F1ψ(η, ξ) = � Rd F1ρ(η − θ, θ + ξ)F1φ(θ, ξ)dθ = � Rd F1φ(θ − η, η + ξ)F1φ(θ, ξ)dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As F1φ is compactly supported, we can choose C > 0 such that F1φ(θ, ξ) = 0 if |θ| ≥ C or |ξ| ≥ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then by definition F1ψ(η, ξ) = 0 if |ξ| ≥ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Given |η| ≥ 2C and |θ| ≤ C we conclude |θ − η| ≥ |η| − |θ| ≥ C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' F1φ(θ − η, η + ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In conclusion we have proven that F1ψ is in fact compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In particular ψ ∈ S(Rd × Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Next we introduce a method to decompose symbols in Sm 1,1 into an infinite sum of infinitely smoothing symbols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' First, choose a function ρ ∈ D(Rd) even and monotonically decaying along rays such that ρ(Rd) ⊂ [0, 1] and ρ(ξ) = � 1, |ξ| ≤ 1 2 0, |ξ| ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For ν ∈ N0 define ρν, ζν ∈ D(Rd) by ρν(ξ) := ρ(ξ/2ν), ζν(ξ) = ρν+1(ξ) − ρν(ξ), ξ ∈ Rd Additionally set ζ−1 := ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='22 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For a function a : Rd × Rd → Cn×n and ν ≥ −1 define aν(x, ξ) := a(x, ξ)ζν(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Note that a = � ν≥−1 aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It is straightforward to show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='23 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let a ∈ Sm 1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then aν ∈ S−r for all r ∈ R and for any α, β ∈ N0 x, ξ ∈ Rd |∂β x∂α ξ aν(x, ξ)|⟨ξ⟩r ≤ C2ν(r+m−|α|+|β|) � γ≤α Cγβ(a), where Cγβ(a) are semi-norms of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='24 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let s > d/2, u ∈ Hs+2 and F ∈ Sm(U) such that there exists an R > 0 with F(y, ξ) + F(y, ξ)∗ ≥ 0 for all y ∈ U and ξ ∈ Rd with |ξ| > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then there exists C = C(∥u∥s+2, F) > 0 and for all q ∈ R there exists c = c(∥u∥s+2, F, q) > 0, both increasing functions of ∥u∥s+2, such that for all v ∈ S(Rd, Cn) ⟨(Op[Fu] + Op[Fu]∗)v, v⟩L2 ≥ −C∥u∥ 1 2 s+2∥v∥2 (m−1)/2 − c∥v∥2 −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the following it is straightforward to see that all constants can be chosen to be increasing functions of ∥u∥s+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' First note that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19 for all l ∈ R ∥ Op[Fu] + Op[Fu]∗ − Op[Fu + F ∗ u]∥L(Hl+m−1,Hl) ≤ Cl∥u∥s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus ⟨(Op[Fu] + Op[Fu]∗)v, v⟩L2 ≥ ⟨Op[Fu + F ∗ u]v, v⟩L2 − C∥u∥s+1∥v∥2 (m−1)/2, v ∈ S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence it is sufficent to prove the result for Op[Fu] + Op[Fu]∗ replaced by Op[Fu + F ∗ u], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' we can assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g F(u, ξ) = F(u, ξ)∗ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It holds R(Fu) = R(F ∗ u) = R(Fu)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By assumption this gives pointwise in Rd × {|ξ| ≥ R} for all v ∈ Cn ⟨(R(Fu))v, v⟩Cn ≥ ⟨(R(Fu) − Fu)v, v⟩Cn ≥ −|R(Fu) − Fu||v|2 ≥ −(|R(Fu − F0) − (Fu − F0)| + |R(F0) − F0|)|v|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18 Fu−F0 ∈ Γm 2 with all semi-norms bounded by a positive constant depending on F times ∥u∥s+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8 (i) this yields R(Fu − F0) − (Fu − F0) ∈ Γm−1 1 with semi- norms bounded in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus |R(Fu − F0) − (Fu − F0)| ≤ C0∥u∥s+2⟨ξ⟩m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Using also that R(F0) − F0 has compact support we conclude that for all q ∈ R |R(Fu − F0) − (Fu − F0)| + |R(F0) − F0| ≤ C0∥u∥s+2⟨ξ⟩m−1 + c0 q⟨ξ⟩−q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Therefore on Rd × {|ξ| ≥ R} a := R(Fu) + C0∥u∥s+2⟨ξ⟩m−1 + c0⟨ξ⟩−r ≥ 0 and a = a∗, a ∈ Sm,1−ǫ2 1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As Op[Fu] = op[R(Fu)] = op[a] − C0∥u∥s+2 op[⟨ξ⟩m−1] − c0 op[⟨ξ⟩−r] it is now sufficient to show ⟨op[a]v, v⟩L2 ≥ −C∥u∥1/2 s+2∥v∥2 (m−1)/2 − c∥v∥−q for all q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' To this end we proceed similarly as in the proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 in [22] but with a crucial modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' First, decompose a = � ν≥−1 aν according to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As for all ν0 ¯aν0 := �ν0 ν=−1 aν ∈ S−q for any q ∈ R with norm depending on µ, ν0 according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='23, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ∥ op[¯aν0]v∥ ≤ cν0,µ∥v∥−r, we only need to consider � ν≥ν0 aν for some ν0 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Naturally, in a first step we choose ν0 large enough to obtain 2ν0−2 > R and thus by assumption 15 aν(x, ξ) ≥ 0 for all x, ξ ∈ Rd, ν ≥ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' But we will later see that we may have to choose ν0 even larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' assume u ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Otherwise the result readily follows as F0 ≥ 0 is constant with respect to x and Op[F0] − op[F0] is infinitely smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Choose an even function ψ ∈ S(Rd × Rd) with unit integral such that op[ψ] = op[ψ]∗, ⟨op[ψ]v, v⟩ ≥ 0 (v ∈ S(Rd)) and F1ψ compactly supported as constructed in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For ν ∈ N0 set qν := 2ν/2 and write aν = bν + hν with bν(x, ξ) := � Rd � Rd ψ((x − y)qνµ, (ξ − θ)/(qνµ))aν(y, θ) dy dθ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='11) = � Rd � Rd ψ(y, θ)aν(x − y/(qνµ), ξ − θqνµ) dy dθ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='12) where µ := ∥u∥s+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As aν ≥ 0 and op[ψ] is a positive operator it is straightforward to obtain the positivity of bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence the theorem is proven provided ⟨op[h]v, v⟩L2 ≥ −Cµ∥u∥ 1 2 k+1∥v∥(m−1)/2, v ∈ S(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='13) To this end we show h ∈ Sm−1,L 1,1 for some L ∈ (0, 1) and that all semi-norms of h are bounded by a constant times ∥u∥ 1 2 s+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='13) follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' First we verify h ∈ Sm−1 1,1 and the estimate on the semi-norms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' | � ν≥ν0 ∂β x∂α ξ hν(x, ξ)| ≤ Cαβ∥u∥ 1 2 s+2⟨ξ⟩m−1−|α|+|β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='14) Let α = β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Fix ξ ∈ Rd and consider ν ∈ N0 with |ξ| < 2ν−2 or |ξ| > 2ν+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As aν(y, θ) = 0 for 2ν−1 ≤ |θ| ≤ 2ν+1 we then have hν(x, ξ) = −bν(x, ξ) and it follows by basic estimates (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [22]) that in the support of the first integrand in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='11) |ξ − θ| ≥ 1 5(2ν + |ξ|) and thus |ξ − θ|/qν = 2−ν/2|ξ − θ| ≥ 1 5(2ν + |ξ|) 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='15) As a ∈ Sm 1,1 and supp aν ⊂ {(x, θ) ∈ Rd × Rd : 2ν−1 ≤ |θ| ≤ 2ν} |aν(y, θ)| ≤ C⟨θ⟩m ≤ C(1 + 2ν)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence ψ ∈ S(Rd × Rd) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='15) yield |hν| ≤ Cm(1 + 2ν)m � � (|ξ − θ|/(qνµ))−2(|m|+1)(1 + |ξ − θ|/(qνµ))−n−1(1 + |(x − y)|qνµ)−n−1dy dθ ≤ Cm,nµ2(|m|+1)(1 + 2ν)m(2ν + |ξ|)−2|m|−2 ≤ Cm,nµ2(|m|+1)(1 + |ξ|)m−12−ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='16) 16 Thus � {ν:|ξ|<2ν−2 or |ξ|>2ν+2} |hν| ≤ C∥u∥ 1 2 s+2⟨ξ⟩m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='17) Now consider ν ∈ N0 with 2ν−2 ≤ |ξ| ≤ 2ν+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As ψ is an even function with unit integral we get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='12) hν = aν − bν = � � ψ(y, θ) � aν(x, ξ) − aν(x − y/(qνµ), ξ − θqνµ) � dy dθ = � � ψ(y, θ) � � |α+β|<2 ∂β x∂α ξ aν(x, ξ)(−y)β(−θ)α − aν(x − y/(qνµ), ξ − θqνµ) � dy dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18) By Taylor’s fomula we can estimate (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' assume |θ| ≤ |ξ|) �� � |α+|β|<2 ∂β x∂α ξ aν(x, ξ)(−y)β(−θ)α − aν(x − y/(qνµ), ξ − θqνµ) �� ≤ C � |α|+|β|=2 sup x,ξ∈Rd |∂β x∂α ξ aν(x, ξ)||yβθα|(qνµ)|α|−|β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19) Note that a = R(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18 Fu ∈ Γm 2 and thus ∂β xFu ∈ Γm 2−|β| for |β| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence for each γ ∈ Nd 0, ξ ∈ Rd ∥∂γ ξ ∂β xFu(·, ξ)∥W 2−|β|,∞ ≤ Cγ⟨ξ⟩m−|γ| and for |β| ≥ 1 we also have ∂β xFu|u=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus again by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18 ∥∂γ ξ ∂β xFu(·, ξ)∥W 2−|β|,∞ ≤ Cγ∥u∥s+2⟨ξ⟩m−|γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Clearly ∂β xa = ∂β xR(Fu) = R � ∂β xFu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' and we conclude from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6 that ∂β xa ∈ Sm 1,1 and for all x, ξ ∈ Rd |∂β x∂γ ξ a(x, ξ)| ≤ Cγ⟨ξ⟩m−|γ| � 1, |β| = 0, ∥u∥s+2, 1 ≤ |β| ≤ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='23 sup x,ξ∈Rd |∂β x∂γ ξ aν| ≤ Cγ2ν(m−|α|) ≤ Cγ � 1, |β| = 0, ∥u∥s+2, 1 ≤ |β| ≤ 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='20) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='20) and µ = ∥u∥ 1 4 s+2, qν = 2ν/2 we now get for 2ν−2 ≤ |ξ| ≤ 2ν+2 |hν| ≤ C � ψ(y, θ)(|θ|2 + |θ||y| + |y|2) dy dθ � 2ν(m−2)2ν∥u∥ 1 2 s+2 + 2ν(m−1)∥u∥s+2 + 2νm∥u∥s+22−ν∥u∥ − 1 2 s+2 � ≤ Cµ2ν(m−1)∥u∥ 1 2 s+2 ≤ C∥u∥ 1 2 s+2⟨ξ⟩m−1, 17 where we used ψ ∈ S(Rd × Rd) and 2ν−2 ≤ |ξ| ≤ 2ν+2 in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='17) this shows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='14) for α = β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Now note that ∂β x∂α ξ bν is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='12) with aν replaced by ∂β x∂α ξ aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='14) for α, β ̸= 0 by applying the argumentation above with aν replaced by ∂β x∂α ξ aν and m replaced by m − |α| + |β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' To finish the proof we show that F1h vanishes on NL = {(η, ξ) ∈ Rd × Rd : |η + ξ| < L|ξ|} with L := min{1 − ǫ2, 1 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then the estimate on the operator norm follows by the continuity of op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As a = R(Fu) ∈ Sm,1−ǫ2 1,1 , it suffices to prove that F1b vanishes on N 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By standard arguments on convolution and Fourier transform we have for all g ∈ S(Rd ×Rd) bν(F1g) = (µqν)−d/2 � Rd � Rd aν(y, θ)F1f(y, θ) dθ dy, where f(η, θ) = � Rd F1ψ(η/(qνµ), (ξ − θ)/qνµ)g(η, ξ)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='21) Let supp g ⊂ N1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By construction we have supp F1ψ ⊂ {(ξ, η) ∈ Rd × Rd : |η|, |ξ| ≤ D, } for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Next choose ν0 ∈ N so large that 3Dµ ≤ qν0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then for ν ≥ ν0 on the support of the integrand of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='21) we have |η|, |ξ − θ| ≤ Dqνµ and |ξ + η| + 1 < 1 2|ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The first and third inequality yield |ξ| < 2|η| ≤ 2Dqνµ and thus the second one gives |θ| ≤ Dqνµ + |ξ| < 3Dqνµ ≤ qνqν0/2 ≤ 2ν/22ν0/2−1 ≤ 2ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' But this implies bν(y, θ) = 0 for all y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Therefore we have proven bν(F1g) = 0 for all ν ≥ ν0 and supp g ⊂ {(ξ, η) ∈ Rd × Rd : |ξ + η| < 1 2|ξ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence this also holds for b = � ν≥ν0 bν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 3 Dissipativity Throughout this section we consider (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) with smooth matrix families Aj, Bjk : U → Rn×n, u0, u1 : Rd → U and u : [0, T] × Rd → U for some domain U ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Carrying out the differentiation with respect to xk on the right-hand side and distinguishing between space and time derivatives we write (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) as −B00(u)utt = d � j,k=1 Bjk(u)uxjxk+ d � j=1 (B0j(u)+Bj0(u)ut)xj−A0(u)ut− d � j=1 Aj(u)uxj+Q(u, Dt,xu), 18 where Q is of the form Q(u, Dt,xu) = n � l=1 d � j,k=0 Qljk(u)ul xkuxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We will see in the proofs that the specific form of the matrices Qljk(u) does not play any role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence multiplying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) by (−B00)−1, we can assume −B00 = In without loss of generality, which we will always do in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Next, denote by B(u, ξ) := d � j,k=1 Bjk(u)ξjξk, C(u, ξ) := d � j=1 (B0j(u) + Bj0(u))ξj, A(u, ξ) := d � j=1 Aj(u)ξj, ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , ξn) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' the symbols of the second and first order parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then the hyperbolicity of both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) is expressed by the following conditions: (HA) (a) there exists a smooth bounded family of hermitian uniformly positive definite ma- trices Σ : U → Rn×n such that Σ(u)A0(u) is symmetric and uniformly positive on U, (b) the matrix family A0(u)−1A(u, ξ) permits a symbolic symmetrizer H(u, ξ), (HB) with B(u, ξ) = � 0 |ξ|In −|ξ|−1B(u, ξ) iC(u, ξ) � , ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=', ξd) ∈ Rd, the matrix family iB(u, ξ) permits a symbolic symmetrizer H(u, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Above we use the following notion of a symbolic symmetrizer (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Let K ∈ C∞(U × Rd \\ {0}, Cn×n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' A symbolic symmetrizer for K is a smooth mapping S ∈ C∞(U × Rd \\ {0}, Cn×n) positive homogeneous of degree 0 with respect to the second argument, bounded as well as all its derivatives on U ×Sd−1 such that for some c > 0 and all (u, ξ) ∈ U × Rd \\ {0} S(u, ξ) = S(u, ξ)∗ ≥ cIn, and S(u, ξ)K(u, ξ) = (S(u, ξ)K(u, ξ))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' K admits a symbolic symmetrizer if K is positive homogeneous of degree 1, for all (u, ω) ∈ U ×Sd−1 all eigenvalues of K(u, ξ) are real, semi-simple (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' their geometric and algebraic multiplicities coincide) and their multiplicities do not depend on (u, ω) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [38], Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2 C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If this holds for A0(u)−1A(u, ξ) or B(u, ξ) the respective operator is often called constantly hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 19 We now fix a homogeneous state ¯u ∈ U and assume the following dissipativity conditions on the coefficient matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Condition (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Matrices Aj(¯u), Bjk(¯u) have three properties: (D1) For every ω ∈ Sd−1, all restrictions, as a quadratic form, of W1 = H(¯u, ω)(A0(¯u))−1� − B(¯u, ω) + (A0(¯u))−1(A(¯u, ω))(A0(¯u))−1A(¯u, ω) + C(¯u, ω)(A0(¯u))−1A(¯u, ω) � , on the eigenspaces E = J−1 E (Cn) of W0 = (A0(¯u))−1A(¯u, ω) are uniformly negative in the sense that J∗ E (W1 + W ∗ 1 ) JE ≤ −¯c J∗ EJE with one ¯c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (D2) For every ω ∈ Sd−1, all restrictions, as a quadratic form, of W1 = H(¯u, ω)A(¯u, ω), A(¯u, ω) = � 0 0 −iA(¯u, ω) −A0(¯u) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) on the eigenspaces E = J −1 E (C2n) of W0 = B(¯u, ω) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) are uniformly negative in the sense that J ∗ E (W1 + W∗ 1) JE ≤ −¯c IE with one ¯c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='. (D3) All solutions (λ, ξ) ∈ C × (Rd \\ {0}) of the dispersion relation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) at ¯u = 0 have Re(λ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Note that as (D) is an open condition there exists a neighbourhood of ¯u such that Bjk(u), Aj(u) satisfy (D) with ¯u replaced by u for all u ∈ U0 with ¯c independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The following remark is useful in the proofs below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It is straightforward to show that (D1) and (D2) are equivalent to the same conditions with W0, W1 replaced by ¯W0 := H(¯u, ω) 1 2A(¯u)−1A(0, ω)H(¯u, ω)− 1 2, ¯W1 := H(¯u, ω)− 1 2W1H(¯u, ω)− 1 2 and W0, W1 replaced by ¯ W0 := H(¯u, ω) 1 2B(¯u, ω)H(¯u, ω)− 1 2, ¯ W1 := H(¯u, ω) 1 2A(¯u, ω)H(¯u, ω)− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 20 From now on we always assume (HA), (HB) and (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As we could also consider (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) in the variable u − ¯u, we can w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' restrict our argumentation to the case ¯u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We write (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) as the first-order in time system ut = v vt = d � j=1 (Bj0 + B0j)(u)vxj + d � j,k=1 Bjk(u)uxjxk − A0(u)v − d � j=1 Aj(u)uxj + Q(u, Dt,xu) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) and denote by ¯ M(u, ξ) := � 0 In M(u, ξ) N(u, ξ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4) with M(u, ξ) = −iA(u, ξ) − B(u, ξ), N(u, ξ) = iC(u, ξ) − A0(u), the Fourier symbol of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We also define M(u, ξ) := Z(ξ) ˜ M(u, ξ)Z(ξ)−1 Z(ξ) = �⟨ξ⟩In 0 0 In � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' First we treat the linearization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) at the reference state u = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' d � j=0 Aj(0)uxj = d � j,k=0 Bij(0)uxixj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5) Such linear systems were studied in [14], however under the stronger assumptions, that the coefficient matrices are symmetric and A0 is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then (HA) is clearly satisfied with FA = In and H = A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Also, condition (HB) (b) ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' requires the existence of a matrix family S : Sd−1 → Cn×n such that iS(ω)B(0, ω)S(ω)−1 is real symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' But one can easily check that this can be relaxed to the assumption that iS(ω)B(0, ω)S(ω)−1 is hermitian, which is satisfied in the present context for S(ω) := H(0, ω) 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Lastly, we want to point out that (D1), (D2) ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' were stated in the equivalent form mentioned in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We will make plausible below that the weaker conditions in the present work are still sufficient to retrieve the main result of [14], namely: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' There exist a c > 0 and a family ξ �→ T (ξ), Rd → C2n×2n of linear transformations of C2n which, together with their inverses T (ξ)−1, are uniformly bounded, such that T (ξ)M(0, ξ)T −1(ξ) + (T (ξ)M(0, ξ)T −1(ξ))∗ ≤ −cρ(ξ)I2n, ξ ∈ Rd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6) where ρ(ξ) = |ξ|2/(1 + |ξ|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 21 As outlined in [14] this brings about the pointwise decay of solutions in Fourier space and thus the following decay estimate for the inhomogeneous linear Cauchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6 Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For any s ∈ N0 there exists C > 0 such that the following holds: For all u0 ∈ Hs+1 ∩ L1, u1 ∈ Hs ∩ L1 and f ∈ C([0, T], Hs ∩ L1) the solution u of f + d � j=0 Aj(0)uxj = d � j,k=0 Bij(0)uxixj with u(0) = u0, ut(0) = u1 satisfies ∥u(t)∥s+1 + ∥ut(t)∥s ≤ C(1 + t)− d 4(∥u0∥s+1 + ∥u0∥L1 + ∥u1∥s + ∥u1∥L1) C � t 0 (1 + t − τ)− d 4(∥f(τ)∥s + ∥f(τ)∥L1) dτ for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As stated above the proof can be found essentially in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We just illustrate at which points it has to be slightly modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The existence of a bounded family T (ξ) ⊂ Gl2n2 satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6) is proven separately for the three different regimes |ξ| ≤ r0, r0 ≤ |ξ| ≤ r∞ and |ξ| ≥ r∞ for suitable r0, r∞ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the latter two cases only ((H)B) and conditions (D2), (D3) are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The symmetry of the matrices plays no role whatsoever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For small values of |ξ| writting ξ = ξω for ξ > 0, ω ∈ Sd−1 one finds a bounded family of invertible R(ξ, ω) with R(ξ, ω)−1 also bounded and (supressing the argument u = 0) R(ξ, ω) ¯ M(ξω)R(ξ, ω)−1 = � X(ξ, ω) 0 0 Y (ξ, ω) � , where X(ξ, ω) = iξ(A0)−1A(ω) + ξ2(A0)−1� − B(ω) + (A0)−1(A(ω))(A0)−1A(ω) + C(ω)(A0)−1A(ω) � + O(ξ3) Y (ξ, ω) = −A0 + O(ξ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' This is due to the fact that A0(0) is invertible and again makes no use of the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence for ˇR(ξ, ω) = � H(ω) 1 2 0 0 F 1 2 A � ˇR(ξ, ω) we get ˇR(ξ, ω)M(ξω) ˇR(ξ, ω)−1 = � iξ ¯W0 + ξ2 ¯W1 + O(ξ3) 0 −F 1 2 A A0F − 1 2 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 2For m ∈ N Glm denotes the space of invertible m × m-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 22 with ¯W0, ¯W1 as in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Since F 1 2 A A0F − 1 2 A is positive definite the existence of the family T (ξ) now follows for sufficiently small ξ by condition (D1) and [14], Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3 In Section 4 we will see that, given d ≥ 3, s > d/2 + 1, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6 directly implies the decay of a solution to the quasi-linear problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) in Hs−1 but only provided that its Hs-norm is a-priori known to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' To close this gap we need to show that the Hs-norm of a small solution can be bounded by the initial conditions and L2-norms of lower order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The rest of this section is devoted to a construction preparing such a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the following for ξ ∈ Rd we write ξ = ξω with ξ = |ξ| ∈ [0, ∞), ω = ξ/|ξ| ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For r > 0, u ∈ Rn, ξ ∈ Rd and ω ∈ Sd−1 by Bn(u, r), Bd(ξ, r), BS(ω, r) we denote the balls with radius r and center u, ξ, ω with respect to the metrices on Rn, Rd, Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For some ω∗ ∈ Sd−1 and δ > 0 we use P(ω∗, δ) = Bn(0, δ) × [0, δ) × BS(ω∗, δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' There exist r > 0, c∞ > 0 and a mapping D∞ ∈ C∞(Ω∞, C2n×2n), Ω∞ := ¯U0 × {ξ ∈ Rd : |ξ| ≥ r−1}, ¯U0 := Bn(0, r) ⊂ U, such that: (i) For all (u, ξ) ∈ Ω∞ D∞(u, ξ) = D∞(u, ξ)∗ ≥ c∞In, and D∞(u, ξ)M(u, ξ) + (D∞(u, ξ)M(u, ξ))∗ ≤ −c∞I2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (ii) For any α, β ∈ Nd 0 there exist Cαβ > 0 with |∂β u∂α ξ D∞(u, ξ)| ≤ Cαβ⟨ξ⟩−|α|, (u, ξ) ∈ Ω∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Consider the mapping K : U × (0, ∞) × Sd−1 → C2n×2n defined by K(u, η, ω) = � 0 In −iηA(u, ω) − B(u, ω) −iC(u, ω) − ηA0(u) � , ω ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8) and H(u, ω) denote the symmetrizer of B(u, ω) as in condition (HB) (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set W(u, η, ω) := H(u, ω) 1 2K(u, η, ω)H(u, ω)− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Since K(0, 0, ω) = � 0 In −B(0, ω) iC(0, ω) � = B(0, ω) 3Note that in said Lemma it is sufficient to assume that iM(0, ω) is selfadjoint instead of requiring iM(0, ω) to be real symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 23 and ∂K ∂η (0, 0, ω) = � 0 0 −iA(0, ω) −A0(0) � = A(0, ω) W satisfies W(0, 0, ω) = ¯ W0, ∂W(0, 0, ω) ∂η = ¯ W1, with ¯ W0, ¯ W1 as in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Now fix ω0 ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By virtue of condition (D2) it follows from Lemma 5 in [14] that there exists δ0 > 0, c0 > 0 and T0 ∈ C∞(P(ω∗, δ0), Gl2n) with T −1 0 also bounded such that pointwise on P(ω0, δ0) T0WT −1 0 + (T0WT −1 0 )∗ ≤ −˜cηI2n for some ˜c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence ˜D0 := H 1 2T ∗ 0 T0H 1 2 ∈ C∞(P(δ0, ω0), C2n×2n) satisfies ˜D0(u, ξ, ω) = ˜D0(u, ξ, ω)∗ ≥ cI2n, (u, ξ, ω) ∈ P(δ0, ω0) for some c > 0 and thus ˜D0K + ( ˜D0K)∗ ≤ −c˜cηI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In conclusion we have shown the following: For each ω ∈ Sd−1 there exist δω > 0, cω > 0 and Dω ∈ C∞(P(ω, δω), C2n×2n) such that for all (u, ξ, ¯ω) ∈ P(ω, δω) Dω(u, η, ¯ω) = Dω(u, η, ¯ω)∗ ≥ cωI Dω(u, η, ¯ω)K(u, η, ¯ω) + (Dω(u, η, ¯ω)K(u, η, ¯ω))∗ ≤ −cωξ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9) As Sd−1 is compact we may choose ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , ωr such that ¯l� l=1 BS(ωl, δl/2) = Sd−1 (δl := δωl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set r0 = min{δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , δr}, c0 = min{cω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , cωr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , ¯l and Pl := Bn(0, r0) × [0, r0) × BS(ωl, δl) choose functions φl ∈ C∞(Sd−1, [0, 1]) with supp φl ⊂ BS(ωj, δl), φl = 1 on BS(ωj, δj/2) and extend Dl := Dωl trivially by 0 to a function defined on Bn(0, r0) × [0, r0) × Sd−1 =: Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Define D0 : Ω0 → C2n×2n : (u, η, ω) �→ ¯l � l=1 φl(ω)Dl(u, η, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then D0 ∈ C∞(Ω0, C2n×2n), and D0(u, η, ω) is hermitian for all (u, η, ω) ∈ Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Furthermore for (u, η, ω) ∈ Ω0 we have ω ∈ BS(ωk, δ/2) for some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , ¯l} and thus as Dl(u, η, ω) ≥ 0 D0(u, η, ω) = ¯l � l=1 φl(ω)Dl(u, η, ω) ≥ Dk(u, η, ω) ≥ c0I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' with the same reasoning we see D0(u, η, ω)K(u, η, ω) + (D0(u, η, ω)K(u, η, ω))∗ ≤ −c0ηI2n, (u, η, ω) ∈ Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 24 Now note that for all u, ξ, ω ξK(u, 1/ξ, ω) = � 0 ξIn −iA(u, ω) − ξB(u, ω) −iξC(u, ω) − A0(U) � = ˜Z(ξ)M(u, ξω) ˜Z(ξ)−1, where ˜Z(ξ) = � ⟨ξ⟩ ξ In 0 0 In � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As clearly ˜Z, ˜Z−1 ∈ C∞((r−1 0 , ∞), C2n×2n) are symmetric and positive definite on (r−1 0 , ∞), for r := r0/2, Ω∞ := Bn(0, r) × {ξ ∈ Rd : |ξ| ≥ r−1} the mapping D∞ : Ω∞ → C2n×2n, (u, ξ) �→ ˜Z(|ξ|)D0(u, 1/|ξ|, ξ/|ξ|) ˜Z(|ξ|) is in C∞(Ω∞, C2n×2n) and for all (u, ξ) ∈ Ω∞ D∞(u, ξ) = D∞(u, ξ)∗ ≥ c∞I2n for some c∞ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Since for ξ = ξω ∈ U0 D∞(u, ξ)M(u, ξ) = ξ ˜Z(ξ)D0(u, 1/ξ, ω) ˜Z(ξ)K(u, 1ξ, ω) = ξ ˜Z(ξ) ˜D0(u, 1/ξ, ω)K(u, 1/ξ) ˜Z(ξ), we also have D∞(u, ξ)M(u, ξ) + (D∞(u, ξ)M(u, ξ))∗ ≤ −c∞I2n for some c∞ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It remains to verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' First note that the functions ξ �→ ⟨|ξ|⟩/|ξ|, ξ �→ ξk/|ξ|, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , d and ξ �→ 1/|ξ| are positive homogeneous of degree 0 and −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus for any α ∈ Nd 0 there exists Cα > 0 such that for all ξ ∈ Rd with |ξ| > 2r−1 0 |DαZ(ξ)| + |Dα(ξk/|ξ|)| + |Dα(1/|ξ|)| ≤ Cα⟨ξ⟩−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Since D0 as well as all of its derivatives are bounded on Bn(0, r0/2) × [0, r0/2] × Sd−1 the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7) follows by product and chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 25 4 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 To begin with, we remark that local well-posednes sof (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) follows from the existing theory for hyperbolic systems of any order [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4 Our task thus consists in showing that under an a priori smallness assumption the solution satisfies the decay and energy estimates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4), for, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=', ¯u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then we can extend them globally by standard methods (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [24], proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Consider d ≥ 3, s > d/2 + 1 and assume (HB), (HA) and (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then there exist constants µ > 0, δ = δ(µ) > 0, and C = C(µ, δ) > 0 (all independent of T) such that the following holds: For all u0 ∈ Hs+1, u1 ∈ Hs with ∥u0∥s+1 + ∥u1∥s < δ and all u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs) satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) and sup t∈[0,T] ∥u(t)∥2 s+1 + ∥ut(t)∥2 s + � T 0 ∥u(τ)∥2 s+1 + ∥ut(τ)∥2 s dτ ≤ µ we have for all t ∈ [0, T] ∥u(t)∥s + ∥ut(t)∥s−1 ≤ C(1 + t)− d 4(∥u0∥s + ∥u0∥L1 + ∥u1∥s−1 + ∥u1∥L1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) ∥u(t)∥2 s+1 + ∥ut(t)∥2 s + � t 0 ∥u(τ)∥2 s+1 + ∥ut(τ)∥2 s ≤ C(∥u0∥2 s+1 + ∥u0∥2 L1 + ∥u1∥2 s1 + ∥u1∥2 L1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) We split the proof into two parts corresponding to the following two assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the situation of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 there exist µ > 0, δ > 0, and C > 0 such that the following holds: For all u0 ∈ Hs+1 ∩ L1, u1 ∈ Hs ∩ L1 with ∥u0∥s+1 + ∥u1∥s, ∥u0∥L1 + ∥u1∥L1 < δ and all u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs) satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) and sup t∈[0,T] ∥u(t)∥2 s+1 + ∥ut(t)∥2 s+1 + � T 0 ∥u(τ)∥2 s+1 + ∥ut(τ)∥2 s dτ ≤ µ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) holds for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3 Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the situation of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 there exist µ > 0, and C > 0 such that the following holds: For all u0 ∈ Hs+1, u1 ∈ Hs and all u ∈ C0([0, T], Hs+1) ∩ C1([0, T], Hs) satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2) and sup t∈[0,T] ∥u(t)∥2 s+1 + ∥ut(t)∥2 s+1 + � T 0 ∥u(τ)∥2 s+1 + ∥ut(τ)∥2 s dτ ≤ µ we have for all t ∈ [0, T] ∥u(t)∥2 s+1 + ∥ut(t)∥2 s + � t 0 ∥u(τ)∥2 s + ∥ut(τ)∥2 s−1dτ ≤ C(∥u0∥2 s+1 + ∥u1∥2 s) + C � t 0 ∥u(τ)∥2 s + ∥ut(τ)∥2 s−1 dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) 4For example, the recent result in [3], which applies to the class we study in Section 5, is of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 26 From there Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1 clearly follows by multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) with a sufficiently small factor integrating, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) with respect to t, and adding the resulting inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For notational reasons we write the first order representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) in the compact form Ut = L(u)U + (0, Q(u, Dx,tu))t (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4) with U = (u, ut), L(u) = � 0 In �d j,k=1 Bjk(u)∂xj∂xk − �d j=1 Aj(u)∂xj �d j=1( ¯Bj0 + ¯B0j)(u)∂xj − A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' � Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As s > d/2+1 we find by Moser type inequalities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [4] Appendix C and the references therein) ∥(L2(u) − L2(0))U∥s−1 + ∥(L2(u) − L2(0))U∥L1 ≤ Cµ∥u∥s−1(∥u∥s+1 + ∥ut∥s), where L(u)U = (U2, L2(u)U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Furthermore ∥Q(u, Dx,tu))∥s−1 + ∥Q(u, Dx,tu)∥L1 ≤ Cµ∥u∥s∥ut∥s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Now writing system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4) as L(0)U = (0, L2(0)−L2(u)+Q(u, Dx,tu)) and applying Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6 to f = (L2(0) − L2(u)) + Q(u, Dx,tu) with s replaced by s − 1 yields ∥u(t)∥s + ∥ut(t)∥s−1 ≤ C(1 + t)− d 4(∥u0∥s + ∥u0∥L1 + ∥u1∥s−1 + ∥u1∥L1) + Cµ sup τ∈[0,t] (∥u(τ)∥s+1 + ∥ut(τ)∥s) � t 0 (1 + t − τ)− d 4(∥u(τ)∥s + ∥ut∥s−1)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5) As t → (1 + t)− d 4 is square-integrable over [0, ∞) for d ≥ 3 this gives (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3) as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [24], proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' From now Cµ always denotes some constant depending monoton- ically increasing on µ, whose concrete value may change at every instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For 0 < ǫ < 1 let Jǫ be the Friedrichs mollifier and set V = (Λu, ut), W := Wǫ := ΛsJǫ(Λu, ut) and Mu(x, ξ) = Mu(t)(x, ξ) = M(u(t, x), ξ) = � 0 ⟨ξ⟩In � − B(u, ξ) − A(u, ξ) � ⟨ξ⟩−1 C(u, ξ) − A0(u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We start with the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='4 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' W satisfies the differential equation Wt = Op[Mu]W + R1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6) for some R1 ∈ L2 satisfying ∥R1∥ ≤ Cµ∥V ∥2 s + C∥V ∥s−1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set ˜L(u) := �ΛIn 0 0 In � L(u) �Λ−1In 0 0 In � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then Vt = Op[Mu]V + ˜R1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8) where ˜R1 = (˜L(u) − Op[Mu])V + (0, Q(u, Dx,tu)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As we have already seen in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2 (now with s − 1 replaced by s) ∥Q(u, Dx,tu)∥s ≤ Cµ∥V ∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9 ∥(˜L(0) − Op[M0])V ∥s ≤ C∥V ∥s−1 and due to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9 (iii) all terms appearing in (˜L(u) − ˜L(0) − Op[Mu − M0])V are of the form (a(u) − Op[au])f, where a is a smooth function with a(0) = 0 and f ∈ {∂l t∂β xu| l ≤ 1, l + |β| ≤ 2} ⊂ Hs−1 ֒→ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='10 yields ∥(˜L(u) − ˜L(0) − Op[Mu − M0])V ∥s ≤ Cµ(∥u∥s∥V ∥s + ∥V ∥s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In conclusion we have shown ∥ ˜R1∥s ≤ Cµ(∥V ∥2 s + ∥V ∥s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9) Now apply ΛsJǫ to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='8) and obtain Wt = Op[Mu]W + R1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='10) where R1 = [ΛsJǫ, Op[Mu]]V + ΛsJǫ ˜R1 Note that (Jǫ)ǫ∈(0,1) is a family of pseudo-differential operators, constant with respect to x, with symbols uniformly bounded in S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus we get from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9) ∥ΛsJǫR1∥ ≤ Cµ∥V ∥2 s + C∥V ∥s−1 and from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19 (iii) ∥[ΛsJǫ, Op[Mu]]V ∥ ≤ Cµ∥u∥s∥V ∥s + C∥V ∥s−1, which proves the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 28 Next, let D∞ ∈ C∞(Bnr (0) × {ξ ∈ Rd : |ξ| ≥ r}, C2n×2n) be the mapping constructed in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7 and extend it trivially by zero to a function defined on Bn r (0)×Rd := U0×Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Choose a function φ ∈ C∞(Rd), with 0 ≤ φ ≤ 1, φ(ξ) = 0 for |ξ| ≤ 2r and φ(ξ) = 1 for |ξ| ≥ 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set D(v, ξ) := φ(ξ)D∞(v, ξ), (v, ξ) ∈ U0 × Rd Let µ be sufficiently small such that u(t, x) ∈ Bnr (0) for all (t, x) ∈ [0, T] × Rd and define Du(x, ξ) := Du(t)(x, ξ) = D(u(t, x), ξ), (t, x, ξ) ∈ [0, T] × Rd × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Choose another function ψ ∈ C∞(Rd), with 0 ≤ ψ ≤ 1, ψ(ξ) = 0 for |ξ| ≥ 5r, ψ(ξ) = 1 for |ξ| ≤ 4r and define ˜Du(x, ξ) = Du(x, ξ) + ψ(ξ)I2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The family of operators (Gu(t))t∈[0,T] defined by Gu(t) := 1 2(Op[ ˜Du(t)] + Op[ ˜Du(t)]∗) + op[ ˜D0] − Op[ ˜D0] is self-adjoint and uniformly positive definite in L(L2) for µ sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Furthermore 1 2 d dt � GuW, W⟩ = Re⟨Gu Op[Mu]W, W⟩ + R2, for some R2 ∈ R with |R2| ≤ Cµ∥W∥(∥V ∥2 s + ∥V ∥s∥W∥ + ∥V ∥s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7 ˜D, D ∈ S0(U) and ˜Du = ˜D∗ u is uniformly positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In particular, op[ ˜D0] = op[ ˜D0]∗ is a self-adjoint and uniformly positive definite operator on L(L2) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Due to ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' also Op[ ˜D0]∗ = Op[ ˜D0], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Gu = op[ ˜D0] + 1 2(Op[ ˜Du − ˜D0] + Op[ ˜Du − ˜D0]∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19 (i) gives ∥ Op[ ˜Du − ˜D0]∥L(L2) ≤ Cµ∥u∥s, which yields the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Now apply Gu to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6), take the L2 scalar product with W and consider the real part to find Re⟨GuWt, W⟩ = Re⟨Gu Op[Mu]W, W⟩ + Re⟨GuR1, W⟩ := Re⟨Gu Op[Mu]W, W⟩ + R21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='11) Due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7) and ∥Gu∥L(L2) ≤ Cµ, ∥R21∥ ≤ Cµ∥W∥(∥V ∥2 s + ∥V ∥s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='12) 29 As Gu is self-adjoint we get Re⟨GuWt, W⟩ = 1 2 d dt � GuW, W⟩ − Re �� d dtGu � W, W⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='13) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='20 (iv) yields 2∥ d dtGu∥L(L2) ≤ �� d dt Op[ ˜Du] �� L(L2) ≤ Cµ∥ut∥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='14) The second statement then clearly follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='11)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The last step consists in showing the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6 Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It holds Re⟨Gu Op[Mu]W, W⟩ ≤ −c∥W∥2 + Cµ∥W∥2(∥u∥ 1 2 s+1 + ∥u∥s) + Cµ∥W∥2 −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' From Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6 we obtain 1 2 d dt⟨GuW, W⟩+c∥W∥2 ≤ Cµ∥W∥(∥V ∥2 s+∥V ∥s∥W∥+∥V ∥ 1 2)+Cµ(∥V ∥2 s−1+∥W∥2 −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='15) As Λ−kW = Λ−kWǫ → V as ǫ → 0 uniformly with respect to t for 0 ≤ k ≤ s and Gu is uniformly postive definite, we find by integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='15) ∥V (t)∥2 s + � t 0 ∥V ∥2 s dτ ≤ Cµ(∥V (0)∥ + � t 0 ∥V (τ)∥3 s + ∥V (τ)∥ 5 2s + ∥V (τ)∥s−1)dτ, t ∈ [0, T], which yields the assertion since ∥V ∥2 s = ∥u∥2 s+1 + ∥ut∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Set κ := op[ ˜D0] − Op[ ˜D0], which is infinitely smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Then Gu = 1 2(Op[ ˜Du] + Op[ ˜Du]∗) + κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As ∥Mu∥L(Hl,Hl−1) ≤ Cµ, l ∈ R, due to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19 (i) we find Re⟨κ Op[Mu]W, W⟩ ≤ Cµ∥W∥2 −1 By construction ˜Du = ˜D∗ u and thus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19 (ii) yields Re � (1 2(Op[ ˜Du]∗ − Op[Du]) Op[Mu]W, W � ≤ Cµ∥u∥s∥W∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Next note that ˜Du(x, ·) − Du(x, ·) is compactly supported with support not depending on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Therefore Op[ ˜Du − Du] is infinitely smoothing and Re⟨Op[ ˜Du − Du] Op[Mu]W, W⟩ ≤ Cµ∥W∥2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 30 In conclusion Re⟨Gu Op[Mu]W, W⟩ = Re⟨Op[Du] Op[Mu]W, W⟩ + Re⟨κ Op[Mu]W, W⟩ + 1 2 Re⟨(Op[ ˜Du]∗ − Op[ ˜Du]) Op[Mu])W, W⟩ + Re⟨Op[ ˜Du − Du]MuW, W⟩ ≤ Re⟨Op[Du] Op[Mu]W, W⟩ + Cµ(∥u∥s∥W∥2 + ∥W∥2 −1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='16) By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19 (iii) ∥(Op[Du] Op[Mu] − Op[DuMu])W∥ ≤ Cµ∥u∥s∥W∥ + C∥W∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence Re⟨Op[Du] Op[Mu]W, W⟩ ≤ Re⟨Op[DuMu]W, W⟩ + Cµ∥u∥s∥W∥2 + C∥W∥2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='17) Set Xu := DuMu + c∞/2I2n with c∞ as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Note that c∞ does not depend on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Since Op[I2n] − IdL2 is infinitely smoothing we conclude Re⟨Op[DuMu]W, W⟩ ≤ Re⟨Op[Xu]W, W⟩ − c∞ 2 ∥W∥2 + C∥W∥2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='7 Xu(x, ξ) + X ∗ u(x, ξ) = DuMu(x, ξ) + (DuMu)(x, ξ)∗ + c∞ ≤ 0, for x ∈ Rd und ξ ∈ Rd with |ξ| ≥ 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Since u ∈ Hs+1 and s + 1 ≥ d/2 + 2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='24 applied to −Xu gives Re⟨Op[Xu]W, W⟩ ≤ Cµ(∥u∥ 1 2 s+1∥W∥2 + ∥W∥−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='19) lead to Re⟨Op[DuMu]W, W⟩ ≤ −c∥W∥2 + Cµ(∥u∥ 1 2 s+1∥W∥2 + ∥W∥2 −1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='20) for c independent of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Clearly the assertion follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='17), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 5 A class of examples from dissipative relativistic fluid dynamics We consider the Euler-augmented Navier-Stokes formulation of dissipative relativistic fluid dynamics on flat Minkowski space-time derived in [12] as a generalization of a model proposed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For barotropic fluids it consists of a system of four equations which, using Einstein’s summation convention, read Aαβγ(ψǫ)∂ψγ ∂xδ = ∂ ∂xβ � Bαβγδ(ψǫ)∂ψγ ∂xδ � , α = 0, 1, 2, 3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) 31 where all Greek indices run from 0 to 3, Aαβγ, Bαβγδ are contravariant tensors and the unknown function ψǫ = (ψ0, ψ1, ψ2, ψ3)t determining the state of the fluid is a 4-vector with respect to the Minkowski-metric of flat space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' More specifically ψǫ = uǫ/θ with uǫ being the 4-velocity, θ the temperature of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We show that the results of the present work imply non-linear stability of the homogeneous reference state ¯ψ = ¯uǫ/¯θ, where ¯uǫ = (1, 0, 0, 0) represents the fluid’s rest frame and ¯θ > 0 is a constant temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' For a fluid with equation of state p = ρ/r, 1 ≤ r < ∞, p being the pressure, ρ the specific internal energy, the coefficent matrices evaluated at ¯ψ are given by [12] (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' assume ¯θ = 1)5 A0( ¯ψ) = � r 0 0 I3 � , Aj( ¯ψ) = � 0 (ej)t ej 0 � , B00( ¯ψ) = � −r2µ 0 0 −νI3 � , B0j( ¯ψ) = Bj0( ¯ψ) = 1 2 � 0 −(µr + ν)(ej)t −(µr + ν)ej 0 � , Bij( ¯ψ) = � −νδij 0 0 ηδij + 1 2(−µ + 1 3η + ζ)(ei ⊗ ej + ej ⊗ ei) � , i, j = 1, 2, 3, where η, ζ > 0 quantify the fluid’s viscosity, ν, µ > 0 with µ > ˜η := 4 3η + ζ reflect a frame change and Aβ(ψǫ) := (Aαβγ(ψǫ))0≤α,γ≤3, Bβγ(ψǫ) := (Bαβγδ(ψǫ))0≤α,γ≤3, β, δ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We do not give the detailed non-linear formulation at this point and just refer to [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The only information we need for the argumentation below is the fact that for all β, δ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , 3 and all states ψǫ the coefficient matrices Aβ(ψǫ), Bβδ(ψǫ), β, δ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , 3 are symmetric (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We show (HA), (HB), (D) for the matrices (−B00)−1Bβδ, (−B00)−1Aβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (HA) is straightforward: As −B00(ψǫ), A0(ψǫ) are positive definite at ψǫ = ¯ψ and symmetric for all states they are symmetric positive definite also in a neighbourhood of ¯ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus (HA) (a) is satisfied with FA(u) = −B00(u) and (HA) (b) with H(u) = A0(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Regarding (HB) Freist¨uhler proved ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' that at the reference state ψǫ = ¯ψ for each ω ∈ S2 the matrix ˜B(ψǫ, ω) = � 0 I4 −(−B00)− 1 2B(ψǫ, ω)(−B00)− 1 2 i(−B00)− 1 2C(ψǫ, ω)(−B00)− 1 2 � , where B(ψǫ, ω) = d � ij=0 Bij(ψǫ)ωiωj, C(ψǫ, ω) = 2 d � j=0 B0j(ψǫ)ωj, ω = (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' , ω2) ∈ S2, 5Here e1, e2, e3 and δij denote the conanical basis of R3 and the Kronecker symbol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 32 has four simple and two semi-simple purely imaginary eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' This is then also true for B(ψǫ, ω) := � 0 I4 (−B00)−1B(ψǫ, ω) i(−B00)−1C(ψǫ, ω), � = T −1 ˜B(ψǫ, ω)T with T = diag((−B00) 1 2, (−B00) 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Now in the present context the geometric multiplicities of purely imaginary eigenvalues of B(ψǫ, ω) are state invariant properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Therefore there exists a symbolic symmetrizer of B due to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' To see this invariance note that (even in the general setting in Section 3) the eigenvectors v = v(u, ω) ∈ C2n \\ {0} to an eigenvalue λ = λ(u, ω) ∈ C of B(u, ω) are exactly of the form v = (v1, λv1) with v1 ∈ Cn such that eλt+iξv1 is a plane wave solution to the linearization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) is a covariant expression, eλt+iξv being a plane wave solution with λ ∈ iR is also a covariant property (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It remains to show (D1), (D2), (D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' In the following we only consider matrices evaluated at ¯ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The Fourier-symbols correpsonding to the differential operators in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='1) are given by A(ω) = d � j=1 Ajωj = �0 ωt ω 0 � , B(ω) = d � j,k=1 Bjkωjωk = �−r2µ 0 0 η + (−µ + 1 3η + ζ)ω ⊗ ω � , C(ω) = 2 d � j=1 B0jξj � 0 −(µr + ν)ωt −(µr + ν)ω 0 � , ω = (ω1, ω2, ω3) ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It is straightforward to see that for any ω ∈ Sd−1 the matrices A0, Aj(ω), B00, Bjk(ω), C(ω) all decompose in sense of linear operators as A0 = A0 l ⊕ A0 t, A(ω) = Al ⊕ At, B00 = B00 l ⊕ B00 t , B(ω) = Bl ⊗ Bt, C(ω) = Cl ⊕ Ct with respect to the orthogonal decomposition C4 = (C×ωC)⊕({0}×{ω}⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Thus we can verify the conditions for A0 l , Al, B00 l , Bl, Cl and A0 t, At, B00 t , Bt, Ct separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' We have A0 t = I2, At = 0, B00 = −νI2, Bt = ηI2, Ct = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' As η > 0, these matrices correspond to coefficients of damped wave equations and it is well-known that such equations satisfy (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' One can also check this easily by virtue of [14], Theorem 4 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Next A0 l = �r 0 0 1 � , Al = �0 1 1 0 � , B00 l = �−r2µ 0 0 −ν � , Bl = �−ν 0 0 ˜η − µ � , Ct = � 0 −(µr + ν) −(µr + ν) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' It was shown in [14] that ˜Aj = (−B00 l )− 1 2Aj l (−B00 l )− 1 2, ˜Bjk l = (−B00 l )− 1 2Bjk l (−B00 l )− 1 2 satisfy (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' But then also ˇAj := (−B00 l )−1Aj l , ˇBjk := (−B00 l )−1Bjk l satisfy (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' 33 To see this note that ˇAj = S ˜AjS−1, ˇBjk = S ˜BjkS−1 with S = (−B00) 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Hence we have ˇW0 = S ˜W0S−1 for ˇW0 and ˜W0 as in (D1) for the matrices ˇAj, ˇBjk and ˜Aj, ˜Bjk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Further the symbolic symmetrizer ˜H = ˜A0 of ( ˜A0)−1 ˜A(ω) the matrix ˇH = S−1 ˜A0S−1 is a symbolic symmetrizer for ( ˇA0)−1 ˇA(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' This yields ˇW1 = S−1 ˜W1S−1 with ˇW1, ˜W1 as in (D1) for the respective matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' If now v is an eigenvector of ˇW0, S−1v is an eigenvector of ˜W0 and as ˜Aj, ˜Bjk satisfy (D1) we get ⟨( ˜W1 + ˜W ∗ 1 )S−1v, S−1v⟩ ≤ −c|S−1v|2 ≤ −ˇc|v|2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' ⟨( ˇW1 + ˇW ∗ 1 )v, v⟩ ≤ −ˇc|v|2, which proves (D1) for ˇAj, ˇBjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' (D2) follows analogously since with S = diag((−B00) 1 2, (−B00) 1 2) the matrix S−1 ˜H(ω)S−1 is a symbolic symmetrizer for ˇB(ω) if ˜H(ω) is a symbolic symmetrizer for ˜B(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Lastly, (D3) is satisfied trivially, as the matrices introduce equivalent systems of PDEs and thus solutions to the dispersion relation are identical for the two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Statements and declarations Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' This work was supported by DFG Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' FR 822/10-1, 10-1/2) Competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The author has no competing interests to declare that are relevant to the content of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} +page_content=' The author would like to sincerely thank Heinrich Freist¨uhler for his highly helpful suggestions and comments as well as many fruitful discussions.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAzT4oBgHgl3EQft_25/content/2301.01685v1.pdf'} diff --git a/29AzT4oBgHgl3EQf9P5O/content/tmp_files/2301.01916v1.pdf.txt b/29AzT4oBgHgl3EQf9P5O/content/tmp_files/2301.01916v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3103b842328db6a4f35f14d3bf1ebaeb6652bb0a --- /dev/null +++ b/29AzT4oBgHgl3EQf9P5O/content/tmp_files/2301.01916v1.pdf.txt @@ -0,0 +1,708 @@ +arXiv:2301.01916v1 [math.CV] 5 Jan 2023 +THE SHARP BOUND OF THE THIRD HANKEL +DETERMINANT FOR INVERSE OF CONVEX FUNCTIONS +BISWAJIT RATH1, K. SANJAY KUMAR 2, D. VAMSHEE KRISHNA3 +Abstract. The objective of this paper is to find the best possible upper bound +of the third Hankel determinant for inverse of convex functions. +1. Introduction +Denote H the family all analytic functions in unit disk D = {z ∈ C : |z| < 1} +and A be the subfamily of functions f normilized by the conditions +f(0) = f ′(0) − 1 = 0, i.e, of the type +(1.1) +f(z) = +∞ +� +n=1 +anzn, a1 := 1, +and S be the subfamily of A, possessing univalent (schlicht) mappings. For f ∈ S, +has an inverse f −1 given by +(1.2) +f −1(w) = w + +∞ +� +n=2 +tnwn, |w| < ro(f); +� +ro(f) ≥ 1 +4 +� +. +A typical problem in geometric function theory is to study a functional made up of +combination of the coefficients of the original functions. For the positive integers +r, n, Pommerenke [16] characterized the rth- Hankel determinant of nth-order for +f given in (1.1), defined as follows: +(1.3) +Hr,n(f) = +an +an+1 +· · · +an+r−1 +an+1 +an+2 +· · · +an+r +... +... +... +... +an+r−1 +an+r +· · · +an+2r−2 +. +The problem of finding sharp estimates of the third Hankel determinant obtained +for r = 3 and n = 1 in (1.2), given by +(1.4) +H3,1(f) := +a1 = 1 +a2 +a3 +a2 +a3 +a4 +a3 +a4 +a5 += 2a2a3a4 − a3 +3 − a2 +4 + a3a5 − a2 +2a5, +is technically much tough than r = n = 2. +In recent years, many authors are working on obtaining upper bounds (see [2, 8, +17, 18, 20]) and a few papers were devoted to the estimation of sharp upper bound +to H3,1(f), for certain subclasses of analytic functions (see [3, 6, 7, 9, 10, 19]). +2020 Mathematics Subject Classification. 30C45, 30C50. +Key words and phrases. Holomorphic function, univalent function, Hankel determinant, Inverse +of Convex, Carath´eodory function. +1 + +2 +B. RATH, K. S. KUMAR, D. V. KRISHNA +Recently Lecko et al. [6] obtained the sharp bound for the class of convex function +denoted as Sc, defined by +(1.5) +Re +� +1 + zf ′′(z) +f ′(z) +� +> 0. +Motivated by these results, in this paper we obtain sharp estimate for H3,1(f −1) +when f ∈ Sc as 1/36. +The collection P, of all functions p, each one called as Carath´eodory function [5] +of the form, +(1.6) +p(z) = 1 + +∞ +� +t=1 +ctzt, +having a positive real part in D. In view of (1.4) and (1.5), the coefficients of the +functions in Sc can be expressed in terms of coefficients of functions in P. We then +obtain the upper bound of H3,1(f −1), buliding our analysis on the familiar formulas +of coefficients c2 (see, [15, p. 166]), c3 (see [11, 12]) and c4 can be found in [10]. +The foundation for proofs of our main results is the following lemma and we +adopt the procedure framed through Libera and Zlotkiewicz [12]. +Lemma 1.1. If p ∈ P, is of the form (1.5) with c1 ≥ 0, such that c1 ∈ [0, 2] then +2c2 = c2 +1 + νµ, +4c3 = c3 +1 + 2c1νµ − c1νµ2 + 2ν +� +1 − |µ|2� +ρ, +and +8c4 = c4 +1 + 3c2 +1νµ + +� +4 − 3c2 +1 +� +νµ2 + c2 +1νµ3 + 4ν +� +1 − |µ|2� � +1 − |ρ|2� +ψ ++ 4ν +� +1 − |µ|2� � +c1ρ − cµρ − ¯µρ2� +, +where ν := 4 − c2 +1, for some µ, ρ and ψ such that |µ| ≤ 1, |ρ| ≤ 1 and |ψ| ≤ 1. +2. Main result +Theorem 2.1. If f ∈ Sc, then +��H3,1(f −1) +�� ≤ 1 +36 +and the inequality is sharp for p0(z) = (1 + z3)/(1 − z3). +Proof. For f ∈ Sc, there exists a holomorphic function p ∈ P such that +(2.1) +� +1 + zf ′′(z) +f ′(z) +� += p(z) ⇔ {f ′(z) + zf ′′(z)} = p(z)f ′(z) +Using the series representation for f and p in (2.1), a simple calculation gives +a2 = c1 +2 , a3 = c2 +1 + c2 +6 +, a4 = 1 +12 +�1 +2c3 +1 + 3 +2c1c2 + c3 +� +and a5 = 1 +20 +�1 +6c4 +1 + c2 +1c2 + 1 +2c2 +2 + 4 +3c1c3 + c4 +� +(2.2) +Now from the defination (1.2), we have +(2.3) +w = f(f −1) = f −1(w) + +∞ +� +n=2 +an(f −1(w))n. + +THE SHARP BOUND OF THE HANKEL DETERMINANT +3 +Further, we have +(2.4) +w = f(f −1) = w + +∞ +� +n=2 +tnwn + +∞ +� +n=2 +an(w + +∞ +� +n=2 +tnwn)n. +Upon simplification, we obtain +(t2 + a2)w2 + (t3 + 2a2t2 + a3)w3 + (t4 + 2a2t3 + a2t2 +2 + 3a3t2 + a4)w4 ++(t5 + 2a2t4 + 2a2t2t3 + 3a3t3 + 3a3t2 +2 + 4a4t2 + a5)w5 + ...... = 0. +(2.5) +Equating the coefficients of like power in (2.5), upon simplification, we obtain +t2 = −a2; t3 = {−a3 + 2a2 +2}; t4 = {−a4 + 5a2a3 − 5a3 +2}; +t5 = {−a5 + 6a2a4 − 21a2 +2a3 + 3a2 +3 + 14a4 +2}. +(2.6) +Using the values of an(n = 2, 3, 4, 5) from (2.2) in (2.6), upon simplification, we +obtain +t2 = −c1 +2 , t3 = 1 +6 +� +2c2 +1 − c2 +� +, t4 = 1 +24 +� +−6c3 +1 + 7c1c2 − 2c3 +� +and t5 = +1 +120 +� +−6c4 + 22c1c3 − 46c2 +1c2 + 7c2 +2 + 24c4 +1 +� +. +(2.7) +Now, +H3,1(f −1) = +t1 = 1 +t2 +t3 +t2 +t3 +t4 +t3 +t4 +t5 +, +(2.8) +Using the values of tj, (j = 2, 3, 4, 5) from (2.7) in (2.8), it simplifies to give +H3,1(f −1) = +1 +8640 +� +4c6 +1 − 24c4 +1c2 + 12c3 +1c3 + 39c2 +1c2 +2 − 44c3 +2 + 36c1c2c3 +−36c2 +1c4 − 60c2 +3 + 72c2c4 +� +. +(2.9) +In view of (2.9), using the values of c2, c3 and c4 from lemma 1.1, gives +24c4 +1c2 =12 +� +c6 +1 + c4 +1νµ +� +; +12c3 +1c3 =3 +� +c6 +1 + 2c4 +1νµ − c4 +1νµ2 + 2c3 +1ν(1 − |µ|2)ρ +� +44c3 +2 =11 +2 +� +c6 +1 + 3c4 +1νµ + 3c2 +1ν2µ2 + ν3µ3� +; +39c2 +1c2 +2 =39 +4 +� +c6 +1 + 2c4 +1νµ + c2 +1ν2µ2� +; +36c1c2c3 =9 +2 +� +c6 +1 + 3c4 +1νµ + 2c2 +1ν2µ2 − c4 +1νµ2 − c2 +1ν2µ3 ++2ν +� +c3 +1 + c1νµ +� � +1 − |µ|2� +ρ +� +; +60c2 +3 =15 +4 +� +c6 + 4c4νµ + 4c4ν2µ2 − 2c4νµ2 − 4c2ν2µ3 + c2ν2µ4 ++4ν(c3 + 2cνµ − cνµ2)(1 − |µ|2)ρ + 4ν2(1 − |µ|2)2ρ2� +; +72c2c4 − 36c2 +1c4 =9 +2 +� +c4 +1νµ + 3c2 +1ν2µ2 + +� +4 − 3c2 +1 +� +ν2µ3 + c2 +1ν2µ4 ++ 4ν2c1µ (1 − µ) +� +1 − |µ|2� +ρ − 4ν2 � +1 − |µ|2� +|µ|2ρ2 ++4ν2 � +1 − |µ|2� � +1 − |ρ|2� +µψ +� +. +(2.10) + +4 +B. RATH, K. S. KUMAR, D. V. KRISHNA +Imputting the values from (2.10) in the expression (2.9), after simplifying, we get +H3,1(f −1) = +1 +8640 +�3 +4c2 +1ν2µ2 − 3c2 +1ν2µ3 + 3 +4c2 +1ν2µ4 − 11 +2 ν3µ3 + 18ν2µ3 +− +� +3c1ν2µ + 3c1ν2µ2� � +1 − |µ|2� +ρ − 3ν2 � +5 + |µ|2� � +1 − |µ|2� +ρ2 ++18ν2µ +� +1 − |µ|2� +(1 − |ρ|2� +ψ +� +. +(2.11) +Putting u := c1 and taking ν = +� +4 − u2� +in (2.11), we obtain +H3,1(f −1) = +� +4 − u2�2 +8640 +�3 +4u2µ2 + 3 +2u2µ3 + 3 +4u2µ4 − (4 − u2)µ3 +− 3uµ (1 + µ) +� +1 − |µ|2� +ρ − 3 +� +5 + |µ|2� � +1 − |µ|2� +ρ2 ++18µ +� +1 − |µ|2� +(1 − |ρ|2� +ψ +� +. +(2.12) +Taking modulus on both sides of (2.12), using |µ| = v ∈ [0, 1], |ρ| = w ∈ [0, 1], +c1 = u ∈ [0, 2] and |ψ| ≤ 1, we obtain +(2.13) +����H3,1(f −1) +���� ≤ ϑ (u, v, w) +8640 +, +where ϑ : R3 → R is defined as +ϑ (u, v, w) = +� +4 − u2�2 �3 +4u2v2 + 3 +2u2v3 + 3 +4u2v4 + +� +4 − u2� +v3 ++ 3uv (1 + v) +� +1 − v2� +w + 3 +� +5 + v2� � +1 − v2� +w2 ++18v +� +1 − v2� +(1 − w2�� +(2.14) +Now, we are making an attempt to maximize the function ϑ (u, v, w) on +Ω := [0, 2] × [0, 1] × [0, 1]. +A. On the vertices of Ω, from (2.14), we get +ϑ (0, 0, 0) = ϑ (2, 0, 0) = ϑ (2, 1, 0) = ϑ (2, 0, 1) = ϑ (2, 1, 1) = 0, +ϑ (0, 0, 1) = 240, ϑ (0, 1, 0) = ϑ (0, 1, 1) = 64. +B. On the edges of Ω, from (2.14), we have +(i) For the edge u = 0, v = 0, 0 < w < 1, we obtain. +ϑ (0, 0, w) = 240w2 ≤ 240. +(ii) For the edge u = 0, v = 1, 0 < w < 1, we obtain +ϑ (0, 1, w) = 64. +(iii) For u = 0, w = 0, 0 < v < 1, +ϑ (0, v, 0) = 32v(9 − 7v2) ≤ 192 +� +3 +7, , for v = +√ +2. +(iv) For u = 0, w = 1, 0 < v < 1, +ϑ (0, v, 1) = 240 − 192v2 + 64v3 − 48v4 ≤ 240. +(v) For v = 0, w = 1, 0 < u < 2, +ϑ (u, 0, 1) = 15(4 − u2)2 ≤ 240. + +THE SHARP BOUND OF THE HANKEL DETERMINANT +5 +(vi) For the edges: v = 1, w = 0, 0 < u < 2 or v = 1, w = 1, 0 < u < 2, we have +ϑ (u, 1, w) = (4 − u2)2(4 + 2u2) ≤ 64. +(vii) For the edges: u = 2, v = 0, 0 < w < 1 or u = 2, v = 1, 0 < w < 1 or +u = 2, w = 0, 0 < v < 1 or c = 2, w = 1, 0 < v < 1 or v = 0, w = 0, 0 < u < 2, +we obtain +ϑ (2, v, w) = 0. +C. Now, we consider the six faces of Ω. +(i) On the face u = 2, from (2.14), we obtain +ϑ (2, v, w) = 0. +(ii) On the face u = 0, v ∈ (0, 1) and w ∈ (0, 1) from (2.14), we get +ϑ (0, v, w) = 288v − 224v3 + (240 − 288v − 192v2 + 288v3 − 48v4)w2 += 288v − 224v3 + 48(5 − v)(−1 + v)2(1 + v)w2 +≤ 288v − 224v3 + 48(5 − v)(−1 + v)2(1 + v) += 240 − 192v2 + 64v3 − 48v4 ≤ 240. +(iii) On the face v = 0 u ∈ (0, 2), w ∈ (0, 1), from (2.14), we obtain +ϑ (u, 0, w) = 15(4 − u2)2w2 ≤ 15(4 − u2)2 ≤ 240. +(iv) On the face v = 1, u ∈ (0, 2), w ∈ (0, 1), from (2.14), we observe that the +function ϑ (u, 1, w) is independent of w, from B(vi), we have ϑ (u, 1, w) ≤ 240. +(v) On the face w = 0, u ∈ (0, 2), v ∈ (0, 1), from (2.14), we obtain +ϑ (u, v, 0) = (4 − u2)2 +�3u2v2 +4 ++ 3u2v3 +2 ++ (4 − u2)v3 + 3u2v4 +4 ++ 18v(1 − v2) +� += (4 − u2)2 +� +18v − 14v3 + u2 +�3v2 +4 ++ v3 +2 + 3v4 +4 +�� +≤ (4 − u2)2 +� +12 +� +3 +7 + 2u2 +� +≤ 192 +� +3 +7, u ∈ (0, 2). +(vi) On the face w = 1, in (2.14), we obtain +ϑ (u, v, 1) = (4 − u2)2 +�3 +4u2v2 + 3 +2u2v3 + 3 +4u2v4 + (4 − u2)v3 ++ 3uv(1 + v)(1 − v2) + 3(5 + v2) +� +1 − v2� � +:= g3(u, v), with (u, v) ∈ R2. +Note that all real solutions (u,v) of the system of equation +∂g3 +∂u = 3 +2(−4 + u2) +� +8(−1 + v)v(1 + v)2 − 10u2(−1 + v)v(1 + v)2 ++u3v2(3 + 2v + 3v2) − 4u(−10 + 9v2 − 2v3 + 3v4) +� += 0 +and + +6 +B. RATH, K. S. KUMAR, D. V. KRISHNA +∂g3 +∂v = 3 +2(−4 + u2)2(−8v(2 − v + v2) + u2v(1 + v + 2v2) ++ u(2 + 4v − 6v2 − 8v3)) = 0 +by a numerical computation are the following +(0, 0), (−2.63625, −1.53087), (−1.0493, 1.14045) and (±2, x), x ∈ R. +Therefore, g3 has no critical point in (0, 2) × (0, 1). +D. Now, consider the interior portion of Ω i.e. (0, 2) × (0, 1) × (0, 1). +Differentiating ϑ(u, v, w) partially with respect w, we obtain +∂ϑ +∂w = 1 +2(4 − u2)2 � +60w2 + 3v2(u2 + 4uw − 16w2) + 12v(6 + uw − 6w2) ++3v4(u2 − 4uw − 4w2) + 2v3(−28 + u2 − 6uw + 36w2) +� +upon solving ∂ϑ +∂w = 0, we get +w0 = − +uv(1 + v) +2(5 − v)(1 − v) /∈ (0, 1) for (u, v) ∈ (0, 2) × (0, 1) +Hence ϑ(u, v, w) has no critical point in the interior of Ω. +In review of cases A, B, C and D, we obtained +(2.15) +max +� +ϑ(u, v, w) : u ∈ [0, 2], v ∈ [0, 1], w ∈ [0, 1] +� += 240. +From expression (2.13) and (2.15), we obtain +(2.16) +���H3,1(f −1) +��� ≤ 1 +36. +For p0 ∈ Sc, we obtain t2 = t3 = t5 = 0, t4 = 1/6, which follows the result. +□ +Data Availability: My manuscript has no associate data +References +[1] M. 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A Mat. ,115(2021), +https://doi.org/10.1007/s13398-020-00977-2. +1.2.3Department of Mathematics, GITAM School of Science, GITAM (Deemed to be +University), Visakhapatnam- 530 045, A.P., India +Email address: brath@gitam.edu1∗,skarri9@gitam.in2,vamsheekrishna1972@gmail.com3 + diff --git a/29AzT4oBgHgl3EQf9P5O/content/tmp_files/load_file.txt b/29AzT4oBgHgl3EQf9P5O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f87f292ce8ecfd0834296d164066bbad1d883742 --- /dev/null +++ b/29AzT4oBgHgl3EQf9P5O/content/tmp_files/load_file.txt @@ -0,0 +1,343 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf,len=342 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='01916v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='CV] 5 Jan 2023 THE SHARP BOUND OF THE THIRD HANKEL DETERMINANT FOR INVERSE OF CONVEX FUNCTIONS BISWAJIT RATH1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' SANJAY KUMAR 2, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' VAMSHEE KRISHNA3 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' The objective of this paper is to find the best possible upper bound of the third Hankel determinant for inverse of convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Introduction Denote H the family all analytic functions in unit disk D = {z ∈ C : |z| < 1} and A be the subfamily of functions f normilized by the conditions f(0) = f ′(0) − 1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='e, of the type (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='1) f(z) = ∞ � n=1 anzn, a1 := 1, and S be the subfamily of A, possessing univalent (schlicht) mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' For f ∈ S, has an inverse f −1 given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='2) f −1(w) = w + ∞ � n=2 tnwn, |w| < ro(f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' � ro(f) ≥ 1 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' A typical problem in geometric function theory is to study a functional made up of combination of the coefficients of the original functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' For the positive integers r, n, Pommerenke [16] characterized the rth- Hankel determinant of nth-order for f given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='1), defined as follows: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='3) Hr,n(f) = an an+1 · · an+r−1 an+1 an+2 · · an+r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' an+r−1 an+r · · an+2r−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' The problem of finding sharp estimates of the third Hankel determinant obtained for r = 3 and n = 1 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='2), given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='4) H3,1(f) := a1 = 1 a2 a3 a2 a3 a4 a3 a4 a5 = 2a2a3a4 − a3 3 − a2 4 + a3a5 − a2 2a5, is technically much tough than r = n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' In recent years, many authors are working on obtaining upper bounds (see [2, 8, 17, 18, 20]) and a few papers were devoted to the estimation of sharp upper bound to H3,1(f), for certain subclasses of analytic functions (see [3, 6, 7, 9, 10, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 30C45, 30C50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Holomorphic function, univalent function, Hankel determinant, Inverse of Convex, Carath´eodory function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 1 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' RATH, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' KUMAR, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' KRISHNA Recently Lecko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' [6] obtained the sharp bound for the class of convex function denoted as Sc, defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='5) Re � 1 + zf ′′(z) f ′(z) � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Motivated by these results, in this paper we obtain sharp estimate for H3,1(f −1) when f ∈ Sc as 1/36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' The collection P, of all functions p, each one called as Carath´eodory function [5] of the form, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='6) p(z) = 1 + ∞ � t=1 ctzt, having a positive real part in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' In view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='5), the coefficients of the functions in Sc can be expressed in terms of coefficients of functions in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' We then obtain the upper bound of H3,1(f −1), buliding our analysis on the familiar formulas of coefficients c2 (see, [15, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 166]), c3 (see [11, 12]) and c4 can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' The foundation for proofs of our main results is the following lemma and we adopt the procedure framed through Libera and Zlotkiewicz [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' If p ∈ P, is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='5) with c1 ≥ 0, such that c1 ∈ [0, 2] then 2c2 = c2 1 + νµ, 4c3 = c3 1 + 2c1νµ − c1νµ2 + 2ν � 1 − |µ|2� ρ, and 8c4 = c4 1 + 3c2 1νµ + � 4 − 3c2 1 � νµ2 + c2 1νµ3 + 4ν � 1 − |µ|2� � 1 − |ρ|2� ψ + 4ν � 1 − |µ|2� � c1ρ − cµρ − ¯µρ2� , where ν := 4 − c2 1, for some µ, ρ and ψ such that |µ| ≤ 1, |ρ| ≤ 1 and |ψ| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Main result Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' If f ∈ Sc, then ��H3,1(f −1) �� ≤ 1 36 and the inequality is sharp for p0(z) = (1 + z3)/(1 − z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' For f ∈ Sc, there exists a holomorphic function p ∈ P such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='1) � 1 + zf ′′(z) f ′(z) � = p(z) ⇔ {f ′(z) + zf ′′(z)} = p(z)f ′(z) Using the series representation for f and p in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='1), a simple calculation gives a2 = c1 2 , a3 = c2 1 + c2 6 , a4 = 1 12 �1 2c3 1 + 3 2c1c2 + c3 � and a5 = 1 20 �1 6c4 1 + c2 1c2 + 1 2c2 2 + 4 3c1c3 + c4 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='2) Now from the defination (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='2), we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='3) w = f(f −1) = f −1(w) + ∞ � n=2 an(f −1(w))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' THE SHARP BOUND OF THE HANKEL DETERMINANT 3 Further, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='4) w = f(f −1) = w + ∞ � n=2 tnwn + ∞ � n=2 an(w + ∞ � n=2 tnwn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Upon simplification, we obtain (t2 + a2)w2 + (t3 + 2a2t2 + a3)w3 + (t4 + 2a2t3 + a2t2 2 + 3a3t2 + a4)w4 +(t5 + 2a2t4 + 2a2t2t3 + 3a3t3 + 3a3t2 2 + 4a4t2 + a5)w5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='. = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='5) Equating the coefficients of like power in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='5), upon simplification, we obtain t2 = −a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' t3 = {−a3 + 2a2 2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' t4 = {−a4 + 5a2a3 − 5a3 2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' t5 = {−a5 + 6a2a4 − 21a2 2a3 + 3a2 3 + 14a4 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='6) Using the values of an(n = 2, 3, 4, 5) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='2) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='6), upon simplification, we obtain t2 = −c1 2 , t3 = 1 6 � 2c2 1 − c2 � , t4 = 1 24 � −6c3 1 + 7c1c2 − 2c3 � and t5 = 1 120 � −6c4 + 22c1c3 − 46c2 1c2 + 7c2 2 + 24c4 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='7) Now, H3,1(f −1) = t1 = 1 t2 t3 t2 t3 t4 t3 t4 t5 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='8) Using the values of tj, (j = 2, 3, 4, 5) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='7) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='8), it simplifies to give H3,1(f −1) = 1 8640 � 4c6 1 − 24c4 1c2 + 12c3 1c3 + 39c2 1c2 2 − 44c3 2 + 36c1c2c3 −36c2 1c4 − 60c2 3 + 72c2c4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='9) In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='9), using the values of c2, c3 and c4 from lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='1, gives 24c4 1c2 =12 � c6 1 + c4 1νµ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 12c3 1c3 =3 � c6 1 + 2c4 1νµ − c4 1νµ2 + 2c3 1ν(1 − |µ|2)ρ � 44c3 2 =11 2 � c6 1 + 3c4 1νµ + 3c2 1ν2µ2 + ν3µ3� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 39c2 1c2 2 =39 4 � c6 1 + 2c4 1νµ + c2 1ν2µ2� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 36c1c2c3 =9 2 � c6 1 + 3c4 1νµ + 2c2 1ν2µ2 − c4 1νµ2 − c2 1ν2µ3 +2ν � c3 1 + c1νµ � � 1 − |µ|2� ρ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 60c2 3 =15 4 � c6 + 4c4νµ + 4c4ν2µ2 − 2c4νµ2 − 4c2ν2µ3 + c2ν2µ4 +4ν(c3 + 2cνµ − cνµ2)(1 − |µ|2)ρ + 4ν2(1 − |µ|2)2ρ2� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 72c2c4 − 36c2 1c4 =9 2 � c4 1νµ + 3c2 1ν2µ2 + � 4 − 3c2 1 � ν2µ3 + c2 1ν2µ4 + 4ν2c1µ (1 − µ) � 1 − |µ|2� ρ − 4ν2 � 1 − |µ|2� |µ|2ρ2 +4ν2 � 1 − |µ|2� � 1 − |ρ|2� µψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='10) 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' RATH, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' KUMAR, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' KRISHNA Imputting the values from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='10) in the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='9), after simplifying, we get H3,1(f −1) = 1 8640 �3 4c2 1ν2µ2 − 3c2 1ν2µ3 + 3 4c2 1ν2µ4 − 11 2 ν3µ3 + 18ν2µ3 − � 3c1ν2µ + 3c1ν2µ2� � 1 − |µ|2� ρ − 3ν2 � 5 + |µ|2� � 1 − |µ|2� ρ2 +18ν2µ � 1 − |µ|2� (1 − |ρ|2� ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='11) Putting u := c1 and taking ν = � 4 − u2� in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='11), we obtain H3,1(f −1) = � 4 − u2�2 8640 �3 4u2µ2 + 3 2u2µ3 + 3 4u2µ4 − (4 − u2)µ3 − 3uµ (1 + µ) � 1 − |µ|2� ρ − 3 � 5 + |µ|2� � 1 − |µ|2� ρ2 +18µ � 1 − |µ|2� (1 − |ρ|2� ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='12) Taking modulus on both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='12), using |µ| = v ∈ [0, 1], |ρ| = w ∈ [0, 1], c1 = u ∈ [0, 2] and |ψ| ≤ 1, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='13) ����H3,1(f −1) ���� ≤ ϑ (u, v, w) 8640 , where ϑ : R3 → R is defined as ϑ (u, v, w) = � 4 − u2�2 �3 4u2v2 + 3 2u2v3 + 3 4u2v4 + � 4 − u2� v3 + 3uv (1 + v) � 1 − v2� w + 3 � 5 + v2� � 1 − v2� w2 +18v � 1 − v2� (1 − w2�� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14) Now, we are making an attempt to maximize the function ϑ (u, v, w) on Ω := [0, 2] × [0, 1] × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' On the vertices of Ω, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14), we get ϑ (0, 0, 0) = ϑ (2, 0, 0) = ϑ (2, 1, 0) = ϑ (2, 0, 1) = ϑ (2, 1, 1) = 0, ϑ (0, 0, 1) = 240, ϑ (0, 1, 0) = ϑ (0, 1, 1) = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' On the edges of Ω, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14), we have (i) For the edge u = 0, v = 0, 0 < w < 1, we obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' ϑ (0, 0, w) = 240w2 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (ii) For the edge u = 0, v = 1, 0 < w < 1, we obtain ϑ (0, 1, w) = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (iii) For u = 0, w = 0, 0 < v < 1, ϑ (0, v, 0) = 32v(9 − 7v2) ≤ 192 � 3 7, , for v = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (iv) For u = 0, w = 1, 0 < v < 1, ϑ (0, v, 1) = 240 − 192v2 + 64v3 − 48v4 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (v) For v = 0, w = 1, 0 < u < 2, ϑ (u, 0, 1) = 15(4 − u2)2 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' THE SHARP BOUND OF THE HANKEL DETERMINANT 5 (vi) For the edges: v = 1, w = 0, 0 < u < 2 or v = 1, w = 1, 0 < u < 2, we have ϑ (u, 1, w) = (4 − u2)2(4 + 2u2) ≤ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (vii) For the edges: u = 2, v = 0, 0 < w < 1 or u = 2, v = 1, 0 < w < 1 or u = 2, w = 0, 0 < v < 1 or c = 2, w = 1, 0 < v < 1 or v = 0, w = 0, 0 < u < 2, we obtain ϑ (2, v, w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Now, we consider the six faces of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (i) On the face u = 2, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14), we obtain ϑ (2, v, w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (ii) On the face u = 0, v ∈ (0, 1) and w ∈ (0, 1) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14), we get ϑ (0, v, w) = 288v − 224v3 + (240 − 288v − 192v2 + 288v3 − 48v4)w2 = 288v − 224v3 + 48(5 − v)(−1 + v)2(1 + v)w2 ≤ 288v − 224v3 + 48(5 − v)(−1 + v)2(1 + v) = 240 − 192v2 + 64v3 − 48v4 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (iii) On the face v = 0 u ∈ (0, 2), w ∈ (0, 1), from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14), we obtain ϑ (u, 0, w) = 15(4 − u2)2w2 ≤ 15(4 − u2)2 ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (iv) On the face v = 1, u ∈ (0, 2), w ∈ (0, 1), from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14), we observe that the function ϑ (u, 1, w) is independent of w, from B(vi), we have ϑ (u, 1, w) ≤ 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (v) On the face w = 0, u ∈ (0, 2), v ∈ (0, 1), from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14), we obtain ϑ (u, v, 0) = (4 − u2)2 �3u2v2 4 + 3u2v3 2 + (4 − u2)v3 + 3u2v4 4 + 18v(1 − v2) � = (4 − u2)2 � 18v − 14v3 + u2 �3v2 4 + v3 2 + 3v4 4 �� ≤ (4 − u2)2 � 12 � 3 7 + 2u2 � ≤ 192 � 3 7, u ∈ (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (vi) On the face w = 1, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14), we obtain ϑ (u, v, 1) = (4 − u2)2 �3 4u2v2 + 3 2u2v3 + 3 4u2v4 + (4 − u2)v3 + 3uv(1 + v)(1 − v2) + 3(5 + v2) � 1 − v2� � := g3(u, v), with (u, v) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Note that all real solutions (u,v) of the system of equation ∂g3 ∂u = 3 2(−4 + u2) � 8(−1 + v)v(1 + v)2 − 10u2(−1 + v)v(1 + v)2 +u3v2(3 + 2v + 3v2) − 4u(−10 + 9v2 − 2v3 + 3v4) � = 0 and 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' RATH, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' KUMAR, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' KRISHNA ∂g3 ∂v = 3 2(−4 + u2)2(−8v(2 − v + v2) + u2v(1 + v + 2v2) + u(2 + 4v − 6v2 − 8v3)) = 0 by a numerical computation are the following (0, 0), (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='63625, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='53087), (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='0493, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='14045) and (±2, x), x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Therefore, g3 has no critical point in (0, 2) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Now, consider the interior portion of Ω i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' (0, 2) × (0, 1) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Differentiating ϑ(u, v, w) partially with respect w, we obtain ∂ϑ ∂w = 1 2(4 − u2)2 � 60w2 + 3v2(u2 + 4uw − 16w2) + 12v(6 + uw − 6w2) +3v4(u2 − 4uw − 4w2) + 2v3(−28 + u2 − 6uw + 36w2) � upon solving ∂ϑ ∂w = 0, we get w0 = − uv(1 + v) 2(5 − v)(1 − v) /∈ (0, 1) for (u, v) ∈ (0, 2) × (0, 1) Hence ϑ(u, v, w) has no critical point in the interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' In review of cases A, B, C and D, we obtained (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='15) max � ϑ(u, v, w) : u ∈ [0, 2], v ∈ [0, 1], w ∈ [0, 1] � = 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' From expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='15), we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='16) ���H3,1(f −1) ��� ≤ 1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' For p0 ∈ Sc, we obtain t2 = t3 = t5 = 0, t4 = 1/6, which follows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' □ Data Availability: My manuscript has no associate data References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Arif, Mohsan Raza, Huo Tang, Shehzad Hussain and Hassan Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=', 17(1)(2019), 1615–1630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Babalola, On H3(1) Hankel determinant for some classes of univalent functions, Inequal Theory Appl, 6 (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Cho)(Nova Science Publishers, New York, 2010), 1-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Banga and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' Sivaprasad Kumar, The sharp bounds of the second and third Hankel deter- 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='1007/s13398-020-00977-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='3Department of Mathematics, GITAM School of Science, GITAM (Deemed to be University), Visakhapatnam- 530 045, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content=', India Email address: brath@gitam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='edu1∗,skarri9@gitam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='in2,vamsheekrishna1972@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} +page_content='com3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29AzT4oBgHgl3EQf9P5O/content/2301.01916v1.pdf'} diff --git a/29E1T4oBgHgl3EQf5wUG/vector_store/index.faiss b/29E1T4oBgHgl3EQf5wUG/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..443f6298ac15edd49b834e17110f8638b712f528 --- /dev/null +++ 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Physics, +Department of Physics, New York University +New York, NY, 10003, USA +dRudolf Peierls Centre for Theoretical Physics, +Clarendon Laboratory, University of Oxford, +Parks Road, Oxford OX1 3PU, UK +and +All Souls College, University of Oxford, +High Street, Oxford OX1 4AL, UK +Abstract +The 3d Ising model in the low temperature (ferromagnetic) phase describes dynam- +ics of two-dimensional surfaces—domain walls between clusters of parallel spins. The +Kramers–Wannier duality maps these surfaces into worldsheets of confining strings in +the Wegner’s Z2 gauge theory. We study the excitation spectrum of long Ising strings by +simulating the Z2 gauge theory on a lattice. We observe a strong mixing between string +excitations and the lightest glueball state and do not find indications for light massive +resonances on the string worldsheet. +arXiv:2301.00034v1 [hep-lat] 30 Dec 2022 + +Contents +1 +Introduction +1 +2 +Ising Model and Z2 Gauge Theory +3 +3 +Effective String Theory +5 +4 +Review of Lattice Techniques +7 +4.1 +Lattice gauge theory and Monte-Carlo simulations . . . . . . . . . . . . . . . . +7 +4.2 +Extracting spectra +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +4.3 +Constructing flux tube operators +. . . . . . . . . . . . . . . . . . . . . . . . . +11 +5 +Results +15 +5.1 +The absolute ground state and the string tension +. . . . . . . . . . . . . . . . +16 +5.2 +Glueball States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +5.3 +Excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +5.4 +Finite volume corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +5.5 +Including multitrace operators . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +6 +Concluding Remarks +34 +A Compilation of energy spectra +36 +References +43 +1 +Introduction +The Ising model has been a fruitful area of research since its discovery in 1920’s [1]. The +3d Ising universality class is realized in a number of physical systems such as 3d uni-axial +magnets [2] and liquid-vapor critical points [3]. On the theoretical side, a lot of work has +been devoted over the years to the physics of the 3d Ising model and to calculations of +its observables, such as critical exponents. A celebrated example of a successful approach +is provided by the ϵ-expansion [4]. Over the last decade, an impressive progress has been +achieved by the numerical conformal bootstrap [5–7], which fixes critical exponents and OPE +coefficients of the 3d Ising model to the greatest precision. Monte-Carlo simulations also give +very precise results for the critical exponents of the 3d Ising model (see, e.g., [8]). +Still this leaves one wondering whether a better analytical control is possible over the +3d Ising model, especially given that the 2d Ising model is exactly solvable. A particularly +intriguing set of ideas [9,10] is related to the possibility of rewriting the 3d Ising model as a +theory of (super)strings. In this description the string worldsheet corresponds to a boundary +between clusters of positive and negative spins. +In the 2d Ising model the corresponding +boundaries describe worldlines of free Majorana particles, which gives rise to an expectation +1 + +for fermionic excitations to be present on the string worldsheet in the 3d case. This idea has +been realized explicitly in the lattice phase of the Ising model [11], however, the continuum +description of the Ising strings is still missing. The corresponding string theory is expected +to be strongly coupled, however see [12] for an interesting recent proposal towards a weakly +coupled description. +Given this state of affairs it is natural to explore the structure of the Ising strings exper- +imentally, where by experiments we mean lattice Monte–Carlo simulations. For this purpose +it is convenient to use the 3d version of the Kramers–Wannier duality, which maps the low +energy ferromagnetic phase of the 3d Ising model into a confining phase of the Z2 lattice +gauge theory [13]. Under this duality, Ising domain walls are mapped into worldsheets of Z2 +confining strings. To gain insight into the worldsheet dynamics it is natural to focus on the +so-called long strings (or torelons). These are strings wrapped around one of the compact +spatial dimensions. The ground state energy and the first few lowest-lying states of Ising +strings in the long string sector have been previously studied in [14–16]. +In this work, we aim to extend these results with a more precise spectrum calculation +and to determine energies of a larger number of excited states. Excitations of closed flux +tubes wrapped around one of the spatial dimensions are characterized by their longitudinal +momentum q along the flux tube. In addition, one may also define two parity transformations. +The longitudinal parity Pl corresponds to a reflection along the string and maps q to −q. The +transverse parity Pt corresponds to a reflection in the transverse direction. The main goal of +our study is to check whether Ising strings carry massive resonant states on their worldsheet. +Our initial results seem to indicate the presence of a massive resonance in the parity (++) +sector (at q = 0). The same state is also present at the lowest non-vanishing q1. However, a +careful analysis shows that this state is a bulk glueball rather than a new worldsheet state. +Similar string spectrum computations were previously performed in the 3d U(1) gauge +theory [17] and in the 3d and 4d SU(N) Yang-Mills theories [18–21]. In these studies, massive +resonances are observed in some cases, such as for the fundamental 4d SU(N) confining string +and confining strings in higher representations. Quite surprisingly though, fundamental con- +fining strings in 3d SU(N) gluodynamics don’t show any sign of additional massive resonant +modes on the string worldsheet. +We see that Ising strings are in some sense in between these two options. On one side, +we observe a well-pronounced resonant state in the spectrum of torelon excitations. On the +other hand, this is not a new state, but rather a bulk glueball. This strong mixing between +torelon excitations and glueballs is possible due to the absence of large N suppression in the +Ising case. +The rest of the paper is organized as follows. In section 2, we review properties of the +3d Ising model and its duality to the Z2 lattice gauge theory. In section 3 we review the +basics of the effective string theory, which provides a good approximation for the lowest-lying +spectrum. In section 4 we summarize the basics of the lattice gauge theory and of the Monte- +Carlo simulations. We describe the algorithm for computing the closed flux tube spectrum, +and discuss how we reduce the systematic and statistical errors and improve the projection +1Recall that the values of q are quantized as a result of a compactification on a circle. +2 + +onto low-lying states. In section 5 we present our results for some of the basic parameters +such as the string tension and the lightest glueball mass. We present and analyze the closed +flux tube spectra in 3d Z2 gauge theory for a wide range of string lengths. We start with the +absolute ground state and continue onto excited states in different sectors. In particular, we +identify a massive resonance state that is not described by the Nambu-Goto theory. Then +we describe the checks which we performed, which indicate that the observed state is not in +fact a novel worldsheet state but rather a scattering state of a long string with an additional +unbound glueball. In section 6, we present our conclusions and discuss future directions. +2 +Ising Model and Z2 Gauge Theory +The 3d Ising model is one of the simplest spin models of (anti-)ferromagnetism. Its partition +function is given by +Z = +� +si +e +−H(si) +T +, +(1) +where the Ising Hamiltonian is given by +H(si) = −J +� +⟨i,j⟩ +sisj − h +� +i +si . +(2) +Here the first sum runs over all neighboring pairs of spins si = ±1 on a cubic lattice. In the +present paper we are interested in the Ising model with a vanishing external magnetic field +h = 0 . +Then the theory enjoys a global Z2 symmetry, which flips signs of all spins. Positive val- +ues of the coupling constant J correspond to ferromagnetism and negative ones to anti- +ferromagnetism. Indeed, for positive J the Hamiltonian is smaller for spins pointing in the +same direction making it energetically favorable for spins to be aligned. On the other hand, +thermal fluctuations tend to randomize the spins. Which effect wins depends on the temper- +ature, so the model exhibits a (second order) phase transition at a critical temperature Tc. +As a consequence of the bipartite property of the square lattice the ferromagnetic and anti- +ferromagnetic models are equivalent at h = 0. Namely, they can be mapped into each other +by taking J → −J and flipping half of the spins, which correspond to one of the sublattices. +In what follows we assume +J > 0 . +At a critical temperature T = Tc the spins develop long range correlations which are described +by a conformal field theory. At temperatures below the critical one the global Z2 symmetry +is spontaneously broken and a typical spin configuration describes clusters of positive and +negative spins separated by domain walls of positive tension. In the vicinity of the critical +temperature, +T ≲ Tc +3 + +this phase is described by a continuous gapped Ising field theory. As reviewed in the intro- +duction, it is a longstanding question whether it is possible to rewrite the Ising dynamics as a +tractable continuum string theory, where the string worldsheet describes the dynamics of the +domain walls. Our goal here is to study the structure of the Ising strings through the lattice +Monte-Carlo simulation. +To study the string dynamics it is instructive to map the Ising model into a Z2 gauge theory. +This map has been constructed by Wegner [13] and can be considered as a generalization of +the Kramers–Wannier duality of the 2d Ising model (see, e.g., [22] for a review). Unlike in the +2d Ising model which is self-dual, the duality maps the 3d Ising model into a different theory +defined by the following partition function +Zgauge(β) = +� +{σl=±1} +exp +� +β +� +□ +σ□ +� +. +(3) +Here σl variables define a Z2 gauge connection which lives on the links of the dual lattice. The +coupling constant β of the dual theory is related to the Ising model parameters via +β = −1 +2 log tanh J +T . +(4) +This Abelian gauge theory exhibits a number of properties characteristic of the non-Abelian +SU(N) Yang–Mills theory. +First, it enjoys a global 1-form Z2 center symmetry (see [23] +for a modern introduction). Similarly to the SU(N) case, upon compactification on a circle +the Z2 center symmetry is realized by (pseudo)gauge transformations with twisted boundary +conditions. A Polyakov loop operator, defined as a Wilson loop wound around the circle, +carries a negative Z2 charge. +As a result, in the phase with unbroken center symmetry +a sector with a Polyakov loop insertion is orthogonal to a trivial sector with no operators +wound around the circle. Analogously to the SU(N) case we will refer to the states created +by topologically trivial operators as glueballs. Deformed Polyakov loops acting on a vacuum +produce “long” flux tube states, which are the main target of our study. +The phase with unbroken center symmetry, which describes the confined phase of the Z2 +gauge theory, is realized at [24] +β < βc ≈ 0.7614133(22) , +where the critical value β = βc corresponds to the conformal Ising point. In addition, Ising +strings exhibit a roughening transition at [25] +β = βr = 0.47542(1) , +so we are interested in the range βr < β < βc, where the string dynamics is described by a +continuum theory in the scaling limit β → βc. +The deconfining phase transition at β = βc needs to be distinguished from the one that +happens when the circumference R of the spatial circle gets sufficiently small, namely at [14] +R = Rc ≈ 0.82ℓs , +(5) +4 + +where ℓ−2 +s +is the tension of a confining string. The latter transition corresponds to the finite +temperature deconfining phase transition of the Z2 gauge theory understood as a (2 + 1)- +dimensional quantum field theory. The parameter β is a coupling constant of this theory, +which also has an interpretation as the inverse temperature, if one understands the Z2 gauge +theory as a 3-dimensional classical statistical model. The Polyakov loop plays a role of the +order parameter for both phase transitions. +In principle, both Ising and Z2 descriptions can be used for Monte-Carlo studies of Ising +strings (see, e.g., [14–16,26] for some previous work). In the Ising description this is achieved +by introducing “interfaces”, i.e., by flipping the sign of the coupling J on the links which +intersect the string worldsheet. To study the spectrum of string excitations, which is our main +goal here, the gauge theory description appears more convenient. Indeed, in this description +excited strings states are created by deformed Polyakov loops. As reviewed in section 4 this +makes it straightforward to produce a large basis of excited states by changing the shape of +the Polyakov loop. Furthermore, a precision mass determination requires a good overlap of +the operator basis with the low lying string states. The gauge theory formulation allows this +to be achieved by the well-developed techniques of blocking and smearing. +For future reference, note that in addition to the string tension ℓ−2 +s , the Z2 gauge theory +in the confining phase has another characteristic energy scale—the inverse correlation length +ξ−1, which is set by the lightest glueball mass. Given that the parity invariant Ising model +has a single relevant deformation, in the scaling limit the ratio of the two scales is universal. +Its numerical value is [27] +ξ2 +ℓ2 +s +≈ 0.1056(19) . +(6) +3 +Effective String Theory +In the absence of additional symmetries confining strings are not expected to carry any mass- +less states on the worldsheet apart from the (D − 2) gapless translational Goldstone bosons +describing transverse oscillations of a string. Here D is the total number of space-time dimen- +sions. In particular, one expects to find a single massless mode on the worldsheet of D = 3 +Ising strings. Then the spectrum of low lying long string excitations is strongly constrained by +the non-linearly realized target space Poincar´e symmetry and can be calculated using the ef- +fective string theory (see, e.g., [28,29] for a review). Effective string theory provides a natural +reference point to be compared with the actual string spectrum, so let us briefly summarize +properties of the effective string spectrum. +The most straightforward approach for calculating the effective string theory predictions +is based on the perturbative expansion which uses the ratio ℓs/R as a small parameter. As +a consequence of the non-linearly realized Poincar´e symmetry all terms in this expansion up +to (and including) O(1/R5) are universal. This means that those terms are insensitive to the +microscopic theory as soon as no additional massless degrees of freedom are present on the +worldsheet. This universality provides a powerful self-consistency check for lattice results. On +the other hand it makes it quite challenging to probe the underlying microscopic theory by +5 + +high precision measurements of the string ground state for which the ℓs/R expansion has good +convergence properties. +Furthermore, the ℓs/R expansion exhibits poor convergence for excited string states. An +efficient technique to calculate the effective string theory predictions for these states is based +on the Thermodynamic Bethe Ansatz [30,31], which can also be reformulated as an undressing +method based on the T ¯T deformation [32]. In this approach one calculates perturbatively the +worldsheet S-matrix, and then makes use of a non-perturbative relation between the S-matrix +and the finite volume spectrum to predict the latter. This technique is a close cousin of the +familiar L¨uscher method [33] combined with the TBA method [34] for calculating the leading +order winding corrections, which is possible due to an approximate integrability of the effective +string theory. The leading order TBA string spectrum is given by +EGGRT(Nl, Nr) = +� +4π2(Nl − Nr)2 +R2 ++ R2 +ℓ4 +s ++ 4π +ℓ2 +s +� +Nl + Nr − D − 2 +12 +� +, +(7) +which is nothing but the Goddard–Goldstone–Rebbi–Thorne (GGRT) spectrum [35] of a +bosonic string in a winding sector. Here Nl and Nr are non-negative integers called levels, +which count the total left- and right-moving momenta along the string. The total longitudinal +momentum is given by +p = 2π(Nl − Nr) +R +. +(8) +In what follows it will be instructive to compare the Ising string spectrum with the GGRT +one. +Note that at D = 26 the GGRT spectum (7) coincides with the exact spectrum of +critical bosonic strings. At D ̸= 3, 26 this spectrum is not compatible with the D-dimensional +Poincar´e symmetry and should be considered as a leading order approximation only. The +D = 3 case is somewhat special, and an integrable theory of a single massless boson with the +spectrum given by (7) appears to be a consistent candidate for the worldsheet theory of a long +D = 3 string. Motivated by the lattice data, the confining string of D = 3 Yang–Mills theory +was conjectured to describe a single massless bosons, however, the corresponding spectrum +deviates from the D = 3 GGRT formula. As we will see, for the Ising string the deviations +are even more pronounced. +The GGRT states are completely characterized by the occupation numbers nl(k), nr(k), +where k is a positive integer labeling longitudinal momenta. +These string excitations are +generated by creation operators ak and a−k2. +We will denote the corresponding state as +|nl(k), nr(k)⟩, which is a shorthand notation of |nl(k), nr(k); k = 1, 2, . . . ⟩. Given such a state +its levels can be computed as +Nl = +� +k +nl(k)k, +Nr = +� +k +nr(k)k . +(9) +In what follows we will refer to effective string excitations as phonons. +For instance, the +N = ˜N = 2 GGRT level corresponds to two degenerate states. +One of these states is a +2For convenience we omit the †. Because the annihilation operators will not appear in this paper, it should +cause no confusion. +6 + +two-phonon excitation with +nl(2) = nr(2) = 1 , +and another a four-phonon excitation with +nl(1) = nr(1) = 2 , +where in both cases all other phonon occupation numbers vanish. +As discussed in the Introduction, the long string spectrum is invariant under longitudinal +and transverse parity transformations Pl and Pt. It is straightforward to determine the cor- +responding transformation properties of the GGRT states. Namely, as far as the transverse +parity is concerned, its action depends only on the total number of excitations and all GGRT +state are eigenvalues of Pt, +Pt|nl(k), nr(k)⟩ = (−1) +� +k(nl(k)+nr(k))|nl(k), nr(k)⟩ . +(10) +On the other hand, the longitudinal parity acts by exchanging the left- and right-moving +excitations, +Pl|nl(k), nr(k)⟩ = |nr(k), nl(k)⟩ . +(11) +Finally, note that in our discussion of the GGRT spectrum we implicitly set the total +transverse momentum pt to zero. By restoring the pt dependence we obtain the full set of the +GGRT states |nl(k), nr(k), pt⟩, with the energies given by the conventional relativistic formula, +E(pt) = +� +p2 +t + E(0)2 . +For convenience in Table 1 we present the states created by phonon creation operators in +different sectors with q = 0, 1 and Nl + Nr ≤ 6. We will discuss more about the quantum +numbers that define the sectors in section 4.3. +4 +Review of Lattice Techniques +4.1 +Lattice gauge theory and Monte-Carlo simulations +A general lattice gauge theory (LGT) is described by a set of fields associated with the links +of a lattice. Lattice links may be labeled by a pair (n, µ), where n labels the lattice site, and +µ is a direction. Each lattice link is then mapped to an element Uµ(n) of the gauge group. +For Z2 gauge theory these elements are simply ±1. For a cubic lattice the action is given by +S = β +� +plaq +{1 − Re(Tr Uplaq)} , +(12) +where the sum is over elementary squares (“plaquettes”) of the lattice which may be labeled +as (n, µ, ν) and +Uplaq(n, µ, ν) = Uµ(n) · Uν(n + ˆµ) · U † +µ(n + ˆν) · U † +ν(n) , +7 + +q = 0 +Nl, Nr +Pt, Pr +GGRT States +Nl = Nr = 0 +++ +|0⟩ +Nl = Nr = 1 +++ +a1a−1|0⟩ +Nl = Nr = 2 +++ +a2a−2|0⟩ +++ +a1a1a−1a−1|0⟩ +−+ +(a2a−1a−1 + a1a1a−2)|0⟩ +−− +(a2a−1a−1 − a1a1a−2)|0⟩ +Nl = Nr = 3 +++ +a3a−3|0⟩ +++ +a2a1a−2a−1|0⟩ +++ +a1a1a1a−1a−1a−1|0⟩ +++ +(a1a1a1a−3 + a3a−1a−1a−1)|0⟩ ++− +(a1a1a1a−3 − a3a−1a−1a−1)|0⟩ +−+ +(a3a−2a−1 + a2a1a−3)|0⟩ +−− +(a3a−2a−1 − a2a1a−3)|0⟩ +−+ +(a2a1a−1a−1a−1 + a1a1a1a−2a−1)|0⟩ +−− +(a2a1a−1a−1a−1 − a1a1a1a−2a−1)|0⟩ +q = 1 +Nl, Nr +Pt +GGRT States +Nl = 1, Nr = 0 +− +a1|0⟩ +Nl = 2, Nr = 1 ++ +a2a−1|0⟩ +− +a1a1a−1|0⟩ +Nl = 3, Nr = 2 ++ +a3a−2|0⟩ ++ +a2a1a−1a−1|0⟩ ++ +a1a1a1a−2|0⟩ +− +a3a−1a−1|0⟩ +− +a2a1a−2|0⟩ +− +a1a1a1a−1a−1|0⟩ +Table 1: Table with the states of the lowest GGRT levels with q = 0, 1 and Nl +Nr ≤ 6. +8 + +is an ordered product of gauge fields around a plaquette. The action (12) is gauge invariant +and can be used to generate Monte-Carlo simulations. +Periodic lattices are used in this +work. In principle, one can generate millions of configurations using Markov Chain Monte- +Carlo algorithms. After achieving thermalization, we compute statistical quantities through +importance sampling. +Different algorithms may have different thermalization speeds and +different step sizes between configurations. In this paper we only use the Metropolis algorithm. +For each lattice system, we created 200000 configurations to perform measurements, with 25 +sweeps between two measurements in order to reduce auto-correlation. +Statistical quantities calculated in this work are correlation functions +⟨φi(U)φj(U) · · · ⟩ = +� � +dUφi(U)φj(U) · · · e−S , +(13) +of gauge invariant operators φi(U)’s. Two-point correlators calculated at different times can +be used to extract the spectrum of different physical states such as glueballs and flux tubes. +The corresponding procedure is further discussed in section 4.2. +The lattice spacing a has units of length, but in numerical simulations we only deal with +numbers, so we have to choose units where everything is dimensionless. A common choice is to +use lattice units, which sets a = 1. This choice is implicitly assumed in the action expression +(12). This choice is convenient during the simulations, but the cost is that the continuum limit +becomes obscure. So it is also common to express physical observables using the units defined +by a certain characteristic energy scale of interest. In this work we are mostly interested in +confining strings, so we will use string units which set the string tension to one, ℓs = 1. +Independently of the units, the continuum limit is achieved when +a2 +ℓ2 +s +→ 0 . +(14) +Of course, in practice this is impossible to achieve on a finite lattice. At the fixed lattice size +the quality of the continuum limit is controlled by the difference between the Z2 coupling +constant β and its critical value βc = 0.7614133(22). In order to stay in the confined phase we +need to keep β < βc. Note that we cannot take the difference β − βc too small, because the +string width ℓs then becomes larger than an overall size of the lattice, making it impossible +to observe confinement. +4.2 +Extracting spectra +In this work we use the framework of [18,20,36] to measure the spectrum. Namely we construct +a set of operators φi in a sector characterized by certain quantum numbers and acting on +constant time slices3. Then a two-point correlator of two operators separated by nt lattice +units in the time direction, which corresponds to the physical time t = ant, can be written in +the following form +Cij(t) = ⟨φ† +i(t)φj(0)⟩ = +� +k +⟨v|φ† +ie−Ht|k⟩⟨k|φj|v⟩ = +� +k +cikc∗ +kje−Ekt, +(15) +3Of course, we work on an Euclidean lattice, so a choice of the “time” direction is a matter of convention. +9 + +where the sum goes over a complete set |k⟩ of energy eigenstates with the chosen quantum +numbers, |v⟩ is the absolute vacuum state and cik’s are the overlap coefficients +cik = ⟨v|φ† +i|k⟩ . +As the time separation increases, higher energy contributions decay faster and only lowest +energy states survive. It can be shown [37] that at large times the eigenvalues λa(t) of the +matrix C−1(0)C(t) are given by the spectrum, +λa(t) ≈ e−tEa, +t → ∞ , +(16) +if the basis of operators is large enough. To determine the energies in practice one first con- +structs the approximate eigenstates Φi by diagonalizing the correlation matrix C−1(0)C(t = a) +at early times, and then extracts the corresponding energy eigenvalues from the exponential +falloff of the diagonal correlation functions ⟨Φ† +i(t)Φi(0)⟩. To illustrate this procedure, let us +consider the simplest case of a single operator, which allows to determine the ground state +energy in the corresponding sector. In this case the diagonalization is trivial, so one simply +studies the correlator +⟨φ†(t)φ(0)⟩ = +� +n +|⟨v|φ|n⟩|2e−Ent → +t→∞ |⟨v|φ|0⟩|2e−E0t. +(17) +To analyze its behavior it is convenient to define an effective mass +ameff(t) = − ln +� +⟨φ†(t)φ(0)⟩ +⟨φ†(t − a)φ(0)⟩ +� +. +(18) +In the limit of an infinite statistics it decreases monotonically over time and asymptotes to +the actual ground state energy in the φ sector, +ameff(t) → +t→∞ aE0. +(19) +In practice one plots the effective mass as a function of time and extracts E0 from the position +of a plateau, which is followed by statistical fluctuations. For the ground state, the effective +mass sets an upper bound on the actual energy and it is possible to observe the plateau up +to rather late times. +A general strategy for measuring energies of excited states is similar, but the practicalities +become more and more challenging for highly excited states. Indeed, statistical noise in the +measured effective mass is an unavoidable feature of the Monte-Carlo simulations using the +importance sampling to compute correlators. The amplitude of the noise stays constant in +time, while correlators exhibit an exponential decay. +Inevitably, at large enough time tn +statistical noise becomes larger than the signal and the effective mass needs to be measured +before this happens. Correlators corresponding to heavier excited states decay faster, so that +the critical time tn is shorter for them. +Clearly, this implies that one needs to achieve a maximal possible overlap of the approxi- +mate eigenstates Φi with the true energy eigenstates, so that the plateau can be measured as +10 + +early as possible. On the other hand, given that we perform a diagonalization in an artificially +truncated finite dimensional Hilbert space, every approximate eigenstate necessarily has an +admixture of heavier states which needs to decay before the plateau can be observed. This +problem becomes more and more severe for highly excited states. +To overcome this problem one needs to maximize the projection of an approximate eigen- +state on the true energy eigenstates. This projection can be estimated by the gap between +the value of the effective mass at t = a and the plateau. Typically, for us this projection +drops below ∼ 0.5 around level Nl, Nr = 3, so we do not expect the corresponding energy +determinations to be reliable. +There are several ways to improve a quality of the plateau. First, one may try to minimize +the measured energies in lattice units. This can be achieved by choosing the values of the +parameters such that the string tension is smaller in the lattice units. In the Ising model this +can be achieved by picking the value of β close to the critical point. However, other issues arise +as one approaches the critical point. First, as one does this, one needs to take a larger lattice +to model a system of the same physical size (i.e., as measured in string units). Given that we +work on a three-dimensional lattice, the simulation time grows as a cube of the lattice size. +Also, close to the critical point, correlations between gauge field configurations created by the +Metropolis algorithm become higher. To overcome this one needs to increase the sampling +interval, which also results in a longer simulation time. All in all, a limited computing power +prevents one from approaching the critical point too closely. +The second way to reduce statistical errors is by creating a larger size of samples. This is +also limited by the computing resource. +Finally, one can improve the quality of the operators, so that the overlaps of the approxi- +mate eigenstates to the exact ones are closer to unity. This can be achieved both by starting +with a larger set of operators, and also by suppressing the overlap of the operators with the +highly energetic microscopic states using blocking and smearing techniques. We will discuss +this more in section 4.3. +4.3 +Constructing flux tube operators +In this paper we work in the confining phase of the Z2 gauge theory. Equivalently, this is +the phase with an unbroken center symmetry. Recall that given a gauge theory compacti- +fied on a circle, the center symmetry may be defined4 by making use of the “twisted gauge +transformation” generated by gauge functions satisfying +g(R) = Λg(0) , +(20) +where Λ is a center element of the gauge group. The Yang-Mills action functional is invariant +under such a transformation. However, given that the gauge function (20) is not periodic, this +transformation defines a global (rather than a gauge) symmetry of the theory. On the other +hand, any two transformations satisfying (20) with the same Λ can be related to each other by +4A modern definition of the center symmetry as a 1-form symmetry does not require to consider a com- +pactification [23]. A traditional and less general discussion presented here is enough for our purposes. +11 + +a conventional gauge transformations. Hence, after dividing out over the conventional gauge +transformations, one obtains a global symmetry transformation which is isomorphic to the +center subgroup of the gauge group. For SU(N) gauge theory it is the ZN center symmetry, +and Λ = e +2πik +N . For the Z2 gauge theory the center symmetry is Z2 itself. +This definition makes it clear that an arbitrary Wilson loop +WC = Tr +� +P exp(i +� +C +Aµ(x)dxµ) +� +, +(21) +corresponding to the contour C with a trivial winding along the chosen compact direction is +neutral under the center symmetry. Indeed, such a loop necessarily crosses any transverse slice +an equal number of times from both sides and all factors of Λ cancel out. On the other hand, +a Polyakov loop is wound around the periodic dimension, so it crosses any transverse slice in +one direction one time more than in the opposite direction. As a result, it is charged under +the center symmetry transformation. This also shows that its vacuum expectation value(vev) +plays a role of the order parameter for the center symmetry. In the confining phase Polyakov +loops have zero vev, and a long string sector is generated by acting on the vacuum by (an +arbitrarily deformed) Polyakov loop. Of course, in addition one may add also any number of +topologically trivial Wilson loops creating additional glueball states. The center symmetry +ensures that this sector does not mix with the topologically trivial one, which is generated by +the glueball operators only. +Before describing the set of operators which we used to probe long strings, let us describe +conserved quantum numbers in these sector. First, there is a longitudinal momentum p along +the flux tube. Flux tubes are wound around a circle of a circumference R, so the longitudinal +momentum is quantized +p = 2πq +R , +with q being an integer. The ground state is translationally invariant, which corresponds to +q = 0. +In addition, there are two parity transformations Pt and Pl, which we already introduced +in our discussion of the GGRT spectrum in section 3. It is straightforward to describe how +they act on the gauge theory operators, without any reference to effective strings. Let us +consider a long string winding around the x direction. Then the transverse parity is a mirror +transformation acting on the transverse y direction, +(x, y) +Pt +−→ (x, −y) . +Similarly, the longitudinal parity Pl acts as a mirror transformation of the longitudinal x- +direction, +(x, y) +Pl +−→ (−x, y) . +Note that in general the longitudinal parity does not commute with the longitudinal momen- +tum, +Pl p Pl = −p , +12 + +Figure 1: Increasing the blocking level of a link by one. +so that only q = 0 states may be simultaneous eigenstates of p and Pl. +Finally, long string states may also carry a non-vanishing transverse momentum pt. It does +not convey any useful information about the worldsheet dynamics and we will always set it +to zero by averaging over transverse positions of all operators. +Let us describe now the set of operators, which we use to probe the long string sector. The +simplest operator charged under the center symmetry associated to the compact x direction +is the straight Polyakov loop +φP(x, t) = +R/a +� +n=1 +Ux(x + na, y, t) . +(22) +where R = La is the string length. In principle, this operator can be used to measure the +ground state energy of a long flux tube. However, its overlap with the ground state of the flux +tube is quite poor. Indeed, the Polyakov loop (22) creates a string with a width of order the +lattice spacing a. On the other hand, a physical string close to its ground state is expected +to have width of order the characteristic string scale ℓs. +The overlap can be improved by applying a combination of smearing and blocking proce- +dures [38]. One starts with the usual link field, which corresponds to blocking level Nbl = 1. +Then one replaces an original link with a sign of a weighted average over the link itself and +two staples attached to it (see Fig. 1). In our simulations we chose the averaging weight to be +0.75. Finally, one constructs a twice longer link by multiplying two consecutive smeared links. +The result is what one calls a level 2 blocked link. To construct the links at Nbl-th blocking +level one applies the same procedure using the blocking level Nbl − 1 links as an input. +Using the blocked links we can now create a basis of Polyakov loop operators of different +shapes. In Fig. 3 we present the shapes used in our simulation. Note that some of these +operators look like creating a flux tube and an additional glueball rather than just a flux +tube excitation. Equivalently, using the SU(N) language, they look like multi trace opera- +tors. However, for the Z2 theory there is no sharp distinction between single trace and multi +trace operators, because any operator can be formally presented in the single trace form by +connecting different components by going back and forward along some path between them +(see Fig. 2), given the Abelian nature. +Finally, to obtain operators with a definite set of quantum numbers one performs averaging +over the action of the corresponding symmetry transformation. +For example, in order to +13 + +For an Abelian gauge group +Figure 2: For an Abelian gauge group there is no sharp distinction between string +excitations and additional glueballs. +Figure 3: The set of operators used in our simulation. +construct an operator with a definite longitudinal momentum p, one sums over all longitudinal +translations with a phase +φ(p) = +L +� +k=1 +φ(x + ak)eipak . +(23) +In the same way one constrains pt = 0 by summing over all the translations in the transverse +y direction without a phase. +Similarly, one may obtain operators with definite value of transverse and longitudinal +parities (Pt, Pl). For example, as we discussed, at p = 0 both parities can be be assigned, +so we get four different sectors (++), (+−), (−+) and (−−). To construct the corresponding +operators one starts with a Wilson line operator UC corresponding to a certain path C, and +14 + +defines the following eigenstate combination +˜UC = (UC ± UPlC) ± (UPtC ± UPtPlC) . +(24) +Here the signs inside the brackets correspond to the eigenvalue of Pl, and the sign in the +middle corresponds to the eigenvalue of Pt. +5 +Results +Let us now present results of our simulations. In this work we performed Z2 lattice gauge +theory simulations at β = 0.756321. This value corresponds to the rough and confining phase. +It is sufficiently close to the critical value βc = 0.7614133(22) [24], to allow for sufficiently long +and clear plateaux in the effective mass. Namely, as follows from the results presented later, +for this value of β the correlation length ξ (which is set by the inverse mass of the lightest +glueball ξ = m−1 +G ) is equal to +ξ = 4.631(8)a . +Unless specified otherwise, the results presented are obtained on lattices of a size +l⊥ = lt = 70a , +in the transverse and time directions, and the lattice size along the string is varied in the +range +R ∈ [20a, 80a] , +which corresponds to the range +R ∈ [1.38ℓs, 5.53ℓs] , +in string units, where the string length is obtained by fitting the absolute ground state energy +of the flux tube to the GGRT formula. Recall that the finite temperature deconfinement +transition corresponds to R ∼ 0.82ℓs. In order to estimate finite volume corrections and for +some other checks we also used lattices with other transverse sizes in the range from 55a to +300a. These values of lattice parameters and the corresponding basic physical observables are +summarized in Table 2. +β +βc +R/a +Rc/a +a/ℓs +amG +0.756321 +0.7614133(22) +[20,80] +∼ 11.8 +0.0691(1) +0.2159(4) +Table 2: Basic parameters of our simulation: the value of the coupling and its critical +value, the range of the string circumference and its critical value, the string tension and +the lightest glueball mass. +Let us now present results of simulations with these parameters. We start with the absolute +flux tube ground state, and continue to excited states in different sectors. Comparing the result +15 + +to the GGRT spectrum we find that the most pronounced qualitative difference is the presence +of an extra state in the parity (++) sector at q = 0. This state can naturally be interpreted +as a massive scalar resonance on the string worldsheet. We identify the corresponding state +also in the q = 1 sector. Later we present results of an additional dedicated analysis which +indicates that this resonance is actually caused by the bulk glueball rather than by a genuine +worldsheet state. +5.1 +The absolute ground state and the string tension +The flux tube ground state is translationally invariant, has q = 0, and belongs to the (++) +sector. Understandably, of all the string states this one is the most straightforward to identify. +As illustrated in Fig. 4, the corresponding effective mass exhibits a well pronounced plateau +even for the longest string circumference R = 80a considered in our simulations, which allows +for a high precision determination of the ground state energy as a function of R. very well +In Fig. 5 we present the ground state energy as a function of circumference R. The solid +line shows the GGRT ground state energy. These results are plotted in string units with the +string length parameter ℓs determined by fitting the data to the GGRT ground state energy. +For the ℓs extraction we used the data in the range R ∈ [25a, 80a], where the quality of the +GGRT fit is the best. The resulting value of ℓs in lattice units is presented in Table 2. We +observe that the GGRT approximation reproduces very well the ground state energy of the +Ising string all the way down to R ∼ 1.4ℓs. On the other hand, the measured ground state +energy significantly deviates from the GGRT formula at shorter values of R. In particular, +the GGRT ground state energy vanishes at R ≈ 1.02ℓs, while the Ising ground state energy +stays positive (and approximately linear) down to a smaller critical value given by (5). +To quantify the agreement of the measured ground state energy with the GGRT approxi- +mation, we also fitted the observed energies at the short string regime [1.4ℓs, 2.8ℓs] using the +following ansatz +E0(R) = EGGRT(R) + cγ +ℓs +�ℓs +R +�γ +, +(25) +for different values of γ and using the string length ℓs and the coefficient cγ as the fitting +parameters. To interpret the results it is instructive to compare the obtained values of cγ with +the corresponding coefficients of the ℓs/R expansion of the GGRT ground state energy itself, +E0(R) = R +ℓ2 +s +− π +6 +1 +R − π2 +72 +ℓ2 +s +R3 − π3 +432 +ℓ4 +s +R5 + O(ℓ6 +s) , +(26) +where we listed all the universal terms in the ℓs/R expansion. For γ = 1, the best fit value of +c1 is negligible compared to the value of the corresponding term in (26) +c1 ≈ 0.024(13) ≪ π +6 ≈ 0.52 , +so we conclude that our results provide a quite precise determination of the first universal +term in the ℓs/R expansion (also known as the L¨uscher term). On the other hand for γ = 3 +16 + +0 +2 +4 +6 +8 +10 +12 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +aE(t) +t/a +Figure 4: The effective mass computed as in formula (18) as a function of time for the +absolute ground states at string circumference R/a = 20, 40, 60, 80, represented as blue, +yellow, green and red dots. The horizontal solid lines are the resulting fitted values +of the state’s energies. The shaded bands represent the corresponding 1σ uncertainty +intervals. +17 + +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +Eℓs +R/ℓs +Figure 5: The absolute ground state energy at different string lengths in string units. +The solid line is the GGRT approximation for the ground state energy. +we obtain +c3 ≈ 0.074(28) ≲ π2 +72 ≈ 0.14 , +so that our results are consistent with the 1/R3 universal term, but cannot be considered as +a high precision test of the universality at this order. +As an additional crosscheck of our simulation we also determined the mass of the lightest +glueball mG. When expressed in string units it reads +mG ≈ 3.124(10)ℓ−1 +s , +(27) +which agrees well with earlier measurements (cf. (6)). +It is instructive to take a look at the ground state energy for even shorter strings: R ≲ 1.2ℓs, +as also shown in Figure 5. Here one observes a large deviation from the GGRT formula. Clearly +in this regime the ℓs/R expansion does not converge, so that it cannot be used to measure +the perturbative non-universal corrections to the GGRT formula. It is worth noting that +these data do seem to extrapolate towards the deconfining point and exhibit scaling behavior, +which indicates that it is a second order phase transition. According to the Svetitsky-Yaffe +conjecture [39], this deconfining transition is described by the 2d Ising universality class, of +which the scaling behavior is linear +E0(R) +R→Rc +∝ +(R − Rc) . +(28) +From our measurements, it is plausible to be linear. But to really determine the exponent, we +need results of higher precision and more data points. One difficulty around the critical point +is that the ground state energy goes to zero, so that a larger lattice is needed to perform its +accurate determination. +18 + +5.2 +Glueball States +As a cross-check for our results, we also calculated the low-lying spectrum of Z2 glueballs +in the 0+ sector, which is summarized in Table 3. Here we can observe the finite volume +corrections for low-lying glueball states. +For example, for the lightest glueball, the finite +volume correction becomes observable for R ≤ 30a. One may also wonder whether we can +observe the state corresponding to two parallel flux tubes, which also has the same quantum +numbers. It has the mass of two ground state flux tubes. We do not observe such a state here, +which indicates that the local operators we use for glueball states have poor overlap on these +states. Comparing our results with that in [27], our measurements have higher precision, and +they agree well. The largest deviation is found for the second excited state, for which our +mass is somewhat lower, but still within a 2σ interval. +(ly/a) × (lt/a) +lx/a +aE; 0+ +70 × 70 +25 +0.1978(28) +0.2531(91) +0.3519(75)* +30 +0.2075(30) +0.2992(87) +0.3726(110)* +35 +0.2163(16) +0.3635(51) +0.4528(117)* +40 +0.2170(15) +0.3804(81) +0.5097(95) +45 +0.2144(17) +0.3896(54) +0.5329(100) +47 +0.2118(14) +0.3865(96) +0.5319(115) +50 +0.2159(20) +0.3742(62) +0.5019(166) +52 +0.2131(22) +0.3920(52) +0.5237(93) +54 +0.2182(12) +0.3899(89) +0.5141(93) +55 +0.2141(20) +0.3990(40) +0.5326(108) +56 +0.2169(18) +0.3953(44) +0.5462(59) +58 +0.2152(22) +0.3849(66) +0.4947(158) +60 +0.2178(20) +0.3998(52) +0.5080(154) +65 +0.2138(20) +0.3906(64) +0.5153(182) +70 +0.2168(11) +0.3984(61) +0.5497(67) +75 +0.2159(17) +0.4025(44) +0.5541(66) +80 +0.2175(17) +0.3886(73) +0.5216(144) +Fitted masses +0.2159(4) +0.3937(16) +0.5359(27) +Table 3: The spectrum of Z2 glueballs in the 0+ sector at β = 0.756321 for different +lattice sizes. +5.3 +Excited states +Let us now present our results for the excited state’s energies of the Ising string. We start +with zero momentum states, q = 0. As discussed before, these states split into four subsectors +19 + +with different transverse and longitudinal parities, +(Pt, Pl) = (++), (+−), (−+), (−−) . +(29) +In Fig. 6 we presented the energy differences between the first three excited states in the +(++) sector and the ground state energy. As we will see later, restricting to these three states +somewhat oversimplifies the overall picture. Nevertheless, it provides a good strating point +for interpreting our results. The numerical values of the corresponding energies (and also of +higher excited states) can be found in Table 4 in the Appendix. In addition to two levels, +which are naturally associated with the (1, 1) and (2, 2) GGRT states5, we observe on this +plot an additional level, which is not associated with any of the GGRT states. Given that the +energy gap between this exotic level and the absolute ground state is approximately constant +over the large range of R, it is natural to associate this state with a massive (++) resonance +on a string worldsheet. The resonance mass can be estimated by fitting the energy gap to a +constant, which results in +mℓs = 3.825(50) , +(30) +where we performed the fit at the intermediate values of string circumference, R/ℓs ∈ [2.4, 4.2] +to reduce possible effects related to level crossing and winding corrections. The latter can be +incorporated by applying the TBA technique (cf. [31]); we will present results of this analysis +in a separate publication. +There are two subtleties worth mentioning here. First, the resonance exhibits two level +crossings with the GGRT states in the range of R covered by our data. Namely, it crosses +the (1, 1) level at R ∼ 2ℓs , and the (2, 2) level at R ∼ 5ℓs. In the GGRT spectrum the +(2, 2) level corresponds to two degenerate states—a two-phonon and a four-phonon states. By +inspecting Table 4 one indeed observes two nearly degenerate states close to the (2, 2) level at +R ≲ 4ℓs. However, one of these states disappears as one approaches the second level crossing +at R ≳ 4ℓs. The explanation for this is not clear at this point. As follows from the data +presented in Table 4, the energy of the second (2, 2) GGRT state starts to increase away from +the GGRT spectrum at around R ≳ 3.8ℓs. As we will see later the (++) resonance is actually +a glueball state mixed with the flux tube. It is possible that these large deviations from the +GGRT formula appear above the glueball threshold, due to interactions between the unbound +glueball and the flux tube. +The second subtlety, which is likely related to the first one, is that the energy gap (30) is +larger than the mass of the lightest glueball (27) in the infinite volume theory. This implies +that (30) is not a strictly localized worldsheet state, but rather a metastable bound state +between a flux tube and a glueball. In particular, in addition to decaying into a two-phonon +flux tube excitation it may also decay into a flux tube and a glueball state. Note that the +Ising model does not have a parameter which would suppress mixing between genuine flux +tube excitations and flux tube states with additional glueballs. This is different from the +Yang–Mills case, where such a mixing is suppressed in the ’t Hooft large-N limit. As a result, +one may doubt whether the state (30) is really due to intrinsic worldsheet dynamics. Perhaps, +5In the following, for convenience we denote the GGRT levels of states in the format (Nl, Nr). +20 + +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +6 +∆Eℓs +R/ℓs +Figure 6: Energy differences with the ground state for q = 0 excited states in the (++) +parity sector as a function of string circumference at different string lengths. Blue curves +are the (1, 1) and (2, 2) GGRT levels. The red horizontal line is the fitted resonance mass. +this state should be considered instead as an admixture of the flux tube and an unbound bulk +glueball. On the other hand, our basis of operators was designed to have a good overlap with +states localized in the vicinity of the flux tube, so a priori one could expect that it is not +sensitive to the states with additional unbound glueballs. +We performed several checks to clarify the proper interpretation of this state. First, if the +exotic state (30) were due to an additional unbound glueball, then one would expect to find +a state with similar properties also in the (−+) sector. Indeed, in infinite volume adding a +glueball to a flux tube ground state leads to a continuum of states labeled by the asymptotic +transverse momentum. +In a finite volume this continuum turns into a “discretuum”. +In +the absence of interactions between the flux tube and the glueball this discretuum would +correspond to the ground state (++) and a series of degenerate doublets with (++) and (−+) +parities. However, the interaction with the flux tube breaks the degeneracy, so one obtains a +series of alternating (++) and (−+) eigenstates. +Furthermore, energies of all these states, possibly apart from the lowest one, have a rather +strong dependence on the transverse size l⊥, due to the momentum quantization. This depen- +dence may be used to distinguish between strongly bound flux tube excitations and unbound +states from the discretuum. +To probe these states, one may enlarge the set of operators in Fig. 3 by adding operators +which are expected to have a good overlap with unbound flux tube/glueball states, to see +21 + +whether additional states indeed appear. +We will describe the results of this analysis in +section 5.5. As we will see there, our overall conclusion is that the state (30) should indeed +be interpreted as a state with an additional unbound glueball. +Let us turn now to excited states in other sectors. For the q = 0 (+−) sector the effective +string theory predicts that the lowest energy state appears at the (3, 3) GGRT level and +corresponds to a Pl odd linear combination of nl(3) = 1, nr(1) = 3 and nr(3) = 1, nl(1) = 3 +states. Indeed, our analysis does not reveal any low lying states in this sector. We provide the +measured energies of the lightest (+−) state in Table 5 in the Appendix. At R/ℓs ≳ 4 these +energies are in between the (3, 3) and (4, 4) GGRT levels and become significantly heavier at +shorter R. Given how heavy these states are we expect that their energy determinations are +likely to be subject to significant systematic uncertainties. The only robust conclusion one +can draw from these results at the moment is that no anomalous light states appear in this +sector. +Let us discuss now Pt odd states, which are the states with an odd number of phonons. +For both (−+) and (−−) sectors the lowest GGRT states appear at the (2, 2) level, and they +correspond to even and odd linear combinations of nl(2) = 1, nr(1) = 2 and nr(2) = 1, +nl(1) = 2 states. We plot the measured energies of the lightest states in these sectors in +Fig. 7, and present the numerical values of these energies and those of the heavier states +in Tables 6, 7 in the Appendix. We observe that at R ≳ 4ℓs these two states are nearly +degenerate, as expected for the GGRT spectrum. In this range of R their energies are quite +close to the expected (2, 2) GGRT value, with a minor systematic disagreement. It is most +likely due to an overestimate of these rather heavy energies due to an admixture of higher +excited states. +At R ≲ 4ℓs the two states are split, and this splitting becomes very large at R ≲ 3ℓs, +mostly due to a rather dramatic increase in the energy of the (−−) state. Interestingly, the +energy of the lightest (+−) states discussed earlier exhibits a similar feature in the same range +of R. At the moment it is hard to tell what is the cause of this effect. Note that, as discussed +in a similar context in [32] for the SU(N) data from [19], the splitting between three-phonon +(−+) and (−−) cannot be explained by a correction to the two-phonon phase shift. Instead, it +is indicative of a strong inelastic multi-phonon scattering. Interestingly, this splitting appears +to be much more dramatic in the Ising case as compared to the SU(N) flux tubes. +Finally, let us discuss states with nonzero longitudinal momentum q = 1, which are plotted +in Fig. 8 and tabulated in Tables 8, 9. The ground state in this sector, which is parity odd, +agrees exceptionally well with the GGRT (1, 0) prediction. This is expected, given that the +(1, 0) GGRT state corresponds to adding an essentially free (modulo winding corrections) +phonon to the ground state of a flux tube. The first excited parity odd state also agrees very +well with the (2, 1) GGRT level. +To interpret the two lowest energy parity even q = 1 states it is instructive to compare +their energies to the (2, 1) GGRT level and also to the free approximation for the energy of +the boosted resonance state, +∆E = +� +m2 + p2 , +(31) +where p = 2π +R . We observe from Fig. 8 that the two low lying states naturally correspond to +22 + +2 +3 +4 +5 +4 +6 +8 +10 +∆Eℓs +R/ℓs +Figure 7: Energy differences with the ground state for q = 0 excited states in the (−+) +(blue dots) and (−−) (brown dots) parity sectors as a function of string circumference +at different string lengths. The blue curve is the energy of the (2, 2) GGRT level. +a level crossing between the (2, 1) GGRT level and a boosted resonance state. +To illustrate how statistical fluctuations influence our results, especially for higher level +states, it is instructive to take a look at the effective mass plateaux behaviour for different +states and at the corresponding effective mass fits. In Fig. 4 we plotted the effective mass as +a function of time separation for the absolute ground states at different string lengths. As +expected, we see that as the string length increases, which corresponds to the heavier ground +state energy, statistical fluctuations become larger and the uncertainty in the effective mass +determination grows. A generic behavior observed for each of the states is that the effective +mass exhibits a drop at early times and then stabilizes on a plateau. The rate of the initial +drop characterizes the quality of the overlap of our operator basis onto the corresponding +state. Statistical fluctuations increase at larger with time and dominate the measurement at +late times. +All these features are even more pronounced for excited states as illustrated in Fig. 9. Here +we chose the string length such that the non-universal corrections to the GGRT spectrum is +small, and at the same time the resonant state is also well pronounced. As compared to the +ground state we observe that statistical fluctuations start to dominate the plateau at earlier +times. At the energy of around 0.67a−1, which corresponds to the second excited state in the +parity (−+) sector at R = 60a, this effect reaches the point when the position of the plateau +is hard to determine. Also, because statistical fluctuations here dominate so early, they are +23 + +2 +3 +4 +5 +5 +6 +7 +8 +9 +10 +∆Eℓs +R/ℓs +Figure 8: Energy differences with the ground state for q = 1 excited states in the (+) +(blue dots) and (−) (brown dots) parity sectors as a function of string circumference at +different string lengths. Blue curves show energies of the (1, 0), (2, 1) and (3, 2) GGRT +levels. A red curve shows an estimate for the resonance state using the resonance mass +(30). +likely to prevent us from observing the point of the plateau stabilization, leading to a possible +overestimation of the energy. Consequently, a reliable spectrum calculation in this energy +range requires a larger sample size. +5.4 +Finite volume corrections +Let us discuss the finite size dependence of the presented results. To be more precise, in our +simulation we have a finite size lattice system with periodic boundary conditions: R × l⊥ × lt. +The main goal of the simulation is to measure the dependence of string energy levels on the +longitudinal size R. Instead, in this section we will discuss the sensitivity of the presented +results to l⊥ and lt. Our goal is twofold. On the one hand the (in)sensitivity of the measured +string energy levels to l⊥ and lt provide a consistency check for the extrapolation of the +measured energy levels to infinite volume. On the other hand, as was already mentioned, +the scattering states containing additional glueball(s) states are expected to exhibit a strong +dependence on l⊥, which can be used to probe the nature of a massive resonance state observed +in the (++) sector. +In more detail, the spatial finite volume dependence of a single particle or string state +24 + +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +* +0 +2 +4 +6 +8 +10 +12 +0.0 +0.2 +0.4 +0.6 +0.8 +aE(t) +t/a +Figure 9: The effective mass computed as in formula (18) as a function of time, for the +first, second and third excited states in the q = 0 (++) sector and compactification +length R = 40a, represented as blue, yellow, green dots, and for the ground state, first +and second excited states in the q = 0 (−+) sector and compactification length R = 60a, +represented as blue, yellow and green “∗”. The horizontal solid lines in dark colors are +the fitted value of the mass of the corresponding states. The shaded bands in light colors +represents ±1 standard deviations. +25 + +with zero momentum in the transverse direction is related to winding corrections associated +to (virtual) particles propagating around the spatial circle. For massive states, which is always +the case for us6, these corrections are of order O(e−caml⊥), where the constant c depends on +the theory [40]. These corrections are exponentially suppressed, so as we take the transverse +size to be moderately large, it will disappear very quickly. +The story is similar for corrections associated to the finite size of the temporal circle. To +partially account for these corrections we used exponents associated with both directions in +time to fit the two-point correlators instead of a single exponential as written in (17). This +still neglects all the time evolutions that wind around the time circle for more than one round, +but these effects are further exponentially suppressed. +Clearly, winding corrections are most prominent for the lightest states. In particular, l⊥ +and lt need to be sufficiently large for a high precision determination of the low lying string +states at small R. +For multiparticle scattering states there are larger finite volume corrections that go like +O(1/(ml⊥)). These are associated with finite momenta of individual particles in a multiparticle +state. In particular, the infinite volume energy spectrum of multiparticle states is continuous. +Instead, in a finite spatial volume one expects to find a discretuum of states which becomes +more and more dense as the lattice size increases. +To probe the size of finite volume effects in our results we performed simulations at dif- +ferent lattices and compare the corresponding energy spectra. We do not find a significant +dependence of the measured flux tube spectrum on the temporal lattice size, as follows from +the data summarized in Appendix A. These data describe low-lying flux tube spectra mea- +sured on 40 × 55 × 55 and 40 × 55 × 70 lattices. The difference is well within error bars. So +in what follows we fix lt = 70a, where the time windings can be safely ignored. +Let us discuss now a set of plots illustrating how energies of low-lying states depend +on the transverse size. We do not discuss states in the (+−) sector because their energy +determinations are not very reliable due to large statistical uncertainties. In this section we +fix the size of the longitudinal direction to R = 40a = 2.77ls. +The transverse size dependence of the (++) states is illustrated in Fig. 10. Blue, yellow, +green and red dots are natural to identify with the GGRT states. They match the correspond- +ing GGRT energies fairly well, and do not exhibit strong finite volume dependence. This is +also true for the resonance state, which is represented by brown dots. However, there is an +extra state represented by purple dots, which exhibits a very pronounced volume dependence +at smaller values of l⊥. As follows from our earlier discussion, this volume dependence suggests +that this state belongs to a discretuum of scattering states describing a string with an addi- +tional glueball with non-vanishing relative momentum. This suggests that also the resonance +state should be zero relative momentum at the bottom of the string-glueball discretuum rather +than a genuine string excitation. In the next section we present further evidence supporting +this conclusion. +6Note that what matters here is the mass of a string as a whole as it move in the transverse direction. +This should not be confused with the mass of longitudinal string excitations, which is of course zero for the +Goldstone modes. +26 + +0.05 +0.10 +0.15 +0.20 +0.25 +0 +2 +4 +6 +8 +10 +E/√σ +1/l⊥ +√σ +Figure 10: Energies in the q = 0 (++) sector at R = 40a = 2.76ls as a function of the +inverse transverse size. Horizontal lines of different colors represent the GGRT spectrum +starting with N = ˜N = 0. The brown dashed line represents the resonance mass. +The transverse size dependence of the (−+) states is illustrated in Fig. 11. These states +are quite a bit heavier than the lightest ones observed in the (++) sector and it is harder +to interpret what happens here. It looks natural to associate blue and yellow dots with the +proper string excitations. Their agreement with the GGRT predictions is not so good, and +the lightest (blue) state appears to exhibit some volume dependence at the small values of +the transverse size l⊥ +√σ ≲ 4.5. In any case, one also observes two additional states (green +and red) which exhibit a very pronounced volume dependence. As in the (++) case this is +suggestive of the scattering states interpretation. +The transverse size dependence of (−−) states is presented in Fig. 12. +There are no +recognizable scattering states among the low-lying states with Eℓs ≲ 10. This is expected. +Indeed to construct a Pl = − scattering states one can either take a Pl = − flux tube or +glueball state, or consider a state where both flux tube and a glueball carry a non-vanishing +longitudinal momentum. In all cases the resulting state is expected to be quite heavy. +For completeness we also presented the transverse volume dependence of q = 1 states in +Figs. 13, 14. The corresponding scattering states can be obtained by boosting a glueball in +the q = 0 states, so these states can be used as consistency check. We expect to find scattering +states for both Pt = + and Pt = − sectors among q = 1 states. These states with strong finite +27 + +0.05 +0.10 +0.15 +0.20 +0.25 +6 +7 +8 +9 +10 +E/√σ +1/l⊥ +√σ +Figure 11: Energies in the q = 0 (−+) sector at R = 40a = 2.76ls as a function of the +inverse transverse size. Horizontal lines of different colors represent the GGRT spectrum +starting with N = ˜N = 2. +volume dependence are indeed present and represented by purple dots in Fig. 13 and by red +dots in Fig. 14. The green dots in Fig. 13 represent a resonance state, which can be plausibly +reinterpreted as string-glueball discretuum with zero relative momentum. +We conclude that for the coupling β = 0.756321, which we use, a lattice with lt = l⊥ = 70a +is large enough to ignore finite size effects for the GGRT states at the current level of precision +at values of R which is not too close to the deconfining value Rc = 0.82ℓs. +We do see strong finite volume corrections associated both with lt and l⊥ dependence as +we approach the deconfinement transition Rc = 0.82ℓs. A much larger lattice size is needed +to perform accurate measurements in the vicinity of that point. Also, we see evidence for the +existence of the flux tube-glueball scattering states at large transverse size for both values of +the transverse parity Pt. This indicates that our set of operators have a sizable overlap with +these states and calls for a more rigorous look on the nature of the massive state in the (++) +sector. This will be the goal of the next section. +28 + +0.05 +0.10 +0.15 +0.20 +0.25 +8 +9 +10 +11 +12 +E/√σ +1/l⊥ +√σ +Figure 12: Energies in the q = 0 (−−) sector at R = 40a = 2.76ls as a function of inverse +transverse size. Horizontal lines of different colors represent the GGRT spectrum starting +with N = ˜N = 2. +29 + +0.05 +0.10 +0.15 +0.20 +0.25 +7 +8 +9 +10 +E/√σ +1/l⊥ +√σ +Figure 13: Energies in the q = 1 (+) sector at R = 40a = 2.76ls as a function of the +inverse transverse size. Horizontal lines of different colors represent the GGRT spectrum +starting from N = 2, ˜N = 1. +30 + +0.05 +0.10 +0.15 +0.20 +0.25 +0 +2 +4 +6 +8 +10 +E/√σ +1/l⊥ +√σ +Figure 14: Energies in the q = 1 (−) sector at R = 40a = 2.76ls as a function of the +inverse transverse size. Horizontal lines of different colors represent the GGRT spectrum +starting from N = 1, ˜N = 0. +31 + +5.5 +Including multitrace operators +A sizable mixing between flux tube and scattering states is an interesting peculiarity of the +Ising model, not present in the non-Abelian Yang–Mills theory. In the Yang–Mills case the +scattering states are created by multitrace operators whose overlap on the flux tube states +produced by single trace operators is suppressed even at moderately large number of colors N. +As discussed before, in the Ising case there is no distinction between multitrace and single trace +operators. We just saw, this leads to a substantial overlap of our operator basis (which was +intended to create pure flux tube states) on the scattering states. On the other hand, this basis +is definitely not very well suited for an accurate identification and separation of the scattering +states, because one still expects that the corresponding overlap is somewhat suppressed as +a consequence of locality. Hence, it should be instructive to enlarge the operator basis by +introducing additional operators with a good overlap on the scattering states. This will allow +us to better probe the nature of the (++) resonance and to confirm its interpretation as a zero +momentum scattering state. The additional (pseudo) multi trace operators can be constructed +by considering a product of (smeared and blocked) plaquette operators φG producing glueball +states with the straight Polyakov loop (22), +φscattering = +l⊥/a +� +n,m=1 +φP(y + na)φG(y + ma)e +2πiq⊥(n−m)a +l⊥ +. +(32) +The double sum in (32) is needed to project on a state with a vanishing total transverse mo- +mentum, which is also characterized by a relative momentum q⊥7. We include such operators +with q⊥ = 0, 1, 2, 3, 4 and Pt = ± (q⊥ = 0 state only appears in the Pt = + sector). On the +other hand, for these operators Pl = + because this holds for the φG and φP that we use, and +no relative longitudinal momentum is introduced. +We now repeat the analysis of the transverse volume dependence of the spectrum using +this extended basis of operators. This should allow a more thorough determination of the +low-lying spectrum including also the discretuum of scattering states. If the (++) resonance +is a genuine string state, one expects to find two low-lying massive states that don’t receive +pronounced finite volume corrections. +One of these states would then correspond to the +lowest lying glueball scattering state and another to the string excitation (which can also be +interpreted as a bound state of a string and a glueball). +The results for the (++) sector are presented in Fig. 15. Here we chose the compactification +radius R = 55a to ensure that the lowest scattering state is well separated from the GGRT +states. We clearly see that beneath the (2, 2) GGRT level, there is only one non-GGRT state +(represented by cyan dots) whose energy exhibits only a moderate dependence on a transverse +size. In addition, there is a series of non-GGRT states with a strong volume dependence +(represented by yellow, brown, purple and mauve-blue dots) which become very dense at +large transverse size and accumulate around the expected threshold for the continuum of the +scattering states. It is natural to interpret these levels as fluxtube-glueball scattering states +7Note that q⊥ is only an approximate quantum number. +32 + +0.05 +0.10 +0.15 +0.20 +0.25 +0 +2 +4 +6 +8 +10 +E/√σ +1/l⊥ +√σ +Figure 15: Energies in the q = 0 (++) sector at R = 55a = 3.80ls as a function of inverse +transverse size determined using an extended operator basis. Horizontal solid lines of +different colors represent the GGRT spectrum starting from N = ˜N = 0. The lower +dashed blue line represents the energy of the absolute ground state plus the glueball mass. +The upper dashed blue line represents the absolute ground state plus the resonance mass +as given by (30). +33 + +with q⊥ = 1, 2, 3, 4 and the level represented by the cyan dots as a q⊥ = 0 state at the bottom +of the continuum. +Interestingly, this candidate q⊥ = 0 state still exhibits a noticeable transverse size depen- +dence in the range of ℓ⊥ presented in Fig. 15. The corresponding energy gap at the shortest +values of ℓ⊥ is significantly higher than the glueball mass. This is indicative of a considerable +repulsive interaction between the glueball and the flux tube. +These interactions appear to be important also for the states with non-zero relative mo- +mentum q. In particular, a priori one could have expected that the ℓ⊥ dependence of the +corresponding energies can be captured by the free dispersion relation, +E = +� +m2 +flux + p2 +⊥ + +� +m2 +glue + p2 +⊥ , +(33) +with p⊥ = 2πq⊥/ℓ⊥. However, we find that this ansatz does not provide a very good fit, +indicative of considerable interactions with the flux tube. These interactions are expected +also to affect the GGRT states above the continuum threshold. This may actually resolve one +of the puzzles encountered earlier. Namely, one expects to find two states at the (2, 2) GGRT +level. However, only one such state is present in Fig. 15 (the one labeled by green dots). This +phenomenon is also observed in Table 4, where we find out that one of the (2, 2) GGRT states +start to deviate from GGRT spectrum at R ≳ 3.8ℓs. It appears that a strong mixing between +the GGRT and scattering states may provide an explantation for this effect. +We observe similar new states with a strong volume dependence also in the (−+) sector. +The corresponding flux tube spectrum as a function of the transverse size is presented in +Fig. 16. +Here blue and orange dots are plausible candidates for GGRT (2, 2) and (3, 3) +states given that their volume dependence is relatively mild. In addition we find four states +with a strong and monotonic volume dependence, which makes them natural candidates for +q⊥ = 1, 2, 3, 4 states in the discretuum. There is no analogue of the q⊥ = 0 state in this sector. +As in the (++) sector the complicated pattern of the corresponding energies, suggests that a +considerable mixing between flux tube and glueball states is present. +To summarize, we believe that the analysis presented here strongly disfavors the existence +of light massive excitations on the worldsheet of the Z2 confining flux tube. In particular, the +state which appears as a massive resonance in the (++) sector corresponds to the glueball +scattering state. In addition, our results indicate the presence of a significant mixing between +flux tube excitations and scattering glueball states. +6 +Concluding Remarks +To summarize, we calculated the low-lying spectrum of closed flux tube excitations up to the +N = ˜N = 3 GGRT level in the Z2 gauge theory, at a coupling β = 0.756321 which is close to +the critical point βc = 0.7614133(22) [24]. The compactification radius covers a wide range +1.38ℓs ≤ R ≤ 5.53ℓs from moderately short strings to very long ones, but still above the +deconfinement transition at Rc ∼ 0.82ℓs [41]. The resulting spectrum agrees with the GGRT +predictions for N = ˜N ≤ 1 states within most of the range of R, and also for N = ˜N = 2 +states for moderately long strings. +34 + +0.05 +0.10 +0.15 +0.20 +0.25 +7 +8 +9 +10 +E/√σ +1/l⊥ +√σ +Figure 16: Energies in the q = 0 (−+) sector at R = 55a = 3.80ls as a function of the +inverse transverse size determined using an extended operator basis. Horizontal solid +lines of different colors represent the GGRT spectrum starting from N = ˜N = 2. The +dashed green line represents the energy of absolute ground state plus glueball mass. +. +35 + +Somewhat surprisingly, our analysis did not reveal any massive excitations on the world- +sheet of the Ising string. A heuristic argument suggesting the presence of a resonance is based +on realizing the critical Ising model as an IR fixed point of the φ4 theory. Then one may +attempt to study the properties of the Ising strings by analyzing domain walls in a mass- +deformed φ4 theory. Even though this approach is not based on a well-controlled perturbative +expansion at d = 3, it was argued [42] to provide a decent approximation to the ratio of the +lighest glueball mass to the string tension. A domain wall in φ4 theory does support a massive +localized resonance [43], so based on this logic one might have expected to find one also in +the Ising case. It will be interesting to study what happens to this resonance using a more +systematic approach, based on the ϵ-expansion rather than a direct study of the d = 3 φ4 +model. +We did observe a state in the 0++ sector which has an appearance of a massive reso- +nance. However, a detailed analysis revealed that this is a multitrace scattering state with an +additional glueball rather than a genuine flux tube excitation. This is related to another in- +teresting (and expected) aspect of the observed spectra. Namely, they indicate the presence of +a significant mode mixing between string excitations and glueball scattering states related to +the repulsive glueball/string interaction. It will be interesting to perform an analytical anal- +ysis of these spectra using an appropriate generalization of the TBA method and to extract +the scattering amplitudes describing glueball/string interactions. It will also be interesting +to connect this data to the properties of the line defect in the Ising model at the conformal +point, which has been studied in [44]. +Another possible direction to extend this work is to study the dynamics of strings in the +ZN gauge theory. In particular, it will be interesting to study how the 3d U(1) gauge theory +(studied, e.g., in [45–47]) is recovered in the N → ∞ limit. It is natural to expect that strong +glueball/string interactions should be present in this whole family of theories. +Acknowledgements. +We thank Ofer Aharony, Victor Gorbenko, Michele Caselle, Nabil +Iqbal and Yifan Wang for fruitful discussions. This work is supported in part by the NSF grant +PHY-2210349, by the BSF grant 2018068 and by the Simons Collaboration on Confinement +and QCD Strings. The work of CL is partly supported by funding resources from NYU physics +department, and the simulation is run on NYU Greene cluster. +A +Compilation of energy spectra +In this appendix we list all the closed flux tube spectra we’ve computed for the Z2 gauge theory +at the coupling β = 0.756321, with different lattice sizes and different quantum numbers. The +convention for denoting sectors follows (29). +36 + +R/a +l⊥ × lt/a2 +aE(R) ; q = 0 (++) +20 +70 × 70 +0.0668(8) +0.2384(46) +0.3460(49) +0.4893(47) +0.4882(69) +0.4996(199)* +0.6339(109) +25 +0.0966(11) +0.3022(50) +0.3664(58) +0.4873(87) +0.5895(290) +0.5649(175) +0.5338(138) +30 +0.1211(17) +0.3251(65) +0.3917(82) +0.5151(65) +0.5884(62) +0.4929(117) +0.5838(164)* +35 +0.1506(13) +0.3456(134) +0.4167(126) +0.5460(60) +0.5479(135) +0.5542(117) +0.5416(161) +40 +0.1785(14) +0.3766(60) +0.4318(124) +0.5439(80) +0.5123(161) +0.6392(117) +0.5854(80)* +45 +0.2037(19) +0.3827(97) +0.4444(208) +0.5436(87) +0.5533(242) +0.6279(130) +0.5464(177)* +47 +0.2143(12) +0.3969(35) +0.4737(101) +0.5448(69) +0.5596(147) +0.6539(64) +0.6010(69) +50 +0.2255(34) +0.4002(58) +0.4997(108) +0.5599(52) +0.5850(154) +0.6507(96) +0.6282(109) +52 +0.2339(19) +0.4118(65) +0.5009(92) +0.5634(59) +0.5813(198) +0.6407(139) +0.6327(67)* +54 +0.2492(27) +0.4239(61) +0.5285(66) +0.5537(92) +0.6113(122) +0.6650(62) +0.6441(57)* +55 +0.2500(29) +0.4106(100) +0.4715(280) +0.5600(61) +0.5612(242) +0.6571(100) +0.6259(72) +56 +0.2571(26) +0.4214(66) +0.5043(117) +0.5583(45) +0.6008(169) +0.6671(46)* +58 +0.2686(19) +0.4409(33) +0.5278(106) +0.5609(70) +0.6136(139) +0.6375(113)* +60 +0.2819(29) +0.4404(72) +0.5412(138) +0.5701(75) +0.6386(207) +0.6543(88) +0.7048(96) +65 +0.3085(31) +0.4640(60) +0.5683(116) +0.5850(83) +0.6663(122) +0.6567(137)* +0.6680(199)* +70 +0.3238(45) +0.4678(61) +0.5612(149) +0.5935(85) +0.6718(202)* +0.6483(144)* +0.6858(174)* +75 +0.3586(38) +0.5012(68) +0.6167(96) +0.6058(77) +0.7401(114) +0.6921(121)* +0.7561(118)* +80 +0.3745(64) +0.5093(75) +0.6069(132) +0.6197(157) +0.7463(152)* +0.7849(126)* +40 +55 × 55 +0.1731(19) +0.3514(57) +0.4407(90) +0.5443(86) +0.5715(177) +0.5694(118) +0.6240(146) +40 +55 × 70 +0.1766(14) +0.3678(43) +0.4517(82) +0.5497(72) +0.5847(159) +0.5800(81) +40 +65 × 70 +0.1772(17) +0.3662(70) +0.4162(133) +0.5506(69) +0.5665(136) +0.6653(87) +0.5415(176) +40 +80 × 70 +0.1780(17) +0.3817(50) +0.4540(130) +0.5556(40) +0.4818(108) +0.6378(96) +0.6385(98) +40 +160 × 70 +0.1768(15) +0.3772(44) +0.4660(61) +0.5607(41) +0.4972(94) +0.6469(57) +0.5515(104)* +40 +300 × 70 +0.1770(13) +0.3767(21) +0.4557(63) +0.5449(48) +0.4928(96) +0.6332(122) +0.5611(103)* +Table 4: The energies, E(R), of the lightest seven flux tube states with length R in the +sector q = 0 (++). +37 + +R/a +l⊥ × lt/a2 +aE(R) ; q = 0 (+−) +20 +70 × 70 +0.9337(259) +25 +0.8790(60) +30 +0.8055(221) +35 +0.7678(83) +40 +0.7155(172) +45 +0.7316(80)* +47 +0.7392(99)* +50 +0.7520(115) +52 +0.7487(105) +54 +0.6905(208) +55 +0.7501(123)* +56 +0.7272(85) +58 +0.7644(55)* +60 +0.7189(112)* +65 +0.7264(132)* +70 +0.7528(53)* +75 +0.7401(159)* +80 +0.7242(126)* +40 +55 × 55 +0.7458(198) +40 +55 × 70 +0.7513(183) +40 +65 × 70 +0.7518(79) +40 +80 × 70 +0.7478(80) +40 +160 × 70 +0.7215(185) +40 +300 × 70 +0.7389(79) +Table 5: The energies, E(R), of the lightest flux tube state with length R in the sector +q = 0 (+−). +38 + +R/a +l⊥ × lt/a2 +aE(R) ; q = 0 (−+) +20 +70 × 70 +0.4497(43) +0.6023(116) +0.6516(268) +0.9297(72) +25 +0.4572(88) +0.5693(87) +0.7823(110) +0.8336(357) +30 +0.4634(65) +0.5525(107) +0.7294(197) +0.7506(230)* +35 +0.4784(45) +0.5851(41) +0.7281(41) +0.7736(329) +40 +0.4686(65) +0.5666(73) +0.7019(137) +0.7166(222)* +45 +0.4925(54) +0.5682(177) +0.6753(100) +0.7935(123) +47 +0.5095(46) +0.5961(98) +0.6698(121) +0.7487(241)* +50 +0.5261(52) +0.5991(87) +0.6866(71) +0.7826(102) +52 +0.5364(59) +0.6139(58)* +0.6904(88) +0.7770(115) +54 +0.5310(64) +0.6063(100)* +0.6897(69) +0.8138(53) +55 +0.5405(56) +0.6393(63) +0.6994(91) +0.7418(258)* +56 +0.5561(57) +0.6405(58) +0.6923(84) +0.7685(137)* +58 +0.5467(39) +0.6314(62) +0.6979(83) +0.7007(83) +60 +0.5717(44) +0.6331(78) +0.6604(148)* +0.8186(61) +65 +0.5795(60) +0.6652(71)* +0.6964(108)* +70 +0.6059(40) +0.7006(330) +0.7084(104)* +75 +0.6259(38) +0.6874(218)* +0.7141(98)* +80 +0.6177(125) +0.7305(118)* +0.7384(73)* +40 +55 × 55 +0.5278(42) +0.6653(115) +0.7093(101) +40 +55 × 70 +0.5308(62) +0.6988(136) +0.7007(83) +40 +65 × 70 +0.5052(53) +0.5939(80) +0.7149(90) +0.7571(208) +40 +80 × 70 +0.4801(83) +0.5430(71) +0.7136(66) +0.6882(161) +40 +160 × 70 +0.4848(53) +0.4733(55) +0.5400(80)* +0.7147(142) +0.5930(135)* +40 +300 × 70 +0.4836(32) +0.4694(82) +0.5325(92)* +0.6608(146) +Table 6: The energies, E(R), of the lightest four flux tube states (for 40 × 160 × 70 it is +five) with length R in the sector q = 0 (−+). +39 + +R/a +l⊥ × lt/a2 +aE(R) ; q = 0 (−−) +20 +70 × 70 +0.7911(92) +0.8396(220) +0.8715(63) +25 +0.6850(68) +0.7588(50) +0.7775(99) +30 +0.6349(27) +0.7458(84) +0.9011(484) +35 +0.6008(74) +0.6935(170) +40 +0.5763(71) +0.6459(125) +0.6780(171) +45 +0.5709(35) +0.6772(161) +0.7331(108) +47 +0.5615(41) +0.6669(127) +0.7235(166) +50 +0.5664(64) +0.6833(121) +0.7018(168)* +52 +0.5707(69) +0.6595(131) +0.7488(163) +54 +0.5704(51) +0.6867(87) +0.7153(130)* +55 +0.5784(56) +0.6894(150) +0.7056(159)* +56 +0.5827(51) +0.6697(186) +0.7709(76)* +58 +0.5731(61) +0.7214(104) +0.7735(71)* +60 +0.5838(62) +0.6984(169) +0.8113(54)* +65 +0.5877(70) +0.7686(62) +0.7783(365)* +70 +0.6121(77) +0.7115(210)* +75 +0.6286(72) +0.6914(222)* +80 +0.6424(80) +0.7357(326)* +40 +55 × 55 +0.5763(50) +0.6136(100) +0.7731(133) +40 +55 × 70 +0.5597(112) +0.6315(82) +0.7454(217) +40 +65 × 70 +0.5768(47) +0.6218(129) +0.6967(191) +40 +80 × 70 +0.5706(131) +0.6211(192) +0.7599(241) +40 +160 × 70 +0.5805(47) +0.6568(103) +0.7938(270) +40 +300 × 70 +0.5875(34) +0.6772(116) +0.8248(115) +Table 7: The energies, E(R), of the lightest three flux tube states with length R in the +sector q = 0 (−−). +40 + +R/a +l⊥ × lt/a2 +aE(R) ; q = 1 (+) +20 +70 × 70 +0.4899(45) +0.5612(88) +0.6680(73) +0.6693(192) +0.8066(195) +25 +0.4722(37) +0.5236(92) +0.6141(132) +0.6804(73) +0.7865(307) +30 +0.4644(29) +0.4919(111) +0.6029(64) +0.6480(153) +0.7514(132) +35 +0.4585(41) +0.5169(85) +0.5594(197) +0.6343(83) +0.7561(34) +0.7460(141) +40 +0.4778(37) +0.4953(95) +0.5904(106) +0.6632(94) +0.6684(45) +0.7484(61) +45 +0.4820(34) +0.5359(51) +0.6110(73) +0.6414(69) +0.6353(95) +0.7416(71) +47 +0.4801(32) +0.5323(65) +0.6249(42) +0.6377(44) +0.6633(53) +0.7538(74) +50 +0.4919(29) +0.5509(61) +0.6414(48) +0.6300(35) +0.6583(131) +0.7675(98) +52 +0.4898(41) +0.5536(75) +0.6269(80) +0.6236(78) +0.6246(110) +0.7402(138) +54 +0.4996(41) +0.5856(62) +0.6359(64) +0.6267(51) +0.6441(232) +0.7397(84) +55 +0.4938(35) +0.5478(98) +0.6345(95) +0.6167(105) +0.6546(156)* +0.7532(84) +56 +0.5016(51) +0.5732(75) +0.6255(70) +0.6550(48) +0.6540(58)* +58 +0.5071(37) +0.5524(118) +0.6527(44) +0.6361(57) +0.6942(82) +0.7587(69) +60 +0.5228(36) +0.5882(79) +0.6614(143) +0.6439(66) +0.6799(72)* +65 +0.5337(57) +0.5905(93) +0.6771(77) +0.6370(103)* +70 +0.5452(44) +0.6166(94) +0.6734(97)* +75 +0.5595(72) +0.6462(121) +0.7031(59)* +80 +0.5865(47) +0.6628(157) +0.6883(73)* +40 +55 × 55 +0.4621(31) +0.5323(61) +0.6214(79) +0.6969(58) +0.6851(124) +0.7858(88) +40 +55 × 70 +0.4615(53) +0.5312(56) +0.6243(29) +0.6811(99) +0.6838(75) +0.7738(57) +40 +65 × 70 +0.4724(25) +0.5049(53) +0.6065(52) +0.6560(93) +0.6999(53) +0.7679(71) +40 +80 × 70 +0.4745(33) +0.5076(48) +0.5712(97) +0.5977(127) +0.6489(77) +0.6982(99) +40 +160 × 70 +0.4737(27) +0.5317(77) +0.6076(64) +0.5532(106)* +0.6793(85) +40 +300 × 70 +0.4701(26) +0.5394(46) +0.6040(62) +0.5587(79)* +0.6958(42) +0.6950(121) +Table 8: The energies, E(R), of the lightest six flux tube states with length R in the +sector q = 1 (+). +41 + +R/a +l⊥ × lt/a2 +aE(R) ; q = 1 (−) +20 +70 × 70 +0.4025(14) +0.5260(73) +0.6537(90) +0.6300(52) +0.6813(202) +0.7666(70) +25 +0.3621(16) +0.4946(58) +0.6232(78) +0.6326(65) +0.6182(84) +0.7189(66) +30 +0.3448(16) +0.4861(50) +0.5896(73) +0.5737(180) +0.6070(159) +0.6968(108) +35 +0.3382(19) +0.4886(37) +0.5600(89) +0.6152(71) +0.5959(140)* +0.6828(114) +40 +0.3403(14) +0.4855(52) +0.5741(107) +0.6100(70) +0.6154(54) +0.7201(63) +45 +0.3467(23) +0.4857(53) +0.5562(93) +0.6007(69) +0.6247(107) +0.6813(228) +47 +0.3524(21) +0.4953(31) +0.5891(51) +0.6019(50) +0.6478(58) +0.6982(81) +50 +0.3577(24) +0.4911(59) +0.5963(74) +0.5999(59) +0.6490(71) +0.6966(58) +52 +0.3665(23) +0.5012(58) +0.6073(41) +0.6179(61) +0.6777(79) +0.6595(143)* +54 +0.3700(17) +0.5075(33) +0.6171(53) +0.6229(55) +0.6671(73) +0.6819(52) +55 +0.3681(35) +0.5033(59) +0.6040(70) +0.6162(47) +0.6852(65) +0.7166(110)* +56 +0.3741(24) +0.5116(38) +0.6230(35)* +0.6196(68) +0.6727(58) +0.6947(70) +58 +0.3840(22) +0.5163(41) +0.6030(72) +0.6296(59) +0.6812(64) +0.6810(68)* +60 +0.3899(20) +0.5221(54) +0.5940(135) +0.6431(77) +0.6850(56) +0.6740(113)* +65 +0.4058(23) +0.5346(49) +0.6342(53) +0.6606(68) +0.6921(83) +0.6810(170)* +70 +0.4230(26) +0.5555(52) +0.6418(222) +0.6895(86) +0.7063(61)* +75 +0.4357(42) +0.5623(97) +0.6660(86) +0.7014(84)* +80 +0.4572(45) +0.5701(84) +0.7048(69) +0.7088(93)* +40 +55 × 55 +0.3424(17) +0.4991(64) +0.6044(67) +0.6363(98) +0.6713(104) +0.7215(61) +40 +55 × 70 +0.3429(14) +0.4994(45) +0.6061(59) +0.6152(137) +0.6743(124) +0.6909(95) +40 +65 × 70 +0.3420(19) +0.4876(47) +0.5947(123) +0.6052(27) +0.6154(93) +0.7075(118) +40 +80 × 70 +0.3375(23) +0.4890(31) +0.5483(89) +0.6091(53) +0.6414(87) +0.7307(51) +40 +160 × 70 +0.3426(11) +0.4855(31) +0.5294(73) +0.5975(59) +0.5975(32) +0.7095(98) +40 +300 × 70 +0.3375(17) +0.4843(34) +0.5386(47) +0.5863(45) +0.5760(66) +0.7030(81) +Table 9: The energies, E(R), of the lightest six flux tube states with length R in the +sector q = 1 (−). +42 + +References +[1] E. 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Vadacchino, “Width of the flux tube in compact U(1) +gauge theory in three dimensions,” JHEP 02 (2016) 180, 1601.07455. +46 + diff --git a/3dAyT4oBgHgl3EQfP_Yp/content/tmp_files/load_file.txt b/3dAyT4oBgHgl3EQfP_Yp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..540b7900d0fd4197cf9e5ed5587359b8801a862c --- /dev/null +++ b/3dAyT4oBgHgl3EQfP_Yp/content/tmp_files/load_file.txt @@ -0,0 +1,1909 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf,len=1908 +page_content='Excitations of Ising Strings on a Lattice Andreas Athenodoroua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Sergei Dubovskyc,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 10003,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' USA dRudolf Peierls Centre for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Clarendon Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Parks Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Oxford OX1 3PU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' UK and All Souls College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' High Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Oxford OX1 4AL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' UK Abstract The 3d Ising model in the low temperature (ferromagnetic) phase describes dynam- ics of two-dimensional surfaces—domain walls between clusters of parallel spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The Kramers–Wannier duality maps these surfaces into worldsheets of confining strings in the Wegner’s Z2 gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We study the excitation spectrum of long Ising strings by simulating the Z2 gauge theory on a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We observe a strong mixing between string excitations and the lightest glueball state and do not find indications for light massive resonances on the string worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='00034v1 [hep-lat] 30 Dec 2022 Contents 1 Introduction 1 2 Ising Model and Z2 Gauge Theory 3 3 Effective String Theory 5 4 Review of Lattice Techniques 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='1 Lattice gauge theory and Monte-Carlo simulations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 32 6 Concluding Remarks 34 A Compilation of energy spectra 36 References 43 1 Introduction The Ising model has been a fruitful area of research since its discovery in 1920’s [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The 3d Ising universality class is realized in a number of physical systems such as 3d uni-axial magnets [2] and liquid-vapor critical points [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the theoretical side, a lot of work has been devoted over the years to the physics of the 3d Ising model and to calculations of its observables, such as critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A celebrated example of a successful approach is provided by the ϵ-expansion [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Over the last decade, an impressive progress has been achieved by the numerical conformal bootstrap [5–7], which fixes critical exponents and OPE coefficients of the 3d Ising model to the greatest precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Monte-Carlo simulations also give very precise results for the critical exponents of the 3d Ising model (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=', [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Still this leaves one wondering whether a better analytical control is possible over the 3d Ising model, especially given that the 2d Ising model is exactly solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A particularly intriguing set of ideas [9,10] is related to the possibility of rewriting the 3d Ising model as a theory of (super)strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this description the string worldsheet corresponds to a boundary between clusters of positive and negative spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the 2d Ising model the corresponding boundaries describe worldlines of free Majorana particles, which gives rise to an expectation 1 for fermionic excitations to be present on the string worldsheet in the 3d case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This idea has been realized explicitly in the lattice phase of the Ising model [11], however, the continuum description of the Ising strings is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The corresponding string theory is expected to be strongly coupled, however see [12] for an interesting recent proposal towards a weakly coupled description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Given this state of affairs it is natural to explore the structure of the Ising strings exper- imentally, where by experiments we mean lattice Monte–Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For this purpose it is convenient to use the 3d version of the Kramers–Wannier duality, which maps the low energy ferromagnetic phase of the 3d Ising model into a confining phase of the Z2 lattice gauge theory [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Under this duality, Ising domain walls are mapped into worldsheets of Z2 confining strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To gain insight into the worldsheet dynamics it is natural to focus on the so-called long strings (or torelons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These are strings wrapped around one of the compact spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The ground state energy and the first few lowest-lying states of Ising strings in the long string sector have been previously studied in [14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this work, we aim to extend these results with a more precise spectrum calculation and to determine energies of a larger number of excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Excitations of closed flux tubes wrapped around one of the spatial dimensions are characterized by their longitudinal momentum q along the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In addition, one may also define two parity transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The longitudinal parity Pl corresponds to a reflection along the string and maps q to −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The transverse parity Pt corresponds to a reflection in the transverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The main goal of our study is to check whether Ising strings carry massive resonant states on their worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Our initial results seem to indicate the presence of a massive resonance in the parity (++) sector (at q = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The same state is also present at the lowest non-vanishing q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, a careful analysis shows that this state is a bulk glueball rather than a new worldsheet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Similar string spectrum computations were previously performed in the 3d U(1) gauge theory [17] and in the 3d and 4d SU(N) Yang-Mills theories [18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In these studies, massive resonances are observed in some cases, such as for the fundamental 4d SU(N) confining string and confining strings in higher representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Quite surprisingly though, fundamental con- fining strings in 3d SU(N) gluodynamics don’t show any sign of additional massive resonant modes on the string worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We see that Ising strings are in some sense in between these two options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On one side, we observe a well-pronounced resonant state in the spectrum of torelon excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, this is not a new state, but rather a bulk glueball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This strong mixing between torelon excitations and glueballs is possible due to the absence of large N suppression in the Ising case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In section 2, we review properties of the 3d Ising model and its duality to the Z2 lattice gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In section 3 we review the basics of the effective string theory, which provides a good approximation for the lowest-lying spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In section 4 we summarize the basics of the lattice gauge theory and of the Monte- Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We describe the algorithm for computing the closed flux tube spectrum, and discuss how we reduce the systematic and statistical errors and improve the projection 1Recall that the values of q are quantized as a result of a compactification on a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 2 onto low-lying states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In section 5 we present our results for some of the basic parameters such as the string tension and the lightest glueball mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We present and analyze the closed flux tube spectra in 3d Z2 gauge theory for a wide range of string lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We start with the absolute ground state and continue onto excited states in different sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, we identify a massive resonance state that is not described by the Nambu-Goto theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Then we describe the checks which we performed, which indicate that the observed state is not in fact a novel worldsheet state but rather a scattering state of a long string with an additional unbound glueball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In section 6, we present our conclusions and discuss future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 2 Ising Model and Z2 Gauge Theory The 3d Ising model is one of the simplest spin models of (anti-)ferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Its partition function is given by Z = � si e −H(si) T , (1) where the Ising Hamiltonian is given by H(si) = −J � ⟨i,j⟩ sisj − h � i si .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (2) Here the first sum runs over all neighboring pairs of spins si = ±1 on a cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the present paper we are interested in the Ising model with a vanishing external magnetic field h = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Then the theory enjoys a global Z2 symmetry, which flips signs of all spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Positive val- ues of the coupling constant J correspond to ferromagnetism and negative ones to anti- ferromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Indeed, for positive J the Hamiltonian is smaller for spins pointing in the same direction making it energetically favorable for spins to be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, thermal fluctuations tend to randomize the spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Which effect wins depends on the temper- ature, so the model exhibits a (second order) phase transition at a critical temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As a consequence of the bipartite property of the square lattice the ferromagnetic and anti- ferromagnetic models are equivalent at h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Namely, they can be mapped into each other by taking J → −J and flipping half of the spins, which correspond to one of the sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In what follows we assume J > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' At a critical temperature T = Tc the spins develop long range correlations which are described by a conformal field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' At temperatures below the critical one the global Z2 symmetry is spontaneously broken and a typical spin configuration describes clusters of positive and negative spins separated by domain walls of positive tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the vicinity of the critical temperature, T ≲ Tc 3 this phase is described by a continuous gapped Ising field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As reviewed in the intro- duction, it is a longstanding question whether it is possible to rewrite the Ising dynamics as a tractable continuum string theory, where the string worldsheet describes the dynamics of the domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Our goal here is to study the structure of the Ising strings through the lattice Monte-Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To study the string dynamics it is instructive to map the Ising model into a Z2 gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This map has been constructed by Wegner [13] and can be considered as a generalization of the Kramers–Wannier duality of the 2d Ising model (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=', [22] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Unlike in the 2d Ising model which is self-dual, the duality maps the 3d Ising model into a different theory defined by the following partition function Zgauge(β) = � {σl=±1} exp � β � □ σ□ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (3) Here σl variables define a Z2 gauge connection which lives on the links of the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The coupling constant β of the dual theory is related to the Ising model parameters via β = −1 2 log tanh J T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (4) This Abelian gauge theory exhibits a number of properties characteristic of the non-Abelian SU(N) Yang–Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' First, it enjoys a global 1-form Z2 center symmetry (see [23] for a modern introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Similarly to the SU(N) case, upon compactification on a circle the Z2 center symmetry is realized by (pseudo)gauge transformations with twisted boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A Polyakov loop operator, defined as a Wilson loop wound around the circle, carries a negative Z2 charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As a result, in the phase with unbroken center symmetry a sector with a Polyakov loop insertion is orthogonal to a trivial sector with no operators wound around the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Analogously to the SU(N) case we will refer to the states created by topologically trivial operators as glueballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Deformed Polyakov loops acting on a vacuum produce “long” flux tube states, which are the main target of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The phase with unbroken center symmetry, which describes the confined phase of the Z2 gauge theory, is realized at [24] β < βc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='7614133(22) , where the critical value β = βc corresponds to the conformal Ising point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In addition, Ising strings exhibit a roughening transition at [25] β = βr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='47542(1) , so we are interested in the range βr < β < βc, where the string dynamics is described by a continuum theory in the scaling limit β → βc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The deconfining phase transition at β = βc needs to be distinguished from the one that happens when the circumference R of the spatial circle gets sufficiently small, namely at [14] R = Rc ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='82ℓs , (5) 4 where ℓ−2 s is the tension of a confining string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The latter transition corresponds to the finite temperature deconfining phase transition of the Z2 gauge theory understood as a (2 + 1)- dimensional quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The parameter β is a coupling constant of this theory, which also has an interpretation as the inverse temperature, if one understands the Z2 gauge theory as a 3-dimensional classical statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The Polyakov loop plays a role of the order parameter for both phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In principle, both Ising and Z2 descriptions can be used for Monte-Carlo studies of Ising strings (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=', [14–16,26] for some previous work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the Ising description this is achieved by introducing “interfaces”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=', by flipping the sign of the coupling J on the links which intersect the string worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To study the spectrum of string excitations, which is our main goal here, the gauge theory description appears more convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Indeed, in this description excited strings states are created by deformed Polyakov loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As reviewed in section 4 this makes it straightforward to produce a large basis of excited states by changing the shape of the Polyakov loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Furthermore, a precision mass determination requires a good overlap of the operator basis with the low lying string states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The gauge theory formulation allows this to be achieved by the well-developed techniques of blocking and smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For future reference, note that in addition to the string tension ℓ−2 s , the Z2 gauge theory in the confining phase has another characteristic energy scale—the inverse correlation length ξ−1, which is set by the lightest glueball mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Given that the parity invariant Ising model has a single relevant deformation, in the scaling limit the ratio of the two scales is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Its numerical value is [27] ξ2 ℓ2 s ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='1056(19) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (6) 3 Effective String Theory In the absence of additional symmetries confining strings are not expected to carry any mass- less states on the worldsheet apart from the (D − 2) gapless translational Goldstone bosons describing transverse oscillations of a string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Here D is the total number of space-time dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, one expects to find a single massless mode on the worldsheet of D = 3 Ising strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Then the spectrum of low lying long string excitations is strongly constrained by the non-linearly realized target space Poincar´e symmetry and can be calculated using the ef- fective string theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=', [28,29] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Effective string theory provides a natural reference point to be compared with the actual string spectrum, so let us briefly summarize properties of the effective string spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The most straightforward approach for calculating the effective string theory predictions is based on the perturbative expansion which uses the ratio ℓs/R as a small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As a consequence of the non-linearly realized Poincar´e symmetry all terms in this expansion up to (and including) O(1/R5) are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This means that those terms are insensitive to the microscopic theory as soon as no additional massless degrees of freedom are present on the worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This universality provides a powerful self-consistency check for lattice results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand it makes it quite challenging to probe the underlying microscopic theory by 5 high precision measurements of the string ground state for which the ℓs/R expansion has good convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Furthermore, the ℓs/R expansion exhibits poor convergence for excited string states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' An efficient technique to calculate the effective string theory predictions for these states is based on the Thermodynamic Bethe Ansatz [30,31], which can also be reformulated as an undressing method based on the T ¯T deformation [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this approach one calculates perturbatively the worldsheet S-matrix, and then makes use of a non-perturbative relation between the S-matrix and the finite volume spectrum to predict the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This technique is a close cousin of the familiar L¨uscher method [33] combined with the TBA method [34] for calculating the leading order winding corrections, which is possible due to an approximate integrability of the effective string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The leading order TBA string spectrum is given by EGGRT(Nl, Nr) = � 4π2(Nl − Nr)2 R2 + R2 ℓ4 s + 4π ℓ2 s � Nl + Nr − D − 2 12 � , (7) which is nothing but the Goddard–Goldstone–Rebbi–Thorne (GGRT) spectrum [35] of a bosonic string in a winding sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Here Nl and Nr are non-negative integers called levels, which count the total left- and right-moving momenta along the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The total longitudinal momentum is given by p = 2π(Nl − Nr) R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (8) In what follows it will be instructive to compare the Ising string spectrum with the GGRT one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Note that at D = 26 the GGRT spectum (7) coincides with the exact spectrum of critical bosonic strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' At D ̸= 3, 26 this spectrum is not compatible with the D-dimensional Poincar´e symmetry and should be considered as a leading order approximation only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The D = 3 case is somewhat special, and an integrable theory of a single massless boson with the spectrum given by (7) appears to be a consistent candidate for the worldsheet theory of a long D = 3 string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Motivated by the lattice data, the confining string of D = 3 Yang–Mills theory was conjectured to describe a single massless bosons, however, the corresponding spectrum deviates from the D = 3 GGRT formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As we will see, for the Ising string the deviations are even more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The GGRT states are completely characterized by the occupation numbers nl(k), nr(k), where k is a positive integer labeling longitudinal momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These string excitations are generated by creation operators ak and a−k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We will denote the corresponding state as |nl(k), nr(k)⟩, which is a shorthand notation of |nl(k), nr(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Given such a state its levels can be computed as Nl = � k nl(k)k, Nr = � k nr(k)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (9) In what follows we will refer to effective string excitations as phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For instance, the N = ˜N = 2 GGRT level corresponds to two degenerate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' One of these states is a 2For convenience we omit the †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Because the annihilation operators will not appear in this paper, it should cause no confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 6 two-phonon excitation with nl(2) = nr(2) = 1 , and another a four-phonon excitation with nl(1) = nr(1) = 2 , where in both cases all other phonon occupation numbers vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As discussed in the Introduction, the long string spectrum is invariant under longitudinal and transverse parity transformations Pl and Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is straightforward to determine the cor- responding transformation properties of the GGRT states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Namely, as far as the transverse parity is concerned, its action depends only on the total number of excitations and all GGRT state are eigenvalues of Pt, Pt|nl(k), nr(k)⟩ = (−1) � k(nl(k)+nr(k))|nl(k), nr(k)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (10) On the other hand, the longitudinal parity acts by exchanging the left- and right-moving excitations, Pl|nl(k), nr(k)⟩ = |nr(k), nl(k)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (11) Finally, note that in our discussion of the GGRT spectrum we implicitly set the total transverse momentum pt to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' By restoring the pt dependence we obtain the full set of the GGRT states |nl(k), nr(k), pt⟩, with the energies given by the conventional relativistic formula, E(pt) = � p2 t + E(0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For convenience in Table 1 we present the states created by phonon creation operators in different sectors with q = 0, 1 and Nl + Nr ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We will discuss more about the quantum numbers that define the sectors in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 4 Review of Lattice Techniques 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='1 Lattice gauge theory and Monte-Carlo simulations A general lattice gauge theory (LGT) is described by a set of fields associated with the links of a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Lattice links may be labeled by a pair (n, µ), where n labels the lattice site, and µ is a direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Each lattice link is then mapped to an element Uµ(n) of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For Z2 gauge theory these elements are simply ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For a cubic lattice the action is given by S = β � plaq {1 − Re(Tr Uplaq)} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (12) where the sum is over elementary squares (“plaquettes”) of the lattice which may be labeled as (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' ν) and Uplaq(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' ν) = Uµ(n) · Uν(n + ˆµ) · U † µ(n + ˆν) · U † ν(n) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 7 q = 0 Nl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Nr Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Pr GGRT States Nl = Nr = 0 ++ |0⟩ Nl = Nr = 1 ++ a1a−1|0⟩ Nl = Nr = 2 ++ a2a−2|0⟩ ++ a1a1a−1a−1|0⟩ −+ (a2a−1a−1 + a1a1a−2)|0⟩ −− (a2a−1a−1 − a1a1a−2)|0⟩ Nl = Nr = 3 ++ a3a−3|0⟩ ++ a2a1a−2a−1|0⟩ ++ a1a1a1a−1a−1a−1|0⟩ ++ (a1a1a1a−3 + a3a−1a−1a−1)|0⟩ +− (a1a1a1a−3 − a3a−1a−1a−1)|0⟩ −+ (a3a−2a−1 + a2a1a−3)|0⟩ −− (a3a−2a−1 − a2a1a−3)|0⟩ −+ (a2a1a−1a−1a−1 + a1a1a1a−2a−1)|0⟩ −− (a2a1a−1a−1a−1 − a1a1a1a−2a−1)|0⟩ q = 1 Nl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Nr Pt GGRT States Nl = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Nr = 0 − a1|0⟩ Nl = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Nr = 1 + a2a−1|0⟩ − a1a1a−1|0⟩ Nl = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Nr = 2 + a3a−2|0⟩ + a2a1a−1a−1|0⟩ + a1a1a1a−2|0⟩ − a3a−1a−1|0⟩ − a2a1a−2|0⟩ − a1a1a1a−1a−1|0⟩ Table 1: Table with the states of the lowest GGRT levels with q = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 1 and Nl +Nr ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 8 is an ordered product of gauge fields around a plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The action (12) is gauge invariant and can be used to generate Monte-Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Periodic lattices are used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In principle, one can generate millions of configurations using Markov Chain Monte- Carlo algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' After achieving thermalization, we compute statistical quantities through importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Different algorithms may have different thermalization speeds and different step sizes between configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this paper we only use the Metropolis algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For each lattice system, we created 200000 configurations to perform measurements, with 25 sweeps between two measurements in order to reduce auto-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Statistical quantities calculated in this work are correlation functions ⟨φi(U)φj(U) · · · ⟩ = � � dUφi(U)φj(U) · · · e−S , (13) of gauge invariant operators φi(U)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Two-point correlators calculated at different times can be used to extract the spectrum of different physical states such as glueballs and flux tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The corresponding procedure is further discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The lattice spacing a has units of length, but in numerical simulations we only deal with numbers, so we have to choose units where everything is dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A common choice is to use lattice units, which sets a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This choice is implicitly assumed in the action expression (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This choice is convenient during the simulations, but the cost is that the continuum limit becomes obscure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' So it is also common to express physical observables using the units defined by a certain characteristic energy scale of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this work we are mostly interested in confining strings, so we will use string units which set the string tension to one, ℓs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Independently of the units, the continuum limit is achieved when a2 ℓ2 s → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (14) Of course, in practice this is impossible to achieve on a finite lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' At the fixed lattice size the quality of the continuum limit is controlled by the difference between the Z2 coupling constant β and its critical value βc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='7614133(22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In order to stay in the confined phase we need to keep β < βc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Note that we cannot take the difference β − βc too small, because the string width ℓs then becomes larger than an overall size of the lattice, making it impossible to observe confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2 Extracting spectra In this work we use the framework of [18,20,36] to measure the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Namely we construct a set of operators φi in a sector characterized by certain quantum numbers and acting on constant time slices3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Then a two-point correlator of two operators separated by nt lattice units in the time direction, which corresponds to the physical time t = ant, can be written in the following form Cij(t) = ⟨φ† i(t)φj(0)⟩ = � k ⟨v|φ† ie−Ht|k⟩⟨k|φj|v⟩ = � k cikc∗ kje−Ekt, (15) 3Of course, we work on an Euclidean lattice, so a choice of the “time” direction is a matter of convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 9 where the sum goes over a complete set |k⟩ of energy eigenstates with the chosen quantum numbers, |v⟩ is the absolute vacuum state and cik’s are the overlap coefficients cik = ⟨v|φ† i|k⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As the time separation increases, higher energy contributions decay faster and only lowest energy states survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It can be shown [37] that at large times the eigenvalues λa(t) of the matrix C−1(0)C(t) are given by the spectrum, λa(t) ≈ e−tEa, t → ∞ , (16) if the basis of operators is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To determine the energies in practice one first con- structs the approximate eigenstates Φi by diagonalizing the correlation matrix C−1(0)C(t = a) at early times, and then extracts the corresponding energy eigenvalues from the exponential falloff of the diagonal correlation functions ⟨Φ† i(t)Φi(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To illustrate this procedure, let us consider the simplest case of a single operator, which allows to determine the ground state energy in the corresponding sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this case the diagonalization is trivial, so one simply studies the correlator ⟨φ†(t)φ(0)⟩ = � n |⟨v|φ|n⟩|2e−Ent → t→∞ |⟨v|φ|0⟩|2e−E0t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (17) To analyze its behavior it is convenient to define an effective mass ameff(t) = − ln � ⟨φ†(t)φ(0)⟩ ⟨φ†(t − a)φ(0)⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (18) In the limit of an infinite statistics it decreases monotonically over time and asymptotes to the actual ground state energy in the φ sector, ameff(t) → t→∞ aE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (19) In practice one plots the effective mass as a function of time and extracts E0 from the position of a plateau, which is followed by statistical fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For the ground state, the effective mass sets an upper bound on the actual energy and it is possible to observe the plateau up to rather late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A general strategy for measuring energies of excited states is similar, but the practicalities become more and more challenging for highly excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Indeed, statistical noise in the measured effective mass is an unavoidable feature of the Monte-Carlo simulations using the importance sampling to compute correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The amplitude of the noise stays constant in time, while correlators exhibit an exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Inevitably, at large enough time tn statistical noise becomes larger than the signal and the effective mass needs to be measured before this happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Correlators corresponding to heavier excited states decay faster, so that the critical time tn is shorter for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Clearly, this implies that one needs to achieve a maximal possible overlap of the approxi- mate eigenstates Φi with the true energy eigenstates, so that the plateau can be measured as 10 early as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, given that we perform a diagonalization in an artificially truncated finite dimensional Hilbert space, every approximate eigenstate necessarily has an admixture of heavier states which needs to decay before the plateau can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This problem becomes more and more severe for highly excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To overcome this problem one needs to maximize the projection of an approximate eigen- state on the true energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This projection can be estimated by the gap between the value of the effective mass at t = a and the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Typically, for us this projection drops below ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5 around level Nl, Nr = 3, so we do not expect the corresponding energy determinations to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' There are several ways to improve a quality of the plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' First, one may try to minimize the measured energies in lattice units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This can be achieved by choosing the values of the parameters such that the string tension is smaller in the lattice units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the Ising model this can be achieved by picking the value of β close to the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, other issues arise as one approaches the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' First, as one does this, one needs to take a larger lattice to model a system of the same physical size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=', as measured in string units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Given that we work on a three-dimensional lattice, the simulation time grows as a cube of the lattice size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Also, close to the critical point, correlations between gauge field configurations created by the Metropolis algorithm become higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To overcome this one needs to increase the sampling interval, which also results in a longer simulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' All in all, a limited computing power prevents one from approaching the critical point too closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The second way to reduce statistical errors is by creating a larger size of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This is also limited by the computing resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Finally, one can improve the quality of the operators, so that the overlaps of the approxi- mate eigenstates to the exact ones are closer to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This can be achieved both by starting with a larger set of operators, and also by suppressing the overlap of the operators with the highly energetic microscopic states using blocking and smearing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We will discuss this more in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='3 Constructing flux tube operators In this paper we work in the confining phase of the Z2 gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Equivalently, this is the phase with an unbroken center symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Recall that given a gauge theory compacti- fied on a circle, the center symmetry may be defined4 by making use of the “twisted gauge transformation” generated by gauge functions satisfying g(R) = Λg(0) , (20) where Λ is a center element of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The Yang-Mills action functional is invariant under such a transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, given that the gauge function (20) is not periodic, this transformation defines a global (rather than a gauge) symmetry of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, any two transformations satisfying (20) with the same Λ can be related to each other by 4A modern definition of the center symmetry as a 1-form symmetry does not require to consider a com- pactification [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A traditional and less general discussion presented here is enough for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 11 a conventional gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Hence, after dividing out over the conventional gauge transformations, one obtains a global symmetry transformation which is isomorphic to the center subgroup of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For SU(N) gauge theory it is the ZN center symmetry, and Λ = e 2πik N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For the Z2 gauge theory the center symmetry is Z2 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This definition makes it clear that an arbitrary Wilson loop WC = Tr � P exp(i � C Aµ(x)dxµ) � , (21) corresponding to the contour C with a trivial winding along the chosen compact direction is neutral under the center symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Indeed, such a loop necessarily crosses any transverse slice an equal number of times from both sides and all factors of Λ cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, a Polyakov loop is wound around the periodic dimension, so it crosses any transverse slice in one direction one time more than in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As a result, it is charged under the center symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This also shows that its vacuum expectation value(vev) plays a role of the order parameter for the center symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the confining phase Polyakov loops have zero vev, and a long string sector is generated by acting on the vacuum by (an arbitrarily deformed) Polyakov loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Of course, in addition one may add also any number of topologically trivial Wilson loops creating additional glueball states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The center symmetry ensures that this sector does not mix with the topologically trivial one, which is generated by the glueball operators only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Before describing the set of operators which we used to probe long strings, let us describe conserved quantum numbers in these sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' First, there is a longitudinal momentum p along the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Flux tubes are wound around a circle of a circumference R, so the longitudinal momentum is quantized p = 2πq R , with q being an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The ground state is translationally invariant, which corresponds to q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In addition, there are two parity transformations Pt and Pl, which we already introduced in our discussion of the GGRT spectrum in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is straightforward to describe how they act on the gauge theory operators, without any reference to effective strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Let us consider a long string winding around the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Then the transverse parity is a mirror transformation acting on the transverse y direction, (x, y) Pt −→ (x, −y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Similarly, the longitudinal parity Pl acts as a mirror transformation of the longitudinal x- direction, (x, y) Pl −→ (−x, y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Note that in general the longitudinal parity does not commute with the longitudinal momen- tum, Pl p Pl = −p , 12 Figure 1: Increasing the blocking level of a link by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' so that only q = 0 states may be simultaneous eigenstates of p and Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Finally, long string states may also carry a non-vanishing transverse momentum pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It does not convey any useful information about the worldsheet dynamics and we will always set it to zero by averaging over transverse positions of all operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Let us describe now the set of operators, which we use to probe the long string sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The simplest operator charged under the center symmetry associated to the compact x direction is the straight Polyakov loop φP(x, t) = R/a � n=1 Ux(x + na, y, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (22) where R = La is the string length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In principle, this operator can be used to measure the ground state energy of a long flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, its overlap with the ground state of the flux tube is quite poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Indeed, the Polyakov loop (22) creates a string with a width of order the lattice spacing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, a physical string close to its ground state is expected to have width of order the characteristic string scale ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The overlap can be improved by applying a combination of smearing and blocking proce- dures [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' One starts with the usual link field, which corresponds to blocking level Nbl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Then one replaces an original link with a sign of a weighted average over the link itself and two staples attached to it (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In our simulations we chose the averaging weight to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Finally, one constructs a twice longer link by multiplying two consecutive smeared links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The result is what one calls a level 2 blocked link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To construct the links at Nbl-th blocking level one applies the same procedure using the blocking level Nbl − 1 links as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Using the blocked links we can now create a basis of Polyakov loop operators of different shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 3 we present the shapes used in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Note that some of these operators look like creating a flux tube and an additional glueball rather than just a flux tube excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Equivalently, using the SU(N) language, they look like multi trace opera- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, for the Z2 theory there is no sharp distinction between single trace and multi trace operators, because any operator can be formally presented in the single trace form by connecting different components by going back and forward along some path between them (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 2), given the Abelian nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Finally, to obtain operators with a definite set of quantum numbers one performs averaging over the action of the corresponding symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For example, in order to 13 For an Abelian gauge group Figure 2: For an Abelian gauge group there is no sharp distinction between string excitations and additional glueballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Figure 3: The set of operators used in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' construct an operator with a definite longitudinal momentum p, one sums over all longitudinal translations with a phase φ(p) = L � k=1 φ(x + ak)eipak .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (23) In the same way one constrains pt = 0 by summing over all the translations in the transverse y direction without a phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Similarly, one may obtain operators with definite value of transverse and longitudinal parities (Pt, Pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For example, as we discussed, at p = 0 both parities can be be assigned, so we get four different sectors (++), (+−), (−+) and (−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To construct the corresponding operators one starts with a Wilson line operator UC corresponding to a certain path C, and 14 defines the following eigenstate combination ˜UC = (UC ± UPlC) ± (UPtC ± UPtPlC) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (24) Here the signs inside the brackets correspond to the eigenvalue of Pl, and the sign in the middle corresponds to the eigenvalue of Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 5 Results Let us now present results of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this work we performed Z2 lattice gauge theory simulations at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='756321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This value corresponds to the rough and confining phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is sufficiently close to the critical value βc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='7614133(22) [24], to allow for sufficiently long and clear plateaux in the effective mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Namely, as follows from the results presented later, for this value of β the correlation length ξ (which is set by the inverse mass of the lightest glueball ξ = m−1 G ) is equal to ξ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='631(8)a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Unless specified otherwise, the results presented are obtained on lattices of a size l⊥ = lt = 70a , in the transverse and time directions, and the lattice size along the string is varied in the range R ∈ [20a, 80a] , which corresponds to the range R ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='38ℓs, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='53ℓs] , in string units, where the string length is obtained by fitting the absolute ground state energy of the flux tube to the GGRT formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Recall that the finite temperature deconfinement transition corresponds to R ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='82ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In order to estimate finite volume corrections and for some other checks we also used lattices with other transverse sizes in the range from 55a to 300a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These values of lattice parameters and the corresponding basic physical observables are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' β βc R/a Rc/a a/ℓs amG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='756321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='7614133(22) [20,80] ∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='0691(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2159(4) Table 2: Basic parameters of our simulation: the value of the coupling and its critical value, the range of the string circumference and its critical value, the string tension and the lightest glueball mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Let us now present results of simulations with these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We start with the absolute flux tube ground state, and continue to excited states in different sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Comparing the result 15 to the GGRT spectrum we find that the most pronounced qualitative difference is the presence of an extra state in the parity (++) sector at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This state can naturally be interpreted as a massive scalar resonance on the string worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We identify the corresponding state also in the q = 1 sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Later we present results of an additional dedicated analysis which indicates that this resonance is actually caused by the bulk glueball rather than by a genuine worldsheet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='1 The absolute ground state and the string tension The flux tube ground state is translationally invariant, has q = 0, and belongs to the (++) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Understandably, of all the string states this one is the most straightforward to identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 4, the corresponding effective mass exhibits a well pronounced plateau even for the longest string circumference R = 80a considered in our simulations, which allows for a high precision determination of the ground state energy as a function of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' very well In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 5 we present the ground state energy as a function of circumference R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The solid line shows the GGRT ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These results are plotted in string units with the string length parameter ℓs determined by fitting the data to the GGRT ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For the ℓs extraction we used the data in the range R ∈ [25a, 80a], where the quality of the GGRT fit is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The resulting value of ℓs in lattice units is presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We observe that the GGRT approximation reproduces very well the ground state energy of the Ising string all the way down to R ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, the measured ground state energy significantly deviates from the GGRT formula at shorter values of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, the GGRT ground state energy vanishes at R ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='02ℓs, while the Ising ground state energy stays positive (and approximately linear) down to a smaller critical value given by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To quantify the agreement of the measured ground state energy with the GGRT approxi- mation, we also fitted the observed energies at the short string regime [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4ℓs, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='8ℓs] using the following ansatz E0(R) = EGGRT(R) + cγ ℓs �ℓs R �γ , (25) for different values of γ and using the string length ℓs and the coefficient cγ as the fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To interpret the results it is instructive to compare the obtained values of cγ with the corresponding coefficients of the ℓs/R expansion of the GGRT ground state energy itself, E0(R) = R ℓ2 s − π 6 1 R − π2 72 ℓ2 s R3 − π3 432 ℓ4 s R5 + O(ℓ6 s) , (26) where we listed all the universal terms in the ℓs/R expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For γ = 1, the best fit value of c1 is negligible compared to the value of the corresponding term in (26) c1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='024(13) ≪ π 6 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='52 , so we conclude that our results provide a quite precise determination of the first universal term in the ℓs/R expansion (also known as the L¨uscher term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand for γ = 3 16 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5 aE(t) t/a Figure 4: The effective mass computed as in formula (18) as a function of time for the absolute ground states at string circumference R/a = 20, 40, 60, 80, represented as blue, yellow, green and red dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The horizontal solid lines are the resulting fitted values of the state’s energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The shaded bands represent the corresponding 1σ uncertainty intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 17 0 1 2 3 4 5 0 1 2 3 4 5 Eℓs R/ℓs Figure 5: The absolute ground state energy at different string lengths in string units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The solid line is the GGRT approximation for the ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' we obtain c3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='074(28) ≲ π2 72 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='14 , so that our results are consistent with the 1/R3 universal term, but cannot be considered as a high precision test of the universality at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As an additional crosscheck of our simulation we also determined the mass of the lightest glueball mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' When expressed in string units it reads mG ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='124(10)ℓ−1 s , (27) which agrees well with earlier measurements (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is instructive to take a look at the ground state energy for even shorter strings: R ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2ℓs, as also shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Here one observes a large deviation from the GGRT formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Clearly in this regime the ℓs/R expansion does not converge, so that it cannot be used to measure the perturbative non-universal corrections to the GGRT formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is worth noting that these data do seem to extrapolate towards the deconfining point and exhibit scaling behavior, which indicates that it is a second order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' According to the Svetitsky-Yaffe conjecture [39], this deconfining transition is described by the 2d Ising universality class, of which the scaling behavior is linear E0(R) R→Rc ∝ (R − Rc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (28) From our measurements, it is plausible to be linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' But to really determine the exponent, we need results of higher precision and more data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' One difficulty around the critical point is that the ground state energy goes to zero, so that a larger lattice is needed to perform its accurate determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2 Glueball States As a cross-check for our results, we also calculated the low-lying spectrum of Z2 glueballs in the 0+ sector, which is summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Here we can observe the finite volume corrections for low-lying glueball states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For example, for the lightest glueball, the finite volume correction becomes observable for R ≤ 30a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' One may also wonder whether we can observe the state corresponding to two parallel flux tubes, which also has the same quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It has the mass of two ground state flux tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We do not observe such a state here, which indicates that the local operators we use for glueball states have poor overlap on these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Comparing our results with that in [27], our measurements have higher precision, and they agree well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The largest deviation is found for the second excited state, for which our mass is somewhat lower, but still within a 2σ interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (ly/a) × (lt/a) lx/a aE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 0+ 70 × 70 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='1978(28) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2531(91) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='3519(75)* 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2075(30) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2992(87) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} 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+page_content='2159(17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4025(44) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5541(66) 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2175(17) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='3886(73) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5216(144) Fitted masses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2159(4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='3937(16) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5359(27) Table 3: The spectrum of Z2 glueballs in the 0+ sector at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='756321 for different lattice sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='3 Excited states Let us now present our results for the excited state’s energies of the Ising string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We start with zero momentum states, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As discussed before, these states split into four subsectors 19 with different transverse and longitudinal parities, (Pt, Pl) = (++), (+−), (−+), (−−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (29) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 6 we presented the energy differences between the first three excited states in the (++) sector and the ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As we will see later, restricting to these three states somewhat oversimplifies the overall picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Nevertheless, it provides a good strating point for interpreting our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The numerical values of the corresponding energies (and also of higher excited states) can be found in Table 4 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In addition to two levels, which are naturally associated with the (1, 1) and (2, 2) GGRT states5, we observe on this plot an additional level, which is not associated with any of the GGRT states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Given that the energy gap between this exotic level and the absolute ground state is approximately constant over the large range of R, it is natural to associate this state with a massive (++) resonance on a string worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The resonance mass can be estimated by fitting the energy gap to a constant, which results in mℓs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='825(50) , (30) where we performed the fit at the intermediate values of string circumference, R/ℓs ∈ [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2] to reduce possible effects related to level crossing and winding corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The latter can be incorporated by applying the TBA technique (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' [31]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' we will present results of this analysis in a separate publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' There are two subtleties worth mentioning here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' First, the resonance exhibits two level crossings with the GGRT states in the range of R covered by our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Namely, it crosses the (1, 1) level at R ∼ 2ℓs , and the (2, 2) level at R ∼ 5ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the GGRT spectrum the (2, 2) level corresponds to two degenerate states—a two-phonon and a four-phonon states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' By inspecting Table 4 one indeed observes two nearly degenerate states close to the (2, 2) level at R ≲ 4ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, one of these states disappears as one approaches the second level crossing at R ≳ 4ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The explanation for this is not clear at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As follows from the data presented in Table 4, the energy of the second (2, 2) GGRT state starts to increase away from the GGRT spectrum at around R ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='8ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As we will see later the (++) resonance is actually a glueball state mixed with the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is possible that these large deviations from the GGRT formula appear above the glueball threshold, due to interactions between the unbound glueball and the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The second subtlety, which is likely related to the first one, is that the energy gap (30) is larger than the mass of the lightest glueball (27) in the infinite volume theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This implies that (30) is not a strictly localized worldsheet state, but rather a metastable bound state between a flux tube and a glueball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, in addition to decaying into a two-phonon flux tube excitation it may also decay into a flux tube and a glueball state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Note that the Ising model does not have a parameter which would suppress mixing between genuine flux tube excitations and flux tube states with additional glueballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This is different from the Yang–Mills case, where such a mixing is suppressed in the ’t Hooft large-N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As a result, one may doubt whether the state (30) is really due to intrinsic worldsheet dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Perhaps, 5In the following, for convenience we denote the GGRT levels of states in the format (Nl, Nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 20 2 3 4 5 0 1 2 3 4 5 6 ∆Eℓs R/ℓs Figure 6: Energy differences with the ground state for q = 0 excited states in the (++) parity sector as a function of string circumference at different string lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Blue curves are the (1, 1) and (2, 2) GGRT levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The red horizontal line is the fitted resonance mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' this state should be considered instead as an admixture of the flux tube and an unbound bulk glueball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, our basis of operators was designed to have a good overlap with states localized in the vicinity of the flux tube, so a priori one could expect that it is not sensitive to the states with additional unbound glueballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We performed several checks to clarify the proper interpretation of this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' First, if the exotic state (30) were due to an additional unbound glueball, then one would expect to find a state with similar properties also in the (−+) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Indeed, in infinite volume adding a glueball to a flux tube ground state leads to a continuum of states labeled by the asymptotic transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In a finite volume this continuum turns into a “discretuum”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the absence of interactions between the flux tube and the glueball this discretuum would correspond to the ground state (++) and a series of degenerate doublets with (++) and (−+) parities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, the interaction with the flux tube breaks the degeneracy, so one obtains a series of alternating (++) and (−+) eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Furthermore, energies of all these states, possibly apart from the lowest one, have a rather strong dependence on the transverse size l⊥, due to the momentum quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This depen- dence may be used to distinguish between strongly bound flux tube excitations and unbound states from the discretuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To probe these states, one may enlarge the set of operators in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 3 by adding operators which are expected to have a good overlap with unbound flux tube/glueball states, to see 21 whether additional states indeed appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We will describe the results of this analysis in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As we will see there, our overall conclusion is that the state (30) should indeed be interpreted as a state with an additional unbound glueball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Let us turn now to excited states in other sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For the q = 0 (+−) sector the effective string theory predicts that the lowest energy state appears at the (3, 3) GGRT level and corresponds to a Pl odd linear combination of nl(3) = 1, nr(1) = 3 and nr(3) = 1, nl(1) = 3 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Indeed, our analysis does not reveal any low lying states in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We provide the measured energies of the lightest (+−) state in Table 5 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' At R/ℓs ≳ 4 these energies are in between the (3, 3) and (4, 4) GGRT levels and become significantly heavier at shorter R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Given how heavy these states are we expect that their energy determinations are likely to be subject to significant systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The only robust conclusion one can draw from these results at the moment is that no anomalous light states appear in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Let us discuss now Pt odd states, which are the states with an odd number of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For both (−+) and (−−) sectors the lowest GGRT states appear at the (2, 2) level, and they correspond to even and odd linear combinations of nl(2) = 1, nr(1) = 2 and nr(2) = 1, nl(1) = 2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We plot the measured energies of the lightest states in these sectors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 7, and present the numerical values of these energies and those of the heavier states in Tables 6, 7 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We observe that at R ≳ 4ℓs these two states are nearly degenerate, as expected for the GGRT spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this range of R their energies are quite close to the expected (2, 2) GGRT value, with a minor systematic disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is most likely due to an overestimate of these rather heavy energies due to an admixture of higher excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' At R ≲ 4ℓs the two states are split, and this splitting becomes very large at R ≲ 3ℓs, mostly due to a rather dramatic increase in the energy of the (−−) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Interestingly, the energy of the lightest (+−) states discussed earlier exhibits a similar feature in the same range of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' At the moment it is hard to tell what is the cause of this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Note that, as discussed in a similar context in [32] for the SU(N) data from [19], the splitting between three-phonon (−+) and (−−) cannot be explained by a correction to the two-phonon phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Instead, it is indicative of a strong inelastic multi-phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Interestingly, this splitting appears to be much more dramatic in the Ising case as compared to the SU(N) flux tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Finally, let us discuss states with nonzero longitudinal momentum q = 1, which are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 8 and tabulated in Tables 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The ground state in this sector, which is parity odd, agrees exceptionally well with the GGRT (1, 0) prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This is expected, given that the (1, 0) GGRT state corresponds to adding an essentially free (modulo winding corrections) phonon to the ground state of a flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The first excited parity odd state also agrees very well with the (2, 1) GGRT level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To interpret the two lowest energy parity even q = 1 states it is instructive to compare their energies to the (2, 1) GGRT level and also to the free approximation for the energy of the boosted resonance state, ∆E = � m2 + p2 , (31) where p = 2π R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 8 that the two low lying states naturally correspond to 22 2 3 4 5 4 6 8 10 ∆Eℓs R/ℓs Figure 7: Energy differences with the ground state for q = 0 excited states in the (−+) (blue dots) and (−−) (brown dots) parity sectors as a function of string circumference at different string lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The blue curve is the energy of the (2, 2) GGRT level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' a level crossing between the (2, 1) GGRT level and a boosted resonance state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To illustrate how statistical fluctuations influence our results, especially for higher level states, it is instructive to take a look at the effective mass plateaux behaviour for different states and at the corresponding effective mass fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 4 we plotted the effective mass as a function of time separation for the absolute ground states at different string lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As expected, we see that as the string length increases, which corresponds to the heavier ground state energy, statistical fluctuations become larger and the uncertainty in the effective mass determination grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A generic behavior observed for each of the states is that the effective mass exhibits a drop at early times and then stabilizes on a plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The rate of the initial drop characterizes the quality of the overlap of our operator basis onto the corresponding state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Statistical fluctuations increase at larger with time and dominate the measurement at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' All these features are even more pronounced for excited states as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Here we chose the string length such that the non-universal corrections to the GGRT spectrum is small, and at the same time the resonant state is also well pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As compared to the ground state we observe that statistical fluctuations start to dominate the plateau at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' At the energy of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='67a−1, which corresponds to the second excited state in the parity (−+) sector at R = 60a, this effect reaches the point when the position of the plateau is hard to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Also, because statistical fluctuations here dominate so early, they are 23 2 3 4 5 5 6 7 8 9 10 ∆Eℓs R/ℓs Figure 8: Energy differences with the ground state for q = 1 excited states in the (+) (blue dots) and (−) (brown dots) parity sectors as a function of string circumference at different string lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Blue curves show energies of the (1, 0), (2, 1) and (3, 2) GGRT levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A red curve shows an estimate for the resonance state using the resonance mass (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' likely to prevent us from observing the point of the plateau stabilization, leading to a possible overestimation of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Consequently, a reliable spectrum calculation in this energy range requires a larger sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4 Finite volume corrections Let us discuss the finite size dependence of the presented results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To be more precise, in our simulation we have a finite size lattice system with periodic boundary conditions: R × l⊥ × lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The main goal of the simulation is to measure the dependence of string energy levels on the longitudinal size R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Instead, in this section we will discuss the sensitivity of the presented results to l⊥ and lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Our goal is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the one hand the (in)sensitivity of the measured string energy levels to l⊥ and lt provide a consistency check for the extrapolation of the measured energy levels to infinite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, as was already mentioned, the scattering states containing additional glueball(s) states are expected to exhibit a strong dependence on l⊥, which can be used to probe the nature of a massive resonance state observed in the (++) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In more detail, the spatial finite volume dependence of a single particle or string state 24 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='8 aE(t) t/a Figure 9: The effective mass computed as in formula (18) as a function of time, for the first, second and third excited states in the q = 0 (++) sector and compactification length R = 40a, represented as blue, yellow, green dots, and for the ground state, first and second excited states in the q = 0 (−+) sector and compactification length R = 60a, represented as blue, yellow and green “∗”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The horizontal solid lines in dark colors are the fitted value of the mass of the corresponding states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The shaded bands in light colors represents ±1 standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 25 with zero momentum in the transverse direction is related to winding corrections associated to (virtual) particles propagating around the spatial circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For massive states, which is always the case for us6, these corrections are of order O(e−caml⊥), where the constant c depends on the theory [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These corrections are exponentially suppressed, so as we take the transverse size to be moderately large, it will disappear very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The story is similar for corrections associated to the finite size of the temporal circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To partially account for these corrections we used exponents associated with both directions in time to fit the two-point correlators instead of a single exponential as written in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This still neglects all the time evolutions that wind around the time circle for more than one round, but these effects are further exponentially suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Clearly, winding corrections are most prominent for the lightest states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, l⊥ and lt need to be sufficiently large for a high precision determination of the low lying string states at small R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For multiparticle scattering states there are larger finite volume corrections that go like O(1/(ml⊥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These are associated with finite momenta of individual particles in a multiparticle state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, the infinite volume energy spectrum of multiparticle states is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Instead, in a finite spatial volume one expects to find a discretuum of states which becomes more and more dense as the lattice size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To probe the size of finite volume effects in our results we performed simulations at dif- ferent lattices and compare the corresponding energy spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We do not find a significant dependence of the measured flux tube spectrum on the temporal lattice size, as follows from the data summarized in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These data describe low-lying flux tube spectra mea- sured on 40 × 55 × 55 and 40 × 55 × 70 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The difference is well within error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' So in what follows we fix lt = 70a, where the time windings can be safely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Let us discuss now a set of plots illustrating how energies of low-lying states depend on the transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We do not discuss states in the (+−) sector because their energy determinations are not very reliable due to large statistical uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In this section we fix the size of the longitudinal direction to R = 40a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='77ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The transverse size dependence of the (++) states is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Blue, yellow, green and red dots are natural to identify with the GGRT states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' They match the correspond- ing GGRT energies fairly well, and do not exhibit strong finite volume dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This is also true for the resonance state, which is represented by brown dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, there is an extra state represented by purple dots, which exhibits a very pronounced volume dependence at smaller values of l⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As follows from our earlier discussion, this volume dependence suggests that this state belongs to a discretuum of scattering states describing a string with an addi- tional glueball with non-vanishing relative momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This suggests that also the resonance state should be zero relative momentum at the bottom of the string-glueball discretuum rather than a genuine string excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the next section we present further evidence supporting this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 6Note that what matters here is the mass of a string as a whole as it move in the transverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This should not be confused with the mass of longitudinal string excitations, which is of course zero for the Goldstone modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='25 0 2 4 6 8 10 E/√σ 1/l⊥ √σ Figure 10: Energies in the q = 0 (++) sector at R = 40a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='76ls as a function of the inverse transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Horizontal lines of different colors represent the GGRT spectrum starting with N = ˜N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The brown dashed line represents the resonance mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The transverse size dependence of the (−+) states is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These states are quite a bit heavier than the lightest ones observed in the (++) sector and it is harder to interpret what happens here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It looks natural to associate blue and yellow dots with the proper string excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Their agreement with the GGRT predictions is not so good, and the lightest (blue) state appears to exhibit some volume dependence at the small values of the transverse size l⊥ √σ ≲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In any case, one also observes two additional states (green and red) which exhibit a very pronounced volume dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As in the (++) case this is suggestive of the scattering states interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The transverse size dependence of (−−) states is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' There are no recognizable scattering states among the low-lying states with Eℓs ≲ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Indeed to construct a Pl = − scattering states one can either take a Pl = − flux tube or glueball state, or consider a state where both flux tube and a glueball carry a non-vanishing longitudinal momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In all cases the resulting state is expected to be quite heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' For completeness we also presented the transverse volume dependence of q = 1 states in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 13, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The corresponding scattering states can be obtained by boosting a glueball in the q = 0 states, so these states can be used as consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We expect to find scattering states for both Pt = + and Pt = − sectors among q = 1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These states with strong finite 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='25 6 7 8 9 10 E/√σ 1/l⊥ √σ Figure 11: Energies in the q = 0 (−+) sector at R = 40a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='76ls as a function of the inverse transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Horizontal lines of different colors represent the GGRT spectrum starting with N = ˜N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' volume dependence are indeed present and represented by purple dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 13 and by red dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The green dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 13 represent a resonance state, which can be plausibly reinterpreted as string-glueball discretuum with zero relative momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We conclude that for the coupling β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='756321, which we use, a lattice with lt = l⊥ = 70a is large enough to ignore finite size effects for the GGRT states at the current level of precision at values of R which is not too close to the deconfining value Rc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='82ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We do see strong finite volume corrections associated both with lt and l⊥ dependence as we approach the deconfinement transition Rc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='82ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A much larger lattice size is needed to perform accurate measurements in the vicinity of that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Also, we see evidence for the existence of the flux tube-glueball scattering states at large transverse size for both values of the transverse parity Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This indicates that our set of operators have a sizable overlap with these states and calls for a more rigorous look on the nature of the massive state in the (++) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This will be the goal of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='25 8 9 10 11 12 E/√σ 1/l⊥ √σ Figure 12: Energies in the q = 0 (−−) sector at R = 40a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='76ls as a function of inverse transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Horizontal lines of different colors represent the GGRT spectrum starting with N = ˜N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='25 7 8 9 10 E/√σ 1/l⊥ √σ Figure 13: Energies in the q = 1 (+) sector at R = 40a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='76ls as a function of the inverse transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Horizontal lines of different colors represent the GGRT spectrum starting from N = 2, ˜N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='25 0 2 4 6 8 10 E/√σ 1/l⊥ √σ Figure 14: Energies in the q = 1 (−) sector at R = 40a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='76ls as a function of the inverse transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Horizontal lines of different colors represent the GGRT spectrum starting from N = 1, ˜N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5 Including multitrace operators A sizable mixing between flux tube and scattering states is an interesting peculiarity of the Ising model, not present in the non-Abelian Yang–Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In the Yang–Mills case the scattering states are created by multitrace operators whose overlap on the flux tube states produced by single trace operators is suppressed even at moderately large number of colors N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As discussed before, in the Ising case there is no distinction between multitrace and single trace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We just saw, this leads to a substantial overlap of our operator basis (which was intended to create pure flux tube states) on the scattering states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, this basis is definitely not very well suited for an accurate identification and separation of the scattering states, because one still expects that the corresponding overlap is somewhat suppressed as a consequence of locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Hence, it should be instructive to enlarge the operator basis by introducing additional operators with a good overlap on the scattering states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This will allow us to better probe the nature of the (++) resonance and to confirm its interpretation as a zero momentum scattering state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The additional (pseudo) multi trace operators can be constructed by considering a product of (smeared and blocked) plaquette operators φG producing glueball states with the straight Polyakov loop (22), φscattering = l⊥/a � n,m=1 φP(y + na)φG(y + ma)e 2πiq⊥(n−m)a l⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' (32) The double sum in (32) is needed to project on a state with a vanishing total transverse mo- mentum, which is also characterized by a relative momentum q⊥7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We include such operators with q⊥ = 0, 1, 2, 3, 4 and Pt = ± (q⊥ = 0 state only appears in the Pt = + sector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' On the other hand, for these operators Pl = + because this holds for the φG and φP that we use, and no relative longitudinal momentum is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We now repeat the analysis of the transverse volume dependence of the spectrum using this extended basis of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This should allow a more thorough determination of the low-lying spectrum including also the discretuum of scattering states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' If the (++) resonance is a genuine string state, one expects to find two low-lying massive states that don’t receive pronounced finite volume corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' One of these states would then correspond to the lowest lying glueball scattering state and another to the string excitation (which can also be interpreted as a bound state of a string and a glueball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The results for the (++) sector are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Here we chose the compactification radius R = 55a to ensure that the lowest scattering state is well separated from the GGRT states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We clearly see that beneath the (2, 2) GGRT level, there is only one non-GGRT state (represented by cyan dots) whose energy exhibits only a moderate dependence on a transverse size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In addition, there is a series of non-GGRT states with a strong volume dependence (represented by yellow, brown, purple and mauve-blue dots) which become very dense at large transverse size and accumulate around the expected threshold for the continuum of the scattering states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is natural to interpret these levels as fluxtube-glueball scattering states 7Note that q⊥ is only an approximate quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='25 0 2 4 6 8 10 E/√σ 1/l⊥ √σ Figure 15: Energies in the q = 0 (++) sector at R = 55a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='80ls as a function of inverse transverse size determined using an extended operator basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Horizontal solid lines of different colors represent the GGRT spectrum starting from N = ˜N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The lower dashed blue line represents the energy of the absolute ground state plus the glueball mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The upper dashed blue line represents the absolute ground state plus the resonance mass as given by (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 33 with q⊥ = 1, 2, 3, 4 and the level represented by the cyan dots as a q⊥ = 0 state at the bottom of the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Interestingly, this candidate q⊥ = 0 state still exhibits a noticeable transverse size depen- dence in the range of ℓ⊥ presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The corresponding energy gap at the shortest values of ℓ⊥ is significantly higher than the glueball mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This is indicative of a considerable repulsive interaction between the glueball and the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These interactions appear to be important also for the states with non-zero relative mo- mentum q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, a priori one could have expected that the ℓ⊥ dependence of the corresponding energies can be captured by the free dispersion relation, E = � m2 flux + p2 ⊥ + � m2 glue + p2 ⊥ , (33) with p⊥ = 2πq⊥/ℓ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, we find that this ansatz does not provide a very good fit, indicative of considerable interactions with the flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' These interactions are expected also to affect the GGRT states above the continuum threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This may actually resolve one of the puzzles encountered earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Namely, one expects to find two states at the (2, 2) GGRT level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, only one such state is present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 15 (the one labeled by green dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This phenomenon is also observed in Table 4, where we find out that one of the (2, 2) GGRT states start to deviate from GGRT spectrum at R ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='8ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It appears that a strong mixing between the GGRT and scattering states may provide an explantation for this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We observe similar new states with a strong volume dependence also in the (−+) sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The corresponding flux tube spectrum as a function of the transverse size is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Here blue and orange dots are plausible candidates for GGRT (2, 2) and (3, 3) states given that their volume dependence is relatively mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In addition we find four states with a strong and monotonic volume dependence, which makes them natural candidates for q⊥ = 1, 2, 3, 4 states in the discretuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' There is no analogue of the q⊥ = 0 state in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' As in the (++) sector the complicated pattern of the corresponding energies, suggests that a considerable mixing between flux tube and glueball states is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' To summarize, we believe that the analysis presented here strongly disfavors the existence of light massive excitations on the worldsheet of the Z2 confining flux tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, the state which appears as a massive resonance in the (++) sector corresponds to the glueball scattering state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In addition, our results indicate the presence of a significant mixing between flux tube excitations and scattering glueball states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 6 Concluding Remarks To summarize, we calculated the low-lying spectrum of closed flux tube excitations up to the N = ˜N = 3 GGRT level in the Z2 gauge theory, at a coupling β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='756321 which is close to the critical point βc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='7614133(22) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The compactification radius covers a wide range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='38ℓs ≤ R ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='53ℓs from moderately short strings to very long ones, but still above the deconfinement transition at Rc ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='82ℓs [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The resulting spectrum agrees with the GGRT predictions for N = ˜N ≤ 1 states within most of the range of R, and also for N = ˜N = 2 states for moderately long strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='25 7 8 9 10 E/√σ 1/l⊥ √σ Figure 16: Energies in the q = 0 (−+) sector at R = 55a = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='80ls as a function of the inverse transverse size determined using an extended operator basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Horizontal solid lines of different colors represent the GGRT spectrum starting from N = ˜N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The dashed green line represents the energy of absolute ground state plus glueball mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 35 Somewhat surprisingly, our analysis did not reveal any massive excitations on the world- sheet of the Ising string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A heuristic argument suggesting the presence of a resonance is based on realizing the critical Ising model as an IR fixed point of the φ4 theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Then one may attempt to study the properties of the Ising strings by analyzing domain walls in a mass- deformed φ4 theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Even though this approach is not based on a well-controlled perturbative expansion at d = 3, it was argued [42] to provide a decent approximation to the ratio of the lighest glueball mass to the string tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A domain wall in φ4 theory does support a massive localized resonance [43], so based on this logic one might have expected to find one also in the Ising case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It will be interesting to study what happens to this resonance using a more systematic approach, based on the ϵ-expansion rather than a direct study of the d = 3 φ4 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We did observe a state in the 0++ sector which has an appearance of a massive reso- nance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' However, a detailed analysis revealed that this is a multitrace scattering state with an additional glueball rather than a genuine flux tube excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This is related to another in- teresting (and expected) aspect of the observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Namely, they indicate the presence of a significant mode mixing between string excitations and glueball scattering states related to the repulsive glueball/string interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It will be interesting to perform an analytical anal- ysis of these spectra using an appropriate generalization of the TBA method and to extract the scattering amplitudes describing glueball/string interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It will also be interesting to connect this data to the properties of the line defect in the Ising model at the conformal point, which has been studied in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Another possible direction to extend this work is to study the dynamics of strings in the ZN gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' In particular, it will be interesting to study how the 3d U(1) gauge theory (studied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=', in [45–47]) is recovered in the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' It is natural to expect that strong glueball/string interactions should be present in this whole family of theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' We thank Ofer Aharony, Victor Gorbenko, Michele Caselle, Nabil Iqbal and Yifan Wang for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' This work is supported in part by the NSF grant PHY-2210349, by the BSF grant 2018068 and by the Simons Collaboration on Confinement and QCD Strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The work of CL is partly supported by funding resources from NYU physics department, and the simulation is run on NYU Greene cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' A Compilation of energy spectra In this appendix we list all the closed flux tube spectra we’ve computed for the Z2 gauge theory at the coupling β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='756321, with different lattice sizes and different quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' The convention for denoting sectors follows (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 36 R/a l⊥ × lt/a2 aE(R) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' q = 0 (++) 20 70 × 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='0668(8) 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4801(83) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='5430(71) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='7136(66) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='6882(161) 40 160 × 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4848(53) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='4733(55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} 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is five) with length R in the sector q = 0 (−+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 39 R/a l⊥ × lt/a2 aE(R) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' q = 0 (−−) 20 70 × 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='7911(92) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='8396(220) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='8715(63) 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content='6850(68) 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} +page_content=' 46' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAyT4oBgHgl3EQfP_Yp/content/2301.00034v1.pdf'} diff --git a/49E1T4oBgHgl3EQfSwOL/content/tmp_files/2301.03070v1.pdf.txt b/49E1T4oBgHgl3EQfSwOL/content/tmp_files/2301.03070v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f5efe428121f63887f58ef9c043bb2d115fb723 --- /dev/null +++ b/49E1T4oBgHgl3EQfSwOL/content/tmp_files/2301.03070v1.pdf.txt @@ -0,0 +1,3932 @@ +Relegation-free closed-form perturbation theory and the domain +of secular motions in the Restricted 3-Body Problem +Mattia Rossi and Christos Efthymiopoulos +Università degli Studi di Padova +Dipartimento di Matematica “Tullio Levi-Civita” +Via Trieste, 63 - 35121 Padova, Italy +mrossi@math.unipd.it, cefthym@math.unipd.it +January 10, 2023 +Abstract +We propose a closed-form (i.e. without expansion in the orbital eccentricities) scheme +for computations in perturbation theory in the restricted three-body problem (R3BP) when +the massless particle is in an orbit exterior to the one of the primary perturber. Starting +with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a +sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamilto- +nian model without use of relegation. To this end, we introduce a book-keeping analogous to +the one proposed in [1] for test particle orbits interior to the one of the primary perturber, +but here adapted, instead, to the case of exterior orbits. We give numerical examples of +the performance of the method in both the planar circular and the spatial elliptic restricted +three-body problem, for parameters pertinent to the Sun-Jupiter system. In particular, we +demonstrate the method’s accuracy in terms of reproducibility of the orbital elements’ vari- +ations far from mean-motion resonances. As a basic outcome of the method, we show how, +using as criterion the size of the series’ remainder, we reach to obtain an accurate semi- +analytical estimate of the boundary (in the space of orbital elements) where the secular +Hamiltonian model arrived at after eliminating the particle’s fast degree of freedom provides +a valid approximation of the true dynamics. +Keywords: Celestial mechanics – Astrodynamics – R3BP – Closed-form – No relegation – +Secular motion +1 +Introduction +As opposed to the usual (Laplace-Lagrange) theory, closed-form perturbation theory [11] provides +a framework for series calculations in perturbed Keplerian problems without expansions in powers +of the bodies’ orbital eccentricities. This is mainly motivated by the necessity to construct secular +models for sufficiently eccentric orbits, like those of many asteroids, in our solar system, or the +planets in extrasolar planetary systems. +The efficiency of the usual series methods of expansion in the orbital eccentricities is limited +by the fact that the inversion of Kepler’s equation in powers of the eccentricity converges only +up to the so-called Laplace limit eL ≈ 0.66274 [6]. Generally, such convergence slows down way +before this value (around e ∼ 0.3 − 0.4 in many applications). In order to address this issue, +closed form perturbation theory aims at solving in ‘closed-form’ the homological equation by +which the Lie generating function is computed at every perturbative step (see for example [3, +1 +arXiv:2301.03070v1 [math-ph] 8 Jan 2023 + +5]). The process is far from being priceless: a major obstruction appears when the kernel of +the homological equations contains addenda beyond the Keplerian terms. The most common +such addendum ([11]) is the centrifugal term −νH, where ν is the angular frequency in a frame +co-rotating with the primary perturber, and H is the Delaunay action equal to the particle’s +angular momentum in the direction of the axis of rotation. In the case of a planet’s orbiter, ν +is equal to the planet’s rotation frequency, and the problem appears for all non-axisymmetric +terms (tesseral harmonics) of the planet’s multipole potential. In the R3BP, instead, ν represents +the mean motion of the primary perturber (e.g. Jupiter in the Sun-Jupiter system), while the +problem appears in a similar way after introducing a multipole expansion of the disturbing +function in the particle’s Hamiltonian. +An algorithm to overcome the above issue, called the relegation algorithm, has been proposed +in works by Deprit, Palaciań and collaborators [2, 4, 7, 13, 15]. Briefly, given a quasi-integrable +Hamiltonian H = H0 + εH1, where ε is a small parameter, suppose that H0 = H′ +0 + H′′ +0 , where, +in a domain in phase space we have that H′ +0 yields the dominant contribution to the Hamiltonian +flow of H0 versus the H′′ +0 term. In usual perturbation theory, we seek to partly normalize the +perturbation H1 via a sequence of canonical transformations defined by generating functions +χ(r), r = 1, 2, . . . satisfying a homological equation of the form {H0, χ(r)} + h(r) +1 += 0, where {·, ·} +denotes the Poisson bracket between two functions of the canonical variables and h(r) +1 +is a term +in the Hamiltonian to be normalized. In the relegation technique, we use instead the equation +{H′ +0, χ(r)}+h(r) +1 += 0, i.e., letting only the dominant function H′ +0 in the kernel of the homological +equation. Such a choice stems mostly from motives of algorithmic convenience. For example, +identifying H′ +0 with the Keplerian term (when ν is small) leads to a homological equation that +can be solved in closed form (we set, instead, H′ +0 = −νH when ν is large). However, all Poisson +brackets of χ(r) with the part H′′ +0 left out of the kernel lead to terms which need to be ‘relegated’, +i.e., pushed to normalization in subsequent steps. For reasons explained in detail in [14], only +a finite number or relegation steps can be performed before reaching a point beyond which the +scheme generates divergent sequences of terms (see also [13]). This implies that the process +necessarily stops after some steps, leading to a finite, albeit possibly quite small remainder. +Relegation is a technique particularly suitable to the limiting situation of a strongly hierarchi- +cal problem, when the integrable part H0 depends on a frequency vector involving n frequencies +ω = (ω1, . . . , ωn) out of which one, say ωi for some i with 1 ≤ i ≤ n is significantly larger in +absolute value than the rest. In particular, the harmonics cos(k · ϕ) in the Hamiltonian whose +normalization can be ‘relegated’ should satisfy |kiωi| ≫ |kjωj|, j = 1, . . . , n, j ̸= i, for every +integer ki, kj ∈ Z \ {0} (assuming also the non-resonant condition k · ω ̸= 0, k = (k1, . . . , kn)). +For example, as explained in [14] in the simple case with n = 2 and ω2 ≫ ω1, the generating +function χ(N) produced after N relegation steps contains terms with coefficients growing as a +geometric sequence with ratio k1ω1/k2ω2. Thus, relagation is limited to those terms for which +the above ratio is smaller than unity. This includes most harmonics of low Fourier order in +the Hamiltonian perturbation when ω2 ≫ ω1, but only few when the two frequencies become +comparable in size. Hence, by construction, relegation has limited applicability in this latter, +non-hierarchical, case. +Variants of the relegation technique have been discussed in literature to address perturbed +Keplerian problems in which the gravitational potential is due to an extended body expanded in +spherical harmonics (e.g. [7, 10]). To address the non-hierarchical case, a techique similar to the +one of the present paper is discussed in [7], referring to the averaging of the tesseral harmonics +in the case of the Earth’s artificial satellites. In the case of the R3BP, instead, Cavallari and +Efthymiopoulos [1] discuss a relegation-free algorithm for the elimination of short-period terms +in the particle’s Hamiltonian, when the orbit of the particle (e.g. an asteroid) is totally interior +to the orbit of the primary perturber (e.g. Jupiter). We are aware of no relegation-free algorithm +2 + +proposed in literature which addresses, instead, the case when the particle’s orbit is exterior to +the orbit of the primary perturber. Providing such an algorithm, discussing some of its im- +portant differences with past-proposed algorithms, as well as checking its limits of applicability, +constitutes the primary goal of our present paper. +The R3BP is defined by the motion of a body P of negligible mass in the gravitational +field of two massive bodies P0 (the primary or central body) and P1 (the secondary or primary +perturber), which perform a motion r1(t) either elliptic in the more general version (ER3BP) +or circular (CR3BP). The starting point for our analysis in the sequel is the Hamiltonian of +the model, obtained after reduction via Jacobi coordinates (R, P).1. Expressing time through +the secondary’s mean anomaly M1 = n1t, where n1 is the mean motion of the secondary, and +canonically conjugating M1 with a dummy action variable J1 allows to express the Hamiltonian +as +H(R, M1, P, J1) = ∥P∥2 +2 +− +Gm0 +∥R + µr1(M1)∥ − +Gm1 +∥R − (1 − µ)r1(M1)∥ + n1J1 , +(1) +where G is the gravitational constant and +µ = +m1 +m0 + m1 +∈ (0, 1/2] +is the mass parameter; +r1(M1) = a1 +� +cos E1(M1) − e1, +� +1 − e2 +1 sin E1(M1), 0 +� +(2) +is the elliptic revolution of P0 − P1 around their barycenter with eccentricity e1 and semi- +major axis a1, in which the dependence of the system’s eccentric anomaly E1 ∈ T = R/(2πZ) +on the mean anomaly M1 ∈ T is given through Kepler’s equation according to standard two- +body problem setting; (R = (X, Y, Z), P = (PX, PY , PZ)) ∈ T ∗(R3 \ {−µr1, (1 − µ)r1}) is the +position-momentum couple of P and the phase space is endowed with standard symplectic form +dPX ∧ dX + dPY ∧ dY + dPZ ∧ dZ + dJ1 ∧ dM1. +We make use then of Delaunay elements (ℓ, g, h, L, G, H), defined by +L = +� +Gm0a , +ℓ = M , +G = L +� +1 − e2 , +g = ω , +(3) +H = G cos i , +h = Ω , +where a, e, i, M, Ω, ω stand for the semi-major axis, the eccentricity, the inclination, the mean +anomaly, the longitude of the ascending node, the argument of pericenter of the particle. +A key ingredient of the method proposed below is the following: similarly as in [1], we +introduce a book-keeping symbol σ with numerical value equal to 1, whose role is to organize the +perturbative scheme so as to successively normalize terms of similar order of smallness, treating +together all small quantities of the problem, i.e., +– the eccentricities e, e1 (when e1 ̸= 0), +– the mass ratio µ, +– the semi-major axis fluctuation δL around the mean L∗ for a particular particle trajectory. +1In the R3BP problem the Jacobi transformation is implemented when ∥R∥ > ∥r1∥. +3 + +The book-keeping symbol acts by assigning powers σ1 and σν1, σν, σν respectively, for non-zero +natural numbers ν, ν1 defined below, to all the terms in the original Hamiltonian as well as +in the Hamiltonian produced after every normalization step. Given this baseline, we arrive (in +Section 2) to the following result: we demonstrate that, for kµ, kmp ∈ N \ {0} with kµ > 1, +the combination of expansions of (1) up to µkµ and (∥r1∥ / ∥R∥)kmp is canonically conjugate +by ν(kµ − 1) near-identity transformations to a secular model, obtained as a normal form with +respect to the fast angles ℓ, M1 +H (ℓ, g, h, M1, δL, G, H, J1) = H0(g, h, δL, G, H, J1) + R(ℓ, g, h, M1, δL, G, H) , +(4) +with +H0 = n∗δL + n1J1 + +νkµ−1 +� +l=ν +� +p∈Z2 +cl,p(δL, e, i; µ, L∗, a1, e1) cos(p1g + p2h)σl , +(5) +R = +� +s∈Z4 +dνkµ,s(E1, δL, e, i; µ, L∗, a1, e1) cos(s1f + s2g + s3h + s4E1)σνkµ ++ O +� +σνkµ+1; +�∥r1∥ +∥R∥ +�kmp+1� +. +(6) +The dependencies f = f(ℓ, δL, G) for the true anomaly, e = e(δL, G) and i = i(G, H) are implied +in all the above expressions; cl,p, dνkµ,s are real coefficients. A crucial point is the way by which +the positive integers ν = ν(e∗, µ) ≥ 1, ν1 = ν1(e∗, e1) ≥ 1 are chosen. As detailed below, these +integers, which regulate the book-keeping scheme, are suitably tuned on the basis of a selected +reference value e∗ ∈ (0, 1): +ν = +� log10 µ +log10 e∗ +� +, +ν1 = +�log10 e1 +log10 e∗ +� +, +(7) +where ⌈·⌉ is the ceiling function. The normalizing scheme leading to (4) is local: knowing that +the semi-major axis is preserved under the flow of the (secular) normal form, we introduce the +splitting L = L∗ + δL, where L∗ = √Gm0a∗ ≫ δL, n∗ = √Gm0a−3/2 +∗ +is a targeted reference +value for the semi-major axis a∗, and expand the Hamiltonian in powers of δL, rendering δL the +new action variable canonically conjugated to the particle’s mean anomaly. +Given the above, the normalization algorithm provides a sequence of Lie generating functions +χ(j) +ν+j−1 = O(σν+j−1), j = 1, . . . , ν(kµ−1), which yields the Lie canonical transformation allowing +to recursively normalize all terms depending on the angles f and E1 in the Hamiltonian. The +normalizing trasformations are possible to define for values of the frequencies n∗ (mean motion +of the particle at the semi-major axis a∗) and n1 far from mean-motion resonances (see Remark +3). Furthermore, the generating functions are computed as solutions of a homological equation +of the form +{Z0, χ(j) +ν+j−1} + R(j−1) +ν+j−1,ν+j−1 = O(σν+j−1) , +(8) +where Z0 = n∗δL + n1J1 and R(j−1) +ν+j−1,ν+j−1 ∼ σν+j−1 collects the trigonometric monomials +of O(σν+j−1) depending on at least one of the two anomalies. The key to obtaining a closed- +form solution for (8) is, precisely, the appropriate choice of a O(σν+j−1) remainder left in the +second hand of the equation. In words, we do not seek for an exact cancellation of the terms +R(j−1) +ν+j−1,ν+j−1, but only for an approximate cancellation, leading to a remainder, which, however, +is of higher order in book-keeping, and, hence, possible to reduce at subsequent steps. +4 + +As discussed in Section 3, a relevant outcome of the analysis of the behavior of the re- +mainder obtained by the above method stems from an estimation of the optimal number of +normalization steps jopt, where the remainder becomes of order ν + jopt − 1 in the book-keeping +parameter, with jopt ≤ ν(kµ − 1). +The value of jopt is defined as the one where the error +bound E (j)(a∗, e∗) = � +ν+j≤l≤νkµ,s |d(j) +l,s | ≥ ∥R(j) +ν+j∥∞ = sup |R(j) +ν+j| becomes minimum, with +R(j) +ν+j = O(σν+j) and d(j) +l,s as in (6) after j normalization steps. As typical in perturbation the- +ory, the value of jopt depends on the chosen reference values (a∗, e∗). With the present method +one can then obtain a map of the size of the optimal remainder as a function of (a∗, e∗) in the +semi-plane a > a1. Using this information, we compute the limiting locus uniting all points in +(a∗, e∗) such that the normal form computation yields no improvement with increasing number of +normalization steps, i.e., where jopt = 1. Comparing with numerical stability maps obtained with +the Fast Lyapunov Indicator (FLI) [9], one sees that, the limiting locus found semi-analytically +essentially coincides with the numerical (FLI map) limit where no harmonic in the Hamiltonian +associated with one of the exterior mean-motion resonances affects the dynamics. As a conse- +quence, all motions in the sub-domain of the plane (a∗, e∗) below the limiting locus are stable +in the secular sense, i.e., protected against instabilities caused by short-period resonant effects. +For this reason, we identify this locus as the border of the domain of secular motions, and sub- +stantiate the fact that its semi-analytical computation (through the normal forms) yields results +in precise agreement with those found by the heuristic definition of the same border via the fully +numerical (FLI) computation of stability maps. +The paper is structured as follows. Section 2 presents step-by-step the algorithm that gives +rise to (5) and (6), supplemented with the formulas for the Poisson algebra in Keplerian elements +used in all closed-form computations. Section 3 is devoted to a numerical investigation of the +method’s accuracy for an asteroid in the Sun-Jupiter system, first in the spatial ER3BP, and then +in the planar CR3BP; in the latter case, the computations are short enough to allow for a speci- +fication of the optimal normalization order in a grid of values in the (a∗, e∗) plane, leading to the +semi-analytical determination of the border of the domain of secular motions. Section 4 summa- +rizes the basic conclusions of the present study and gives some relevant comments for future work. +2 +The closed-form method for the outermost R3BP +2.1 +Multipole expansion of the perturbation +Referring to section 1, let H be given in barycentric Cartesian coordinates as in (1): +H = ∥P∥2 +2 ++ n1J1 − Gm0R , +(9) +5 + +X +Y +Z +O +r1 +P0 +P1 +−µr1 +(1 − µ)r1 +R +P +Figure 1: +Representation of the R3BP in the barycentric frame (or equivalently in Jacobi +variables) with ∥R∥ > ∥r1∥. +Assuming ∥r1∥ / ∥R∥ < 1, we carry out a multipole expansion of the function R(R, M1) in powers +of the ratio ∥r1∥ / ∥R∥ < 1: +R = +1 +∥R + µr1∥ + +µ +1 − µ +1 +∥R + (1 − µ)r1∥ += +1 +∥R∥ +� +∞ +� +l=0 +�−1/2 +l +� � +2µr1 · R +∥R∥2 ++ µ2 +�∥r1∥ +∥R∥ +�2�l ++ +µ +1 − µ +∞ +� +l=0 +�−1/2 +l +� � +−2(1 − µ)r1 · R +∥R∥2 ++ (1 − µ)2 +�∥r1∥ +∥R∥ +�2�l � += +1 +1 − µ +1 +∥R∥ + O +��∥r1∥ +∥R∥ +�2� +. +(10) +where, for β ∈ R +�β +l +� += β(β − 1) · · · (β − l + 1) +l! +indicates the generalized binomial coefficient (equal to 1 for l = 0). +Remark 1. For l = 1 in Eq.(10) the coefficients of the dipole term (r1 · R)/ ∥R∥3 in the two +sums in the r.h.s. of the equation cancel each other exactly. Thus, no dipole term appears in the +disturbing function. This is a consequence of the choice of Jacobi coordinates. +2.2 +Canonical form of the Hamiltonian +Performing an extra series expansion in powers of µ < 1 yields the standard nearly-integrable +form +H = H0 + µH1 , +(11) +6 + +where the Keplerian part reads +H0 = ∥P∥2 +2 +− Gm0 +∥R∥ + n1J1 +(12) +and the disturbing function becomes +H1 = −Gm0 +∥R∥ +� +∞ +� +l=0 +µl + +∞ +� +l=1 +µl−1 +�−1/2 +l +� � +2r1 · R +∥R∥2 + µ +�∥r1∥ +∥R∥ +�2�l ++ +∞ +� +l=1 +(1 − µ)l−1 +�−1/2 +l +� � +−2r1 · R +∥R∥2 + (1 − µ) +�∥r1∥ +∥R∥ +�2�l � +. +(13) +We now move to Delaunay action-angle variables (3) by replacing into (11) the relationships +H0 = −Gm0 +2a ++ n1J1 , +(14) +∥R∥ = a(1 − e2) +1 + e cos f , +(15) +r1 · R = a1 ∥R∥ +� +(cos E1 − e1) (cos h cos(g + f) − sin h sin(g + f) cos i) ++ +� +1 − e2 +1 sin E1 (sin h cos(g + f) + cos h sin(g + f) cos i) +� +(16) +as well as (2) for the vector r1. We get +H = −Gm0 +2a ++ n1J1 + µH1(f, g, h, E1, a, e, i; µ, a1, e1) . +(17) +Remark 2. Only the square of the norm ∥r1∥2 = r1 · r1 is required in Eq.(13), while the norm +∥R∥ appears only in the denominator of the above equation, in powers equal to or higher than +quadratic. Then equations (15) and (2), respectively dependent on f and E1, lead to a represen- +tation of the disturbing function as a sum of trigonometric polynomials depending on harmonics +of the form cos(s1f + s2g + s3h + s4E1). This is a key ingredient of the closed-form method, i.e., +working with the angles f and E1, instead of the mean anomalies M, M1, no series reversion of +Kepler’s equation is used throughout the whole perturbative scheme. +In order to avoid relegation, our method discussed below works locally, by constructing a +model for the secular Hamiltonian valid for a particle’s semi-major axis varying as a = a∗+δa(t), +i.e., by a small quantity δL around some reference value a∗. By standard secular theory, we +have the estimate δa = O(µ) far from mean-motion resonances. Formally, introducing the new +canonical variable δL as +L = L∗ + δL = +� +Gm0a∗ + 1 +2 +� +Gm0 +a∗ +δa + O(δa2) . +(18) +and expanding the Hamiltonian in powers of the quantity δL around L∗, we obtain +H = −G2m2 +0 +2L2∗ +∞ +� +l=0 +�−2 +l +� �δL +L∗ +�l ++ n1J1 + µ +∞ +� +l=0 +1 +l! +∂lH1 +∂Ll +���� +L=L∗ +δLl += n∗δL + n1J1 + µ +� +H1|δL=0, µ=0 + ∂H1 +∂δL +���� +δL=0, µ=0 +δL +� ++ O(µ2, δL2) , +(19) +7 + +where a constant term −G2m2 +0/(2L2 +∗) was dropped from the expansion. +The constant n∗ = +G2m2 +0/L3 +∗ is equal to the particle’s mean motion under Keplerian orbit at the semi-major axis +a∗. +Remark 3. The choice of the reference value a∗ determines the kind of divisors appearing in +the normalization procedure. In the present paper, we deal only with the ‘non-resonant’ case, in +which the frequencies n∗ and n1 satisfy no-commensurability condition. For example, to be far +from any resonance we may require that n∗ and n1 satisfy a diophantine condition +|k∗n∗ + k1n1| > +γ +|k|τ , +∀k = (k∗, k1) ∈ Z2 \ {0} +(20) +with |k| = |k∗| + |k1| and some suitable γ > 0, τ > 1. +However, the algorithm presented below can be readily extended to cases of mean-motion res- +onance. We leave the details for a future work, noting only that in resonant cases we have the +estimate δL = O(µ1/2), instead of O(µ). The effect of approaching close to a mean-motion +resonance with the present series is seen, instead, as a rise in the value of the series’ remainder, +caused by (non-zero) small divisors in the series (as visible, for example, in Fig. 7 discussed in +section 3 below). +2.3 +Poisson structure and book-keeping +2.3.1 +Poisson bracket formulas +All steps of closed-form perturbation theory involve Poisson brackets between differentiable +functions of the form F(ℓ, g, h, M1, δL, G, H, J1) ∈ C∞(T4 × D), D ⊂ R4 being an open set, +whose dependence on the variables ℓ, M1, G and H is given in implicit form through the func- +tions f(ℓ, δL, G), E1(M1, e(δL, G)), e(δL, G), ιc(G, H) = cos i(G, H), ιs(G, H) = sin i(G, H), +η(δL, G) = +� +1 − e(δL, G)2, ∥r1∥ (M1) = a1(1−e1 cos E1(M1)), and φ1(M1) = E1(M1, e(δL, G))− +M1. The Poisson bracket between two functions F1, F2 of the above form is computed by the +formulas +{F1, F2} = dF1 +dℓ +dF2 +dδL + dF1 +dg +dF2 +dG + dF1 +dh +dF2 +dH + dF1 +dM1 +dF2 +dJ1 +− dF1 +dδL +dF2 +dℓ − dF1 +dG +dF2 +dg − dF1 +dh +dF2 +dH − dF1 +dJ1 +dF2 +dM1 +(21) +implemented to the closed-form version of the functions F1, F2. The closed-form version of a +function F is defined as: +F = F(f, g, h, E1, δL, e, η, ιc, ιs, J1) . +(22) +The derivatives in the canonical variables of a function F as in Eq.(21) are computed by the +chain rule formulas +dF +dℓ = ∂F +∂f +∂f +∂ℓ , +(23) +dF +dg = ∂F +∂g , +(24) +dF +dh = ∂F +∂h , +(25) +dF +dM1 += +� ∂F +∂E1 ++ +∂F +∂ ∥r1∥ +d ∥r1∥ +dE1 ++ ∂F +∂φ1 +� dE1 +dM1 +− ∂F +∂φ1 +, +(26) +8 + +dF +dδL = ∂F +∂f +∂f +∂δL + ∂F +∂δL + ∂F +∂e +∂e +∂δL + ∂F +∂η +∂η +∂δL , +(27) +dF +dG = ∂F +∂f +∂f +∂G + ∂F +∂e +∂e +∂G + ∂F +∂η +∂η +∂G + ∂F +∂ιc +∂ιc +∂G + ∂F +∂ιs +∂ιs +∂G , +(28) +dF +dH = ∂F +∂ιc +∂ιc +∂H + ∂F +∂ιs +∂ιs +∂H , +(29) +dF +dJ1 += ∂F +∂J1 +, +(30) +where +∂f +∂ℓ = (1 + e cos f)2 +η3 +, +(31) +d ∥r1∥ +dE1 += a1e1 sin E1 , +(32) +dE1 +dM1 += +a1 +∥r1∥ , +(33) +∂f +∂δL = 1 +L +�2 sin f +e ++ sin(2f) +2 +� += 1 +L∗ +�2 sin f +e ++ sin(2f) +2 +� � +1 − δL +L∗ +� ++ O(δL2) , +(34) +∂e +∂δL = η2 +eL = η2 +eL∗ +� +1 − δL +L∗ +� ++ O(δL2) , +(35) +∂η +∂δL = − η +L = − η +L∗ +� +1 − δL +L∗ +� ++ O(δL2) , +(36) +∂f +∂G = − 1 +ηL +�2 sin f +e ++ sin(2f) +2 +� += − 1 +ηL∗ +�2 sin f +e ++ sin(2f) +2 +� � +1 − δL +L∗ +� ++ O(δL2) , +(37) +∂e +∂G = − η +eL = − η +eL∗ +� +1 − δL +L∗ +� ++ O(δL2) , +(38) +∂η +∂G = 1 +L = 1 +L∗ +� +1 − δL +L∗ +� ++ O(δL2) , +(39) +∂ιc +∂G = − ιc +ηL = − ιc +ηL∗ +� +1 − δL +L∗ +� ++ O(δL2) , +(40) +∂ιs +∂G = −1 − ι2 +s +ηLιs += −1 − ι2 +s +ηL∗ιs +� +1 − δL +L∗ +� ++ O(δL2) , +(41) +∂ιc +∂H = 1 +ηL = +1 +ηL∗ +� +1 − δL +L∗ +� ++ O(δL2) , +(42) +∂ιs +∂H = − ιc +ηLιs += − +ιc +ηL∗ιs +� +1 − δL +L∗ +� ++ O(δL2) . +(43) +A sketch of the derivation of the above formulas can be found in Appendix A. They are strictly +valid with e ∈ (0, 1), i ∈ (0, π). However, several cancellations lead to no singular behavior of +the Poisson bracket formulas arising throughout the various perturbative steps also when e = 0 +or i = 0. +9 + +2.3.2 +Book-keeping: Hamiltonian +We introduce in the series a book-keeping symbol σ (see [5] for an introduction to the book- +keeping technique), with numerical value σ = 1, whose role is to provide a grouping of all the +various terms in the series according to their ‘order of smallness’. Hence, a group of terms with +common factor σl, l ∈ Z, indicates a term considered as of the ‘l-th order of smallness’. +Since in our series there are several small quantities, we introduce a book-keeping scheme +allowing to simultaneously deal with all small quantities while maintaining the closed-form char- +acter of the series. To this end, we make the following substitutions, called ‘book-keeping rules’, +within the initial Hamiltonian: +• BK-Rule 1: e � σ1e = σe (not applicable to the quantity e2 within η = +√ +1 − e2), +• BK-Rule 2: η � σ0η = η, +• BK-Rule 3: µ � σνµ, with ν as in Eq.(7), +• BK-Rule 4: e1 � σν1e1, with ν1 as in Eq.(7) (not applicable to the quantity e2 +1 within +η1 := +� +1 − e2 +1), +• BK-Rule 5: +1 +η2 � +� +1 +η2 − 1 +� +σ2 + 1, +• BK-Rule 6: η1 � (η1 − 1)σ2ν1 + 1, +• BK-Rule 7: δLλ � σlνδLλ with l = +� +λ , +if δLλ comes from H1 , +λ − 1 , +if δLλ comes from H0 , λ ∈ N \ {0}. +Since σ = 1, the above substitutions affect the structure of the series only at the formal level, +and can be substituted directly into the original Hamiltomian, whereby they propagate at sub- +sequent normalization steps once these steps are organized in successive powers σ, σ2, etc., of the +book-keeping symbol. The BK-Rules 1 to 7 above are justified on physical ground as well as on +motives of algorithmic convenience. In particular: +- BK-Rule 1 implies that, despite the use of closed-form formulas, the basic small quantity +in powers of which the series are organized is the eccentricity of the test particle. +- BK-Rule 3 implies that a factor µ in front of a series term should be treated as of compa- +rable order of smallness as a term of order eν, with ν given by Eq.(7). Similarly, BK-Rule 4 +implies that a term containing a factor e1 raised to some power should be treated as of compa- +rable order of smallness with a term eν +1 raised to the same power. Note that the eccentricity e +is a quantity variable in time, so that to compute the exponents ν, ν1 we need to use, for any +examined trajectory, a reference value e∗ yielding an estimate of the overall level of eccentricity +all along the orbital evolution for that trajectory. Note that, by standard secular theory we have +e(t) = e∗ + O(µ) if e∗ is close to the mean eccentricity (see also discussion at the introduction). +Note finally that we obtain exponents ν, ν1 ≥ 1 in the typical case in which e > µ and e ≥ e1. +These inequalities arise naturally in the case of small bodies in highly eccentric orbits perturbed +by some planet of, say, our solar system, which are the cases of main interest in applying the +present method (see, nevertheless, Remark 4 on the treatment of cases where the above condi- +tions are not met). +- BK-Rule 7 stems from the estimate δL = O(δa) = O(µ) holding for the oscillations in semi- +major axis of trajectories far from mean-motion resonances (as already pointed outin the latter +10 + +case, instead, we have in general δL = O(δa) = O(µ1/2) and the corresponding rule has to be +adapted accordingly). The lowering of the book-keeping power by one for within H0 is intro- +duced for reasons of algorithmic convenience, i.e., in order to maintain n∗δL in the kernel of the +homological equation. +- BK-Rules 5 and 6 imply just a partition of the unity aiming at keeping the perturbative +scheme in closed-form while splitting the corresponding expressions (involving η and η1 respec- +tively) in two parts, of orders O(1) and O(e2), or O(e2 +1). +2.3.3 +Book-keeping: Poisson structure +Some of the formulas in Subsection 2.3.1 imply differentiation with respect to e through the +corresponding partial derivatives in (27), (28), thus yielding a lowering of the power of the +eccentricity in some terms arising through Poisson brackets at consecutive steps of perturbation +theory. To account for this fact, similarly as in [1] we introduce the use of the book-keeping +symbol σ in the formulas of the Poisson algebra as follows: first, we re-write the derivatives with +respect to the angles ℓ, g, h, M1 as +dF +dℓ = ∂F +∂f +∂f +∂ℓ +a1(1 − e1σν1 cos E1) +∥r1∥ +, +(44) +dF +dg = ∂F +∂g +a1(1 − e1σν1 cos E1) +∥r1∥ +, +(45) +dF +dh = ∂F +∂h +a1(1 − e1σν1 cos E1) +∥r1∥ +, +(46) +dF +dM1 += +� ∂F +∂E1 ++ +∂F +∂ ∥r1∥ +d ∥r1∥ +dE1 ++ ∂F +∂φ1 +σ−ν1 +� dE1 +dM1 +− ∂F +∂φ1 +σ−ν1 , +(47) +and with respect to the actions δL, G as +dF +dδL = ∂F +∂f +∂f +∂δL + ∂F +∂δL + ∂F +∂e +∂e +∂δLσ−1 + ∂F +∂η +∂η +∂δL , +(48) +dF +dG = ∂F +∂f +∂f +∂G + ∂F +∂e +∂e +∂Gσ−1 + ∂F +∂η +∂η +∂G + ∂F +∂ιc +∂ιc +∂G + ∂F +∂ιs +∂ιs +∂G . +(49) +Note that in (47) use was made of the identity φ1 = e1 sin E1 (Kepler’s equation). +Finally, +we revise formulas (31), (32), (34)–(43), attributing a book-keeping to all factors involving the +eccentricity function η as +∂f +∂ℓ = 1 + 2e cos f +η3 +σ + +� 1 +η3 − 1 + e2 cos2 f +η3 +� +σ2 , +(50) +d ∥r1∥ +dE1 += a1e1σν1 sin E1 +(51) +∂f +∂δL = 1 +L∗ +�2 sin f +e +σ−1 + sin(2f) +2 +� ++ O(δLσν) , +(52) +∂e +∂δL = 1 +L∗ +�1 +eσ−1 + η2 − 1 +e +σ +� ++ O(δLσν) , +(53) +∂η +∂δL = − 1 +L∗ +� +1 + (η − 1)σ2� ++ O(δLσν) , +(54) +11 + +∂f +∂G = − 1 +L∗ +� +2 sin f +e +σ−1 + sin(2f) +2 ++ 2 sin f +e +�1 +η − 1 +� +σ + sin 2f +2 +�1 +η − 1 +� +σ2 +� ++ O(δLσν) , +(55) +∂e +∂G = − 1 +L∗ +�1 +eσ−1 + η − 1 +e +σ +� ++ O(δLσν) , +(56) +∂η +∂G = 1 +L∗ ++ O(δLσν) , +(57) +∂ιc +∂G = − ιc +L∗ +� +1 + +�1 +η − 1 +� +σ2 +� ++ O(δLσν) , +(58) +∂ιs +∂G = −1 − ι2 +s +L∗ιs +� +1 + +�1 +η − 1 +� +σ2 +� ++ O(δLσν) , +(59) +∂ιc +∂H = 1 +L∗ +� +1 + +�1 +η − 1 +� +σ2 +� ++ O(δLσν) , +(60) +∂ιs +∂H = − ιc +L∗ιs +� +1 + +�1 +η − 1 +� +σ2 +� ++ O(δLσν) . +(61) +Remark 4. The small eccentricity problem consists of the fact that the above-proposed book- +keeping rules are not applicable in the case 0 < e∗ ≲ µ < e1, since, by (7), the exponents ν, +ν1 would be smaller than unity. The simple solution of rounding these exponents to 1, while +maintaining the same book-keeping rules as above, fails, since, at any given normalization order +r, the presence of σ−1, σ−ν1 terms in the formulas of the Poisson algebra leads to the generation +of terms of order lower than r in the normal form’s remainder. Notwithstanding our focus on +a method dealing with large eccentricity orbits (for which the problem does not appear), we +discuss below a variant of the main algorithm that deals with trajectories in the case ν = 1, i.e., +when e∗ ≲ µ. +2.4 +Iterative normalization algorithm +2.4.1 +Preliminary step: Hamiltonian preparation +After implementing BK-Rules 1 to 7 the Hamiltonian (19) resumes the form: +H = n∗δL + n1J1 + +� +s∈Z4 +qs(δL, e, η, ιc, ιs; µ, L∗, a1, e1, η1) cos(s1f + s2g + s3h + s4E1)σs +(62) +where σs ∈ {σν, σν+1, . . .} and, by D’Alembert rules, only cosines and real coefficients qs appear +(invariance under simultaneous change of sign of all angles). Setting Z0 = n∗δL + n1J1, for ob- +taining a closed-form normalization algorithm it turns convenient to re-express the Hamiltonian +according to +H = Z0 + (H − Z0)a1(1 − e1σν1 cos E1) +∥r1∥ +. +(63) +12 + +The Hamiltonian (63) resumes the form: +H = H (0) = Z0 + R(0) +ν +, +(64) +where +R(0) +ν += +� +l≥ν +R(0) +ν,l += +� +l≥ν +a1 +∥r1∥ +� +� +� +� +� +p∈Z2 +q′ +l,p cos(p1g + p2h) + +� +s∈Z4 +(s1,s4)̸=(0,0) +q′′ +l,s cos(s1f + s2g + s3h + s4E1) +� +� +� +� σl ; +(65) +We call R(0) +ν +the remainder at the zero-th normalization step (i.e. in the original Hamiltonian). +The terms R(0) +ν,l contain terms of book-keeping order σl, with l ≥ ν. +2.4.2 +Step 1: normalization of the σν-terms +For a suitable generating function χ(1) +ν +to be determined in a while, we introduce the Lie series +operator as +exp +� +Lχ(1) +ν +� +: Cω(T4 × D) −→ Cω(T4 × D) +exp +� +Lχ(1) +ν +� += +� +n≥0 +1 +n!Ln +χ(1) +ν += I + Lχ(1) +ν ++ 1 +2Lχ(1) +ν +◦ Lχ(1) +ν ++ . . . , +(66) +where Cω(T4 × D) denotes the set of real analytic functions in the phase space and +Lχ(1) +ν · = {·, χ(1) +ν } +(67) +is the time derivative along the Hamiltonian vector field generated by χ(1) +ν +(Lie derivative). +Applying (66) to (63) we get the transformed Hamiltonian +H (1) = Z0 + R(0) +ν ++ {Z0, χ(1) +ν } + {R(0) +ν , χ(1) +ν } + 1 +2{{H, χ(1) +ν }, χ(1) +ν } + . . . , +(68) +in which, with the usual abuse of notation, we still indicate with ℓ, g, h, M1, δL, G, H, J1 the new +canonical variables given by the inverse transformation +exp +� +Lχ(1) +ν +�−1 += exp +� +L−χ(1) +ν +� +. +(69) +Our scope will be to define the Lie generating function χ(1) +ν +in such a way that, after imple- +menting the transformation (68), H (1) contains no terms depending on the angles f and E1 at +order σν. The required generating function χ(1) +ν +is computed as an outcome of the following: +Proposition 1. Define χ(1) +ν +as +χ(1) +ν += φ1 +n1 +σν+ν1 � +p∈Z2 +q′ +ν,p cos(p1g + p2h) ++ σν +� +s∈Z4 +(s1,s4)̸=(0,0) +q′′ +ν,s +s1n∗ + s4n1 +sin(s1f + s2g + s3h + s4E1) . +(70) +13 + +Then, it holds that +{Z0, χ(1) +ν } + R(0) +ν,ν = Z (1) +ν ++ O +� +σν+1� +, +(71) +where +Z (1) +ν += σν � +p +q′ +ν,p cos(p1g + p2h) . +(72) +Furthermore, the function H (1) as computed by Eq.(68) takes the form +H (1) = Z0 + Z (1) +ν ++ R(1) , +(73) +where the remainder R(1) is O(σν+1) ∀ν ≥ 1 independently of the value of ν1. +Proof. Setting +χ(1) +ν (f, g, h, E1, φ1, δL, e, η, ιc, ιs) = σν +� +� +� +�φ1σν1 � +p∈Z2 +ˆq′ +ν,p(δL, e, η, ιc, ιs) cos(p1g + p2h) ++ +� +s∈Z4 +(s1,s4)̸=(0,0) +ˆq′′ +ν,s(δL, e, η, ιc, ιs) sin(s1f + s2g + s3h + s4E1) +� +� +� +� , +and recalling the chain rules (44), (47) and (50), (51), (33), we find +{Z0, χ(1) +ν } + R(0) +ν,ν = −n∗ +� +1 + 2e cos f +η3 +σ + +� 1 +η3 − 1 + e2 cos2 f +η3 +� +σ2 +� +a1(1 − e1σν1 cos E1) +∥r1∥ +σν +� +(s1,s4)̸=(0,0) +s1ˆq′′ +ν,s cos(s1f + s2g + s3h + s4E1) +− n1 +a1 +∥r1∥σν +� +� +� +(s1,s4)̸=(0,0) +s4ˆq′′ +ν,s cos(s1f + s2g + s3h + s4E1) + +� +p +ˆq′ +ν,p cos(p1g + p2h) +� +� ++ n1σν � +p +ˆq′ +ν,p cos(p1g + p2h) + σν a1 +∥r1∥ +�� +p +q′ +ν,p cos(p1g + p2h) ++ +� +(s1,s4)̸=(0,0) +q′′ +ν,s cos(s1f + s2g + s3h + s4E1) +� +. +Requiring that no trigonometric terms depending on f, E1 be present at order σν then leads to +ˆq′′ +ν,s = +q′′ +ν,s +s1n∗ + s4n1 +, +s ∈ Z4 : (s1, s4) ̸= (0, 0) , +ˆq′ +ν,p = q′ +ν,p +n1 +, +p ∈ Z2 , +which implies Eq.(70). At order σν we then obtain immediately the formula +Z (1) +ν += σν � +p +q′ +ν,p cos(p1g + p2h) . +14 + +We now consider the function H (1) computed by replacing (70) into (68). The function H (1) +can be decomposed as in Eq.(73). We shall demonstrate that the remainder R(1) contains no +terms of order lower than σν+1. To this end, it suffices to show that +{R(0) +ν , χ(1) +ν } = O(σ2ν) , +1 +n!{. . . {{H, χ(1) +ν }, χ(1) +ν }, . . . , χ(1) +ν +� +�� +� +n≥2 +} = O(σn(ν−1)+2) , +(74) +since n(ν − 1) + 2 > ν, for all n ≥ 2, ν ≥ 1. +The term R(0) +ν +contains terms of order equal to or larger than σν, while χ(1) +ν +contains only +terms of order σν. Thus, except for the Poisson bracket {Z0, χ(1) +ν }, which only contributes to +the secular terms Z (1) +ν +due to Eq.(71), the first Poisson bracket in (74) contains prefactors of +order σ2ν or higher, while the second contains prefactors σnν or higher. However, the exponent +of σ in these brackets can be lowered due to the negative powers introduced in the book-keeping +formulas in the following three classes of factors: +(i) partial derivatives with respect to the eccentricity in (48), (49) (carrying σ−1) multiplied +by corresponding formulae (53), (56) (another σ−1), hence a total of σ−2; +(ii) differentiations (52), (55) involving f (weighting σ−1) again in (48), (49), thus a pre-factor +σ−1; +(iii) partial derivatives with respect to φ1 in (47) (σ−ν1, ν1 ≥ 1), thus a prefactor at least σ−1. +As regards (iii) φ1 shows up in the numerator of χ(1) +ν +accompanied by a prefactor σν+ν1 +(Eq.(70)), thus the negative powers σ−ν1 are cancelled by the positive powers σν1, implying no +dependence of the minimum order of the remainder on ν1. +As regards (i), we first note that χ(1) +ν +has no explicit dependence on e, but only an implicit +dependence through η, which in the closed-form context is treated as an independent symbol. +This follows from the fact that χ(1) stems from balancing the coefficients of R(0) +ν,ν. The latter +term contains a pre-factor µ, which is already O(σν), thus it cannot contain any further factors +produced by any explicit power of e. In view of the above, setting ∂χ(1)/∂e = 0, we find that +for any F ∈ C∞(T4 × D) the expression in {F, χ(1) +ν } pertaining (i) can be factored out as +{F, χ(1) +ν }(i) = −∂F +∂e σ−1 +� +∂f +∂ℓ +∂e +∂δL +∂χ(1) +ν +∂f ++ ∂e +∂G +∂χ(1) +ν +∂g +� +. +(75) +We now have the following lemma: +Lemma 1. For every term in the Hamiltonian (63) of the form +qs(∥r1∥ , δL, η, ιc, ιs; µ, L∗, a1, e1, η1) cos(s1f + s2g + s3h + s4E1)σs , +(76) +i.e., explicitly independent on e, we have s1 = s2. +Proof. This is a consequence of D’Alembert rules. Using modified Delaunay angular elements +˜λ = ℓ + g + h , +˜p = −g − h , +(77) +˜q = −h , +as well as the formulas f = ℓ + 2e sin ℓ + O(e2), eη(e)−2λ = e + λe3 + O(e5), λ ∈ N, we find that, +after expanding in the eccentricity e, (76) should give the terms +qs cos(s1(˜λ + ˜p) + s2(˜q − ˜p) − s3˜q + s4E1)σs + O(e) . +(78) +15 + +However, according to the D’Alembert rules, in a generic trigonometric monomial of the form +bw(∥r1∥ , δL, η, ιc, ιs; µ, L∗, a1, e1, η1)elσl cos(w1˜λ + w2˜p + w3˜q + w4E1)σw , +l ∈ N , +(79) +appearing after expanding H in the eccentricities e, e1, we necessarily have that l − |w2| must be +non-negative and even. Since for any closed-form term in the Hamiltonian, explicitly independent +of e, the lowermost term in e produced after the expansion satisfies l = 0, we necessarily have +w2 = 0, that is s1 = s2. +In view, now, of (70), the relation s1 = s2 implies ∂χ(1) +ν /∂f = ∂χ(1) +ν /∂g. Therefore, making +use of (50), (53) and (56), Eq.(75) translates into +{F, χ(1) +ν }(i) = −∂F +∂e σ−1 ∂χ(1) +ν +∂f +�σ−1 +L∗e − σ−1 +L∗e + O(σ0) +� += −∂F +∂e σ−1 ∂χ(1) +ν +∂f O(σ0) . +It follows that for any of the functions F = R(0) +ν , {H, χ(1) +ν }, {{H, χ(1) +ν }, χ(1) +ν }, . . ., terms produced +by derivatives of the type (i) in (68) are subject to a lowering of the exponent of σ per Poisson +bracket only by a factor σ−1, instead of σ−2. In particular, in the case F = R(0) +ν,ν (as well as for +any other closed-form function explicitly independent on the eccentricity) we have that (75) is +identically vanishing. +As regards (ii), we find that for any F1, F2 ∈ C∞(T4 × D), the derivative ∂f/∂δL (Eq.(52)) +participates in the Poisson bracket {F1, F2} only through the combination +∂f +∂ℓ +∂f +∂δL +�∂F1 +∂f +∂F2 +∂f − ∂F1 +∂f +∂F2 +∂f +� += 0 . +(80) +On the other hand, the derivative ∂f/∂G (Eq.(55)) participates in the same Poisson bracket +through the combination +∂f +∂G +�∂F1 +∂g +∂F2 +∂f − ∂F1 +∂f +∂F2 +∂g +� +(81) +which, by Lemma 1, is also equal to zero for F1 = R(0) +ν,ν (or any other term O(σν+1) in H not +depending explicitly on e), and F2 = χ(1) +ν . +In conclusion, returning to (74), and taking all the above deductions into account, we arrive +at the expressions +{R(0) +ν , χ(1) +ν } = {R(0) +ν,ν, χ(1) +ν } + +� +� +� +� +l≥ν+1 +R(0) +ν,l , χ(1) +ν +� +� +� = O(σν+ν) + O(σν+1+ν−1) = O(σ2ν) +and similarly, +1 +2{{H, χ(1) +ν }, χ(1) +ν } = 1 +2{{Z0, χ(1) +ν }, χ(1) +ν } + 1 +2{{R(0) +ν , χ(1) +ν }, χ(1) +ν } += O(σ2ν) + O(σ3ν−1) = O(σ2ν) , +since {Z0, χ(1) +ν } satisfies Lemma 1. We then have {Z0, χ(1) +ν } = Z (1) +ν +− R(0) +ν,ν + O(σν+1), with +Z (1) +ν +independent on f, g, e. Proceeding by induction +1 +n!{. . . {{Z0 + R(0) +ν , χ(1) +ν }, χ(1) +ν }, . . . , χ(1) +ν +� +�� +� +n≥3 +} = O(σmin{nν−(n−2), (n+1)ν−(n−1)}) = O(σn(ν−1)+2) +which concludes the proof of the proposition. +16 + +By Proposition 1, computing all Poisson brackets in (68), substituting φ1 = e1 sin E1 where +appropriate, and multiplying all terms missing a factor 1/ ∥r1∥ with the factor a1(1 − σν1e1 +cos(E1))/ ∥r1∥ (equal to 1), the remainder R(1) +ν+1 resumes the standard form +R(1) +ν+1 = +� +l≥ν+1 +R(1) +ν+1,l = +� +l≥ν+1 +� +λ≥1 +a1 +∥r1∥λ +� +s∈Z4 +d(1) +l,λ,s cos(s1f + s2g + s3h + s4E1)σl , +(82) +where the coefficients d(1) +l,λ,s satisfy the relations +d(1) +l,λ,s = d(1) +l,λ,s(δL, e, η, ιc, ιs, ; µ, L∗, a1, e1, η1) = +� +� +� +d′(1) +l,λ,p , +s1 = s4 = 0, (s2, s3) = p , +d′′(1) +l,λ,s , +(s1, s4) ̸= (0, 0) , +∈ R . +These last algebraic operations conclude the first normalization step. +2.4.3 +Loop: normalization of the σν+j−1-terms +The procedure followed in the first step can be repeated iteratively in order to normalize consec- +utively terms of order σν+j−1, with each time an O(σν+j) remainder, for ν, j > 1. As anticipated +in Remark 4, the iterative procedure described below fails in the case ν = 1 at step j = 2, so an +adjustment (involving one more iteration) is required, as discussed in Subsection 2.4.4 below. +The j-th normalization step is carried out as follows from the next proposition. +Proposition 2. Assume ν ≥ 2, ν1 ≥ 1. Assume that the Hamiltonian before the j-th normal- +ization step has the form: +H (j−1) = Z0 + +j−1 +� +l=1 +Z (l) +ν+l−1 + R(j−1) +ν+j−1 +(83) +where +Z (l) +ν+l−1 = σν+l−1 � +λ≥1 +� +p∈Z2 +ζ(l) +ν+l−1,λ,p cos(p1g + p2h) . +(84) +R(j−1) +ν+j−1 = +� +l≥ν+j−1 +R(j−1) +ν+j−1,l = +� +l≥ν+j−1 +� +λ≥1 +a1 +∥r1∥λ +� +� +� +� +� +p∈Z2 +d′(j−1) +l,λ,p +cos(p1g + p2h) ++ +� +s∈Z4 +(s1,s4)̸=(0,0) +d′′(j−1) +l,λ,s +cos(s1f + s2g + s3h + s4E1 +� +� +� +� σl , +(85) +for some real coefficients ζ(l) +ν+l−1,λ,p, d′(j−1) +l,λ,p , d′′(j−1) +l,λ,s +specified at previous steps, where +ζ(1) +ν,λ,p = +� +q′ +ν,p , +λ = 1 +0 , +λ > 1 +by (72). +17 + +Define the j-th step Lie generating function χ(j) +ν+j−1 as +χ(j) +ν+j−1 = φ1 +n1 +σν+j−1+ν1 � +λ≥1 +λ +� +ψ=1 +1 +aψ−1 +1 +∥r1∥λ−ψ +� +p∈Z2 +d′(j−1) +ν+j−1,λ,p cos(p1g + p2h) ++ σν+j−1 � +λ≥1 +1 +∥r1∥λ−1 +� +s∈Z4 +(s1,s4)̸=(0,0) +d′′(j−1) +ν+j−1,λ,s +s1n∗ + s4n1 +sin(s1f + s2g + s3h + s4E1) . +(86) +Then, the Hamiltonian H (j) produced by the Lie operation H (j) = exp +� +Lχ(j) +ν+j−1 +� +H (j−1) has +the form +H (j) = exp +� +Lχ(j) +ν+j−1 +� +H (j−1) = Z0 + +j +� +l=1 +Z (l) +ν+l−1 + R(j) +ν+j , +(87) +where +Z (j) +ν+j−1 = σν+j−1 � +λ≥1 +� +p∈Z2 +ζ(j) +ν+j−1,λ,p cos(p1g + p2h) +(88) +with +ζ(j) +ν+j−1,λ,p = +1 +aλ−1 +1 +d′(j−1) +ν+j−1,λ,p , +(89) +and +R(j) +ν+j = +� +l≥ν+j +R(j) +ν+j,l = +� +l≥ν+j +� +λ≥1 +a1 +∥r1∥λ +� +� +� +� +� +p∈Z2 +d′(j) +l,λ,p cos(p1g + p2h) ++ +� +s∈Z4 +(s1,s4)̸=(0,0) +d′′(j) +l,λ,s cos(s1f + s2g + s3h + s4E1 +� +� +� +� σl , +(90) +with real coefficients d′(j) +l,λ,p, d′′(j) +l,λ,s computed from the known coefficients ζ(l) +ν+l−1,λ,p (l = 1, . . . , j−1), +d′(j−1) +l,λ,p , d′′(j−1) +l,λ,s +. +Proof. We repeat the strategy of Proposition 1 and look for a generating Hamiltonian this time +dependent on ∥r1∥: +χ(j) +ν+j−1(f, g, h, E1, φ1, ∥r1∥ , δL, e, η, ιc, ιs) += σν+j−1 +� +� +� +�φ1σν1 � +λ≥1 +� +p∈Z2 +ˆd′(j−1) +ν+j−1,λ,p(∥r1∥ , δL, e, η, ιc, ιs) cos(p1g + p2h) ++ +� +λ≥1 +� +s∈Z4 +(s1,s4)̸=(0,0) +ˆd′′(j−1) +ν+j−1,λ,s sin(s1 + s2g + s3h + s4E1) +� +� +� +� . +18 + +Requiring {Z0, χ(j) +ν+j−1} + R(j−1) +ν+j−1,ν+j−1 to be O(σν+j) in fast angles we come up with +−n∗ ˆd′′(j−1) +ν+j−1,λ,ss1 − n1 ˆd′′(j−1) +ν+j−1,λ,ss4 + +1 +∥r1∥λ−1 d′′(j−1) +ν+j−1,λ,s = 0 , +−n1 +a1 +∥r1∥ +ˆd′(j−1) +ν+j−1,λ,p + n1 ˆd′(j−1) +ν+j−1,λ,p + +a1 +∥r1∥λ d′(j−1) +ν+j−1,λ,p = +1 +aλ−1 +1 +d′(j−1) +ν+j−1,λ,p , +that is, for λ ≥ 1, +ˆd′′(j−1) +ν+j−1,λ,s = +1 +∥r1∥λ−1 +d′′(j−1) +ν+j−1,λ,s +s1n∗ + s4n1 +, +s ∈ Z4 : (s1, s4) ̸= (0, 0) , +ˆd′(j−1) +ν+j−1,λ,p = +1 +aλ−1 +1 +d′(j−1) +ν+j−1,λ,p +n1 +λ−1 +� +ψ=0 +� a1 +∥r1∥ +�ψ += +d′(j−1) +ν+j−1,λ,p +n1 +λ +� +ψ=1 +1 +aψ−1 +1 +∥r1∥λ−ψ , +p ∈ Z2 , +which proves Eq.(86), and new accumulated addenda in normal form +Z (j) +ν+j−1 = σν+j−1 � +λ≥1 +1 +aλ−1 +1 +� +p +d′(j−1) +ν+j−1,λ,p cos(p1g + p2h) . +which proves Eq.(89). It remains to demonstrate that the expression (90) is O(σν+j). The proof +is done by induction: for j = 2 we get +H (2) = Z0 + Z (1) +ν ++ Z (2) +ν+1 + O(σν+2) + +� +l≥ν+2 +R(1) +ν+1,l + {Z (1) +ν +, χ(2) +ν+1} ++ {R(1) +ν+1, χ(2) +ν+1} + . . . + +� +n≥2 +1 +n!{. . . {{H (1), χ(2) +ν+1}, χ(2) +ν+1}, . . . , χ(2) +ν+1 +� +�� +� +n +} . +(91) +Similarly as in Proposition 1, a lowering of the book-keeping exponents in a Poisson bracket of +the form {F, χ(2) +ν+1} can occur through derivatives of the form (i). However, this time the latter +can only appear in a Poisson bracket via the combination +σ−1 +� +∂f +∂ℓ +∂e +∂δL +� +∂F +∂f +∂χ(2) +ν+1 +∂e +− ∂F +∂e +∂χ(2) +ν+1 +∂f +� ++ ∂e +∂G +� +∂F +∂g +∂χ(2) +ν+1 +∂e +− ∂F +∂e +∂χ(j) +ν+1 +∂g +�� +; +(92) +so we can infer that +{Z (1) +ν +, χ(2) +ν+1} = O(σ2ν+1) , +{R(1) +ν+1, χ(2) +ν+1} = O(σ2ν) , +1 +n!{. . . {{H (1), χ(2) +ν+1}, χ(2) +ν+1}, . . . , χ(2) +ν+1 +� +�� +� +n≥2 +} += O(σmin{n(ν+1)−2(n−1), n(ν+1)+ν−2(n−1), (n+1)(ν+1)−2n}) = O(σn(ν−1)+2) +because (80), (81), (92) vanish when F = F1 = Z (1) +ν +. Now, for all ν ≥ 2, n(ν − 1) + 2 > ν + 1, +n ≥ 2, hence, the proposition is valid for j = 2. For j ≥ 3, we have +H (j) = Z0 + Z (1) +ν ++ . . . + Z (j−1) +ν+j−2 + Z (j) +ν+j−1 + O(σν+j) + +� +l≥ν+j +R(j−1) +ν+j−1,l ++ {Z (1) +ν ++ . . . + Z (j−1) +ν+j−2, χ(j) +ν+j−1} + {R(j−1) +ν+j−1, χ(j) +ν+j−1} + . . . ++ +� +n≥2 +1 +n!{. . . {{H (j−1), χ(j) +ν+j−1}, χ(j) +ν+j−1}, . . . , χ(j) +ν+j−1 +� +�� +� +n +} , +(93) +19 + +and analogously +{Z (1) +ν +, χ(j) +ν+j−1} = O(σ2ν+j−1) , +{Z (j−1) +ν+j−2, χ(j) +ν+j−1} = O(σ2ν+2j−5) , +{R(j−1) +ν+j−1, χ(j) +ν+j−1} = O(σ2ν+2j−4) , +1 +n!{. . . {{H (j−1), χ(j) +ν+j−1}, χ(j) +ν+j−1}, . . . , χ(j) +ν+j−1 +� +�� +� +n≥2 +} += O(σmin{n(ν+j−1)−2(n−1), n(ν+j−1)+ν−2(n−1), n(ν+j−1)+ν+j−2−2n, (n+1)(ν+j−1)−2n}) += O(σn(ν+j−3)+2) . +However, since ν > 1, n ≥ 2, we readily find n(ν + j − 3) + 2 > ν + j − 1, which concludes the +proof. +2.4.4 +The case ν = 1 +Coming to ν = 1, one realizes that (91) produces same order σ2 non-normalized terms via +{R(1) +2 , χ(2) +2 } and {. . . {{Z0 + R(1) +2 , χ(2) +2 }, χ(2) +2 }, . . . , χ(2) +2 }, namely the resulting remainder is R(2) +2 , +so the scheme in Proposition 2 is not directly applicable beyond j = 1. Despite this, it is worth +noticing that if we manage to get rid of these spurious terms, by performing, for instance, an +extra normalization II, such that the new outcome returns R(II) = R(II) +3 +, then the algorithm (87) +will work for j ≥ 3 upon restarting the recursion from iteration II in place of 2. This is precisely +the claim we are about to show to complete the treatment. +Let us write (91) as H (2) = Z0 + Z (1 +1 + Z (2) +2 ++ R(2) +2 . Introduce the extra second normalization +II based on Proposition 2 targeted to R(2) +2,2 with generating function χ(II) +2 +. Then we have the +following. +Proposition 3. For ν = 1 and any ν1 ≥ 1, +H (II) = exp +� +Lχ(II) +2 +� +H (2) = Z0 + Z (1) +1 ++ Z (2) +2 ++ Z (II) +2 ++ R(II) +3 +. +(94) +Moreover the loop composed by (87)–(90) in Proposition 2 holds true for any j ≥ 4 under the +modifications +H (3) = exp +� +Lχ(3) +3 +� +H (II) = Z0 + Z (1) +1 ++ Z (2) +2 ++ Z (II) +2 ++ Z (3) +3 ++ R(3) +4 +, +(95) +H (j) = exp +� +Lχ(j) +j +� +H (j−1) = Z0 + +j +� +l=1 +Z (l) +l ++ Z (II) +2 ++ R(j) +j+1 . +(96) +Proof. We begin with a necessary generalization of Lemma 1. +Lemma 2. Given F1, F2 ∈ Cω(T×D) trigonometric monomials of the form (76), or equivalently +in terms of the sine, fulfilling the property of Lemma 1, addenda of the same type in the Lie +series transformation applied to F1 with respect to F2 preserve such property. +Proof. Since exp (LF2) F1 involves the computation of Poisson brackets of functions explicitly +independent on e, we have that (92), with F1, F2 in place of F, χ(2) +ν+1, is identically null, as well as +(81) because ∂F1/∂f = ∂F1/∂g, ∂F2/∂f = ∂F2/∂g by assumption. Thus, the bracket {F1, F2} +20 + +in the Lie series either does not introduce any eccentricity dependence at all, or only at numerator +through (50) multiplied by cos f or cos2 f; therefore its derivatives contain products of cosines +(sines) whose coefficients are independent on e like +G1(s1f + s2g + s3h + s4E1)G2(u1f + u2g + u3h + u4E1) , +G1, G2 = cos, sin . +The arguments are now either summed or subtracted, hence they clearly satisfy the property +concerned. By cascade reasoning for further nested brackets we conclude. +Remark 5. A straightforward use of the lemma in conjunction with formulae (80), (81), (92) +(χ(2) +ν+1 replaced by generic differentiable function) reveal that any transformed Hamiltonian H (j) +and corresponding generating function χ(j) +ν+j−1 encountered are regular at e = 0 in agreement +with D’Alembert rules, i.e. they never depend on negative powers of e. Furthermore, every time +one of the two entries of {·, ·} does not depend on e, the upshot due to item (i) in the proof of +Proposition 1, as soon as non-zero, is diminished by σ−1 instead of σ−2. +We consider step II: +H (II) = Z0 + Z (1) +1 ++ Z (2) +2 ++ Z (II) +2 ++ O(σ3) + +� +l≥3 +R(2) +2,l + {Z (1) +1 +, χ(II) +2 +} + {Z (2) +2 +, χ(II) +2 +} ++ {R(2) +2 , χ(II) +2 +} + . . . + +� +n≥2 +1 +n!{. . . {{H (2), χ(II) +2 +}, χ(II) +2 +}, . . . , χ(II) +2 +� +�� +� +n +} . +(97) +The analysis of the contributions reports these deductions, by which (94) follows. +• {Z (1) +1 +, χ(II) +2 +} = O(σ3) because Z (1) +1 +is independent on f, g, e. +• {Z (2) +2 +, χ(II) +2 +} = O(σ4) because Z (2) and χ(II) +2 +fulfil Lemma 2. Indeed, R(1) +2,2 depends on e at +most linearly by book-keeping rules, so it does χ(2) +2 +by construction. At this point we show +that for eccentricity dependent terms stemming from R(1) +2,2 (or equivalently χ(2) +2 ) d′(1) +2,λ,p = 0. +Lemma 3. Every trigonometric monomial in R(1) +2,2 explicitly dependent on e carries the +dependence on at least one of the two fast anomalies f, E1 as well, namely corresponding +coefficients in (82) are d(1) +2,λ,s = d′′(1) +2,λ,s, (s1, s4) ̸= (0, 0). +Proof. By Proposition 1, Lemma 1 and 2, the substitution φ1 = e1 sin E1 and the formulas +listed in Subsection 2.3.1, Subsection 2.3.3, we take out of (68) the order σ2 remainder and +it is not restrictive to assume ν1 = 1 in order to include also the e1 cos E1 dependent term +in (71): +R(1) +2,2 = R(0) +1,2 + a1 +∥r1∥ +� +n∗ +� +e1 cos E1 − 2e cos f +η3 +� +σ∂χ(1) +1 +∂f ++ +∂R(0) +1,1 +∂f +∂χ(1) +1 +∂δL +− +∂R(0) +1,1 +∂δL +∂χ(1) +1 +∂f +− 1 +L∗ +∂χ(1) +1 +∂ιc +� +ιc +∂R(0) +1,1 +∂f +− +∂R(0) +1,1 +∂h +� ++ 1 +L∗ +∂R(0) +1,1 +∂ιc +� +ιc +∂χ(1) +1 +∂f +− ∂χ(1) +1 +∂h +� +− 2 sin f +L∗e σ−1 ∂χ(1) +1 +∂f +� +∂R(0) +1,2 +∂g +− +∂R(0) +1,2 +∂f +� � +− a1 +2 +� +1 +∥r1∥ +� +n∗ +� +1 + 2e cos f +η3 +σ +� ∂χ(1) +1 +∂f ++ n1 +∂χ(1) +1 +∂E1 +� +, χ(1) +1 +� +2 +, +21 + +where {·, ·}2 indicates that we retain only σ2 quantities after the operation (in virtue of +Lemma 2 and Remark 5, inductions derived to demonstrate Proposition 1 are a coarser +bound and no other parts of order σ2 come out). Plugging in (70) and (65) for l = 1, 2 and +taking into account Lemma 1, upon simplifications the contributions involving e result +R(0) +1,2e − a1en∗ +η3 ∥r1∥σ2 +� +(s1,s4)̸=(0,0) +s1q′′ +1,s +s1n∗ + s4n1 +(cos((1 − s1)f − s1g − s3h − s4E1) ++ cos((1 + s1)f + s1g + s3h + s4E1)) , +(98) +where +R(0) +1,2e = +a1 +∥r1∥σ2 � +s∈Z4 +q2,s cos(s1f + s2g + s3h + s4E1) , +q2,s = e¯q2,s . +(99) +We employ now all D’Alembert rules to show that only the harmonics of interest can exist. +Following the same argument as in Lemma 1, let us write the cosine input of (99) using +modified Delaunay angles (77) also for P1 in relation to corresponding orbital elements (3) +(subscript ‘1’): +s1˜λ + (s1 − s2)˜p + (s2 − s3)˜q + s4˜λ1 + (s4 − s5)˜p1 + (s5 − s6)˜q1 , +sl ∈ Z , +in which ˜p1 = ˜q1 = 0. For the elimination of the apparent singularity at e = 0, we must +have 1 − |s1 + s2| ≥ 0 and even, hence s2 = s1 ± 1. Then, since R(0) +1,2e is independent on e1 +by book-keeping setting, analogously we must end up with s4 = s5. Regarding instead the +regularity at i1 = 0, because of the absence of i1 we must conclude that 0 − |s5 − s6| ∈ 2N, +namely s5 = s6. At this stage, we invoke the invariance under rotation around the Z axis, +which prescribes +s1 − s1 + s2 − s2 + s3 + s4 − s4 + s5 − s5 + s6 = s3 + s6 = 0 , +and summing up this implies s3 = −s4. Ultimately, concerning the inclination, we must +ensure that l − |s2 − s3| ∈ 2N, with l even as well again being i1 not involved, thus +s2 = s3 ± 2n, n ≤ l/2 natural number. Putting all together we arrive at +s1f + s2g + s3h + s4E1 =⇒ s1f + (s1 ± 1)g + (s1 ∓ 2n ± 1)h + (±2n ∓ 1 − s1)E1 , +which always depends on at least one among f, E1 since the coefficients s1, ±2n ∓ 1 − s1 +never vanish simultaneously. +By means of an identical reasoning and given the preservation of D’Alembert rules under +exp +� +Lχ(1) +1 +� +, we achieve the same outcome for the remaining part of (98) after replacing +s1 �→ 1 ± s1, indeed we find +(1 ± s1) + (1 ± s1 ± 1)g + (1 ± s1 ∓ 2n ± 1)h + (±2n ∓ 1 − 1 ∓ s1)E1 , +and no solutions to 1 ± s1 = 0, ±2n ∓ 1 − 1 ∓ s1 = 0. +Given that the order 2 normal form is sourced from the part of R(1) +2,2 explicitly indepen- +dent on fast angles, it turns out that it is free of e. Finally, R(2) +2,2 is free of e too, being +generated by terms in {R(1) +2,2, χ(2) +2 } and {. . . {{Z0 + R(1) +2,2, χ(2) +2 }, χ(2) +2 }, . . . , χ(2) +2 } subjected +to computation (i) of Proposition 1 (Remark 5). Again by construction, the same applies +to χ(II) +2 +. +22 + +• {R(2) +2 , χ(II) +2 +} = O(σ4) by Remark 5. +• 1 +n!{. . . {{H (2), χ(II) +2 +}, χ(II) +2 +}, . . . , χ(II) +2 +� +�� +� +n≥2 +} = O(σ4) consequently. +In order to conclude, we just need to check that the next step gives rise to an O(σ4) perturbation +and the cycle of normalizations can restart for j ≥ 4 in light of the bounds on σ from (93) at +the end of the proof of Proposition 2. Upon repeating the usual argument, it is easy to see that +the only bracket worth investigating is {Z (II) +2 +, χ(3) +3 }, that is, nevertheless, O(σ4) because Z (II) +2 +is made out of R(2) +2,2 independent on e. +Remark 6. By the above argument it is immediate to realize that even p2 ≡ 0 in (70) and (65) +for l = ν, so q′ +ν,p = 0 for all p ̸= (0, 0). +Serving as an example, a detailed demonstration of the normalization procedure exposed in +the present section for a simple model, containing just few terms of the disturbing function, is +presented in Appendix B. +3 +Numerical tests +3.1 +Computer-algebraic implementation of the normalization algorithm +Implementing the above normalization procedure, e.g. by use of a Computer Algebra System +(CAS), requires working with a finite truncation of the initial Hamiltonian model (11). To this +end, the disturbing function (13) multiplied by µ can be re-arranged as +µH1 = −Gm0µ +∥R∥ +∞ +� +κ1=0 +∞ +� +κ2=0 +κ2̸=1 +∞ +� +κ3=0 +˜hκ1,κ2,κ3µκ1 +�2r1 · R +∥R∥2 +�κ2 �∥r1∥ +∥R∥ +�2κ3 +, +(100) +where ˜hκ1,κ2,κ3 are real coefficients derived from the coefficients of (13). A convenient truncation +of (100) stems from defining two separate truncation orders in powers of µ (truncation order kµ), +and in powers of ∥r1∥ / ∥R∥ (multipole truncation order kmp), through the formula +H≤kµ,kmp +1 += −Gm0µ +∥R∥ +kµ−1 +� +κ1=0 +kmp +� +κ2=0,κ̸=1 +⌊kmp/2⌋ +� +κ3=0 +˜hκ1,κ2,κ3µκ1 +�2r1 · R +∥R∥2 +�κ2 �∥r1∥ +∥R∥ +�2κ3 +, +(101) +where ⌊·⌋ is the integer part function. Working with the truncated Hamiltonian H≤kµ,kmp = +H0 + H≤kµ,kmp +1 +, we then obtain a sequence of secular models Z (j), j = 1, 2, . . ., where j denotes +the normalization step, computed via the formula +Z (j) = Z0 + +j +� +l=1 +Z (l) +ν+l−1 . +(102) +In particular, we implement the following steps of the CAS algorithm: +(i) for a fixed value of µ, choose values for kµ, kmp, perform the corresponding expansions of +the Hamiltonian as in (100) and compute the truncated model H≤kµ,kmp; +23 + +(ii) choose the reference values of a∗ and e∗; +(iii) pass to variables (f, g, h, E1, δL, e, η, ιc, ιs, J1) and parameters L∗, e1, a1, η1 on the basis of +the selected a∗; +(iv) compute ν and ν1 (Eq.(7)); +(v) set the appropriate book-keeping weights following the rules in Subsection 2.3.2 and expand +correspondingly the Hamiltonian in δL up to σνkµ; +(vi) drop constants, perform the identity operation (63), discard book-keeping powers larger +than νkµ and introduce n∗; +(vii) if ν > 1, compute the generating function (70) as well as the first-normalized Hamiltonian +H (1) by the Lie series operation (66) truncated at the maximum book-keeping order Nbk = +νkµ; if ν = 1, compute H (1) (always truncated to the book-keeping order Nbk) via the +procedure of Subsection 2.4.4; +(viii) compute the successive normalizations H (j), truncated at book-keeping order Nbk via the +procedure of Subsection 2.4.3, up to a maximum normalization order ν + jmax − 1 < Nbk, +jmax ≤ ν(kµ − 1); this allows to obtain truncated Hamiltonian models containing a finite +number of normal form terms as well as a finite number of terms provided by the truncated +remainder. +In the CAS implementation of the above algorithm we work with numerical coefficients, +substituting all constants with their corresponding numerical values. Several types of numerical +tests of the precision and overall performance of the method can be carried out as exemplified +in the sequel. +3.2 +Numerical examples in the Sun-Jupiter ER3BP: semi-analytic orbit prop- +agation +For all numerical tests below we refer to the Sun-Jupiter one (µ = 9.5364 · 10−4). We employ +Earth-orbit based units, such that Gm0 = 4π2AU3/y2, a1 = 5.2044AU, so that Jupiter’s period +is T1 = 11.86 y. +Jupiter’s mean motion is n1 = 2π/T1, and eccentricity e1 = 0.0489, used +throughout all computations in the framework of the ER3BP model. +In all tests below, a particle’s orbit is defined by providing the initial conditions a(0), e(0), i(0), +complemented by f(0) = g(0) = h(0) = 0. +Our basic probe of the efficiency of the normalization method in the framework of the ER3BP +is given by comparing the short-period oscillations of the orbital elements a(t), e(t), i(t), g(t), h(t), +as found by two different methods. +Direct Cartesian propagation: the initial conditions z(0) := (a(0), e(0), i(0), f(0), g(0), h(0)) are +mapped into initial conditions for the Cartesian canonical positions and conjugate momenta +(X(0), Y (0), Z(0), PX(0), PY (0), PZ(0)). Using Hamilton’s equations with the full Hamiltonian +(1) (setting also J1(0) = 0, M1(0) = 0), we obtain the numerical evolution (X(t), Y (t), Z(t), +PX(t), PY (t), PZ(t)), which can be transformed to element evolution +z(t) = (a(t), e(t), i(t), f(t), g(t), h(t)) . +Semi-analytical propagation: following the implementation of the normalization algorithm as de- +scribed in the previous subsection, the initial osculating element state vector z(0) is transformed +24 + +0 +500 +1000 +49.98 +49.99 +50.00 +0 +500 +1000 +0.098 +0.099 +0.100 +0 +500 +1000 +9.99992 +9.99996 +10.00000 +0 +500 +1000 +29.98 +29.99 +30.00 +0 +500 +1000 +0.148 +0.149 +0.150 +0 +500 +1000 +9.99974 +9.99987 +10.00000 +Figure 2: First and second example (ER3BP). Data: a∗ = 50AU, e∗ = 0.1 (ν = 3), i(0) = 10◦, +kµ = kmp = 2 (top panels); a∗ = 30AU, e∗ = 0.15 (ν = 4), i(0) = 10◦, kµ = kmp = 2 (bottom +panels). Black curves represent semi-analytic time variations (our method), while red curves +stand for Cartesian series. +into an initial condition for the corresponding ‘mean element’ state vector ξ(j)(z(0)), i.e., the +element vector corresponding to the new canonical variables conjugated to the original ones af- +ter j near-identity normalizing transformations. This is computed by the Lie series composition +formula truncated at book-keeping order Nbk: +ξ(j)(z) = +� +exp +� +L−χ(1) +ν +� +◦ exp +� +L−χ(2) +ν+1 +� +◦ . . . ◦ exp +� +L−χ(j) +ν+j−1 +� +z +�≤Nbk +, +(103) +using Eq.(69) for the inverse series. We then obtain the evolution of the mean element vector +ξ(j)(t) through numerical integration of the secular equations of motion +˙ξ(j) = J∇Z (j)(ξ(j)) +(104) +(J standard symplectic unit). This can be back-transformed to yield the evolution of the oscu- +lating element vector z(t) using the truncated Lie series composition formula +z(ξ(j)) = +� +exp +� +Lχ(j) +ν+j−1 +� +◦ exp +� +Lχ(j−1) +ν+j−2 +� +◦ . . . ◦ exp +� +Lχ(1) +ν +� +ξ(j) +�≤Nbk +. +(105) +Note that both the direct and inverse transformations (Eqs.(103) and (105)), as well as Hamilton’s +secular equations (104), can be computed in closed form, using the Poisson algebra rules of +Subsection 2.3. We then call semi-analytic the evolution of the element vector z(t) obtained via +the formula +z(t) = z(ξ(j)(t)) . +(106) +Fig. 2 shows the comparison between the Cartesian and the semi-analytical propagation of the +elements in ‘easy’ cases, where the particle departs from initial conditions a(0) = 50AU (top left +25 + +0 +200 +400 +600 +800 +1000 +49.70 +49.75 +49.80 +49.85 +49.90 +49.95 +50.00 +0 +200 +400 +600 +800 +1000 +0.00 +0.05 +0.10 +0.15 +0.20 +Figure 3: Third example (ER3BP). Data: a∗ = 50AU, e∗ = 0.7 (ν = 20), i(0) = 20◦, kµ = 2, +kmp = 3. On the left, the black curve represents the semi-analytic time variation of the semi- +major axis (our method) versus the one found by propagation of the Cartesian equations of +motion (red). The right panel shows the evolution of the corresponding percent relative error +E% +a . +panel) or a(0) = 30AU (bottom left panel), with a relatively low value of the eccentricity e(0) = +0.1 or e(0) = 0.15 respectively (middle panels) and inclination i(0) = 10◦ (right panels). In these +cases, the distance ratio ∥r1∥ / ∥R∥ is small (about 0.1-0.2), a fact implying that the quadrupolar +expansion (kmp = 2) suffices to have obtained a relative error of about 0.1% in the representation +of the Hamiltonian perturbation H1. Going to higher multipoles is straightforward, albeit with +a significant computational cost as the number of terms in the Hamiltonian grows significantly. +On the other hand, even with low-order truncations of the Hamiltonian we achieve to have an +accurate semi-analytical representation of the O(µ) short-period oscillations in all three ‘action- +like’ elements (semi-major axis, eccentricity, inclination). Most notably, keeping a(0) = 50AU +but changing the eccentricity to e(0) = 0.7, i.e., beyond the Laplace value, yields an orbit whose +pericenter is at ∥Rp∥ = 15AU, implying a distance ratio ∥r1∥ / ∥R∥ ≈ 0.3 (Fig. 3). This time, an +octupole truncation (kmp = 3) is required to produce an approximation of the Hamiltonian model +at the level of a relative error of 0.1%. Still, however, as shown in Fig. 3 the semi-analytical +propagation of the orbit is able to track the fully numerical one with an error which does not +exceed 0.2% even close to the orbit’s pericentric passages. +In the above examples, the maximum number of normalization steps at which the secular +Hamiltonian is computed was set equal to jmax = 3, jmax = 4 and jmax = 4 respectively, which +corresponds to the best match in all cases. As discussed in the next subsection, an estimate of the +minimum possible error in the semi-analytic propagation of the trajectories requires computing +first the so-called optimal number of normalizations jopt (or equivalently optimal normalization +order ν + jopt − 1) as a function of the reference values (a∗, e∗) within a model given by a preset +fixed multipole truncation order. Owing to the fact that the same divisors appear in the ER3BP +and in the CR3BP, we verify with numerical examples that the error analysis yields essentially +identical results in either case. However, the computation of the optimal normalization is easier to +perform in the CR3BP, owing to the considerably smaller number of terms produced in the CAS +computation of the normal form. Hence, we now turn our attention to this latter computation. +26 + +3.3 +Numerical examples in the Sun-Jupiter planar CR3BP: order and size +of the optimal remainder +3.3.1 +Trajectory propagation: optimal remainder +A considerable reduction of the computational cost occurs in the case of the planar and circular +R3BP. This is due, in particular, to the following: +• the dependence on M1 becomes explicit (M1 = E1 in (2)), while a1 = ∥r1∥. As a conse- +quence, φ1 = 0. +• no terms involving (h, H) appear in the disturbing function, thus ιc, ιs are discarded; +• no terms requiring a book-keeping in terms of the exponent ν1 appear, hence, only ν is +defined, as in (7); +• d′(j) +l,λ,p = 0 for every j, l, λ, p in (90), (86), and consequently p1 = p2 ≡ 0 in (88). This is due +to the fact that the expression (16) reduces to +r1 · R = ∥r1∥ ∥R∥ cos(f + g − M1) , +(107) +which always depends on the difference g − M1 by D’Alembert rules. This implies that, +unlike the ER3BP, the action G (and the corresponding eccentricity e) are integrals of the +secular Hamiltonian; +• as a consequence no lower or equal book-keeping order terms appear in any Poisson bracket +of the first normalization step in the case ν = 1. Hence Proposition 3 is redundant. +Owing to the above, in the planar CR3BP we are able to make normal form computations +in a grid of points in the plane (a∗, e∗) up to a sufficiently high normalization order so that the +asymptotic character of the series computed by the algorithm of Section 2 can show up. To this +end, we introduce an estimate of the size of the series’ remainder after j normalization steps via +the upper norm bound +E (j) = +νkµ +� +l=ν+j +� +s∈Z3 +|d(j) +l,s | ≥ +���R(j) +ν+j +��� +∞ , +j = 1, . . . , ν(kµ − 1) , +(108) +where ∥·∥∞ denotes the sup norm. +Plotting E (j) against the number of normalization steps +j allows then to estimate the error committed at any step (size of the remainder). Figure 4 +yields an example of such computation. The relevant fact is that there is an optimal number of +normalization steps (j = jopt = 6) where the estimate E (j) of the remainder size yields a global +minimum. +Although a systematic investigation of the dependence of the optimal number of normalization +steps jopt on the parameters (a∗, e∗) is beyond our present scope, Figs. 5 and 6 allow to gain some +insight into the question. The most relevant remark concerns the dependence of the behavior of +the curve E (j) (versus j) on how close to the ‘hierarchical’ regime the trajectory with reference +values (a∗, e∗) is. As a measure of the hierarchical character of an orbit we adopt either the ratio +of the semi-major axes a1/a∗, or of the pericentric distances ∥r1∥ / ∥Rp∥ = a1(1 − e1)/(a∗(1 − +e∗)) = a1/(a∗(1 − e∗)). +Fig. +5 (a∗ = 30AU, e∗ = 0.5) implies a pericentric distance ratio +∥r1∥ / ∥Rp∥ ≈ 0.3 smaller than the one of the example of Fig. 4 (∥r1∥ / ∥Rp∥ ≈ 0.4). We observe +that the optimal number of normalization steps in the former case satisfies jopt = 10, i.e., it is +larger than in the latter case. Fig. 6 shows, instead, an example of orbit far from the hierarchical +limit, satisfying the estimate ∥r1∥ / ∥Rp∥ ≈ 0.7. In this case a higher order multipole expansion +27 + +1 +2 +3 +4 +5 +6 +7 +8 +-3.6 +-3.4 +-3.2 +-3.0 +-2.8 +0 +150 +300 +19.97 +19.99 +20.01 +0 +150 +300 +0.3985 +0.3995 +0.4005 +0 +150 +300 +-0.002 +0.001 +0.004 +Figure 4: Fourth example (planar CR3BP). Data: a∗ = 20AU, e∗ = 0.4 (ν = 8), kµ = 2, +kmp = 3. The estimate E (j) is depicted in semi-logarithmic scale on top left panel. The direct +comparison of the semi-analytic (black) evolution vs. +the fully numerical (red) one for the +osculating elements a(t), e(t), g(t) are shown in the top right and bottom panels respectively. +The semi-analytic curves are obtained for j = jopt = 6, where E (j) is minimum. +28 + +1 2 3 4 5 6 7 8 9 10 +-3.4 +-3.3 +-3.2 +-3.1 +-3.0 +-2.9 +-2.8 +0 +150 +300 +29.94 +29.97 +30.00 +0 +150 +300 +0.498 +0.499 +0.500 +0 +150 +300 +-0.002 +0.005 +0.012 +Figure 5: Fifth example (planar CR3BP): a∗ = 30AU, e∗ = 0.5 (ν = 10), kµ = 2, kmp = 3. Plot +types and color conventions are the same as in Fig. 4. The semi-analytic curves are obtained for +j = jopt = 10. +(kmp = 5) is required to obtain a precise truncated Hamiltonian model for this orbit. We note, +however, that the normalization procedure performs well, producing a decreasing remainder as +a function of j up to the point where it is arrested, i.e. j = 6 = ν(kµ − 1). We find numerically +that this performance is deteriorated as we gradually approach the condition ∥r1∥ / ∥R∥ = 1, +beyond which the multipole expansion of the Hamiltonian is no longer convergent. +3.3.2 +Semi-analytical determination of the domain of secular motions +The results shown in the two previous subsections refer to isolated examples of orbits treated +within various multipole truncation orders as well as different choices of the number of nor- +malization steps, searching each time to arrive at the best approximating secular model given +computational restrictions. +In the present subsection, we aim to investigate the behavior of +the remainder in a closed-form normalization with uniform choice of all truncation orders of +the problem, but performed, instead, in a fine grid (100 × 20) of reference values in the plane +(a∗, e∗). To this end, we set kµ = 2 (second order in the mass parameter), and fix kmp = 3 +(octupole approximation). The latter choice, imposed by computational restrictions, yields an +initial model whose error with respect to the full Hamiltonian becomes of the order of 1% only +for a∗ > 2a1. However, for reasons explained below, a computation within the framework of +the octupole approximation becomes relevant to the problem addressed in the sequel also in the +range 1.5a1 < a∗ < 2a1, while higher multipoles are required to address still smaller values of a∗. +29 + +1 +2 +3 +4 +5 +6 +-3.4 +-3.2 +-3.0 +-2.8 +-2.6 +-2.4 +0 +150 +300 +7.98 +8.01 +8.04 +0 +150 +300 +0.088 +0.095 +0.102 +0 +150 +300 +-0.08 +0.05 +0.18 +Figure 6: Sixth example (planar CR3BP): a∗ = 8, e∗ = 0.1 (ν = 3), kµ = 3, kmp = 5. Plot +types and color conventions are the same as in Fig. 4. The semi-analytic curves are obtained for +j = jopt = 6. +The result of the above computation is summarized in Fig. +7: the left panel shows in +logarithmic color scale the size of the remainder, estimated by the value of E (n)(a∗, e∗) computed +as in (108), corresponding to each point in the plane (a∗, e∗), where the number of normalization +steps is set as n = min{ν(kµ − 1), 7} = min{ν, 7}. The maximum value n = 7 is, again, imposed +by computational restrictions, and it implies that n varies with e∗ up to about e∗ = 0.37. +The relevant information in Fig. 7 is provided by the black curve, which corresponds to +the isocontour E (n)(a∗, e∗) = 10−2. +Since in the original Hamiltonian we have the estimate +E (0)(a, e) := H≤kµ,kmp +1 += O(10−2), the black curve provides a rough estimate of the limiting +border dividing the plane (a∗, e∗) in two domains: in the one below the black curve the progressive +elimination of the fast angles by the iterative normalization steps leads to a secular model whose +remainder decreases with the number of normalization steps j at least up to j = n. +A physical interpretation of the border approximated through the isocontour E (n)(a∗, e∗) = +10−2 can be given through a comparison with a numerical stability map obtained, e.g., as in +the right panel of Fig. 7. For each trajectory in a 300 × 900 grid in (a, e), the plot shows in +color scale the value of the Fast Lyapunov Indicator (FLI, see [9] for a review) obtained after +integrating the variational equations of motion together with the equations of motion of the full +Hamiltonian model for a time equal to 50 periods of Jupiter. Thus, deep blue colors indicate +the most regular, and light yellow the most chaotic orbits as identified by the value of the FLI. +Superposed to the FLI cartography are three curves: +30 + +Figure 7: Left panel: computation of log10(E (n)), n = min{ν, 7}, kmp = 3, over a 100 × 20 +(a, e) grid. For every e = e∗, n different normalizations are executed and then evaluated for +each a = a∗. Right panel: short-period FLI map over a 300 × 900 (a, e) grid of initial data +integrated for 50T1. As indicated, the three curves represent, respectively, the line of constant +pericenter of the particle’s trajectory equal to the radius of Jupiter’s orbit ∥r1∥ = ∥rJ∥ (red), +Hill’s stability criterion (brown) and the isolevel E (n) = 1% (black). Each region enclosed by +two consecutive above curves is labeled with the corresponding regime of motion. The main +mean-motion resonances are reported below the pictures. +(i) the ‘perihelion crossing curve’ (red) yields the locus of values satisfying the condition +a(1 − e) = ∥rJ∥ = aJ (in the circular case), that is the points where the pericenter of +the test particle’s orbit comes at distance equal to the radius of Jupiter’s orbit; +(ii) the Hill limit [12] (brown) is based on the relationship CJac(a, e) = CJac(L1), where CJac +is the particle’s Jacobi constant as function of the orbital elements and CJac(L1) its value +at the Lagrangian point L1; +(iii) the isocontour E (n)(a, e) = 10−2 (black, same as in the left panel of Fig. 7). +Of the above three curves, the perihelion crossing curve is analogous, in the R3BP, of the +so-called Angular Momentum Deficit criterion (AMD, [8]) used to separate systems protected +from perihelia crossings in the case of the full planetary three-body problem. As indicated by the +FLI cartography data, Hill’s curve gives an overall better approximation separating the domain +of strong chaos (yellow) from the domain of regular or weakly-chaotic orbits (all blue nuances). +This is expected, since the Hill’s curve separates orbits for which Jupiter’s gravitational effect +becomes (at least temporarily) dominant from those for which it does not. Nevertheless, through +the FLI cartography we note the presence of a large domain between the curves (ii) and (iii), +where the trajectories, while protected from close encounters, are subject to the long term effects +on dynamics produced by resonant multiplets associated with the mean-motion resonances of +the problem (the most important of which are marked in the figure). Note that in the octupole +approximation, the Hamiltonian contains harmonics including all combinations of the fast angles +of the form cos(s1f + s2(g − M1)), with +(s1, s2) = (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (7, 3), +31 + +0.6 +0 +0.6 +0.55 +Apsidal +-0.5 +0.55 +Hill +Apsidal +Close encounter regime +Method +Hill +4.5 +0.5 +-1 +0.5 +Method +0.45 +-1.5 +0.45 +Crdssing orbit +4 + Resonant regime +0.4 +-2 +0.4 +regime +0.35 +-2.5 +e +e +0.35 +3.5 +0.3 +-3 +0.3 +3 +0.25 +-3.5 +0.25 +0.2 +-4 +0.2 +Secular regime +2.5 +0.15 +-4.5 +0.15 +0.1 +5 +0.1 +1.5 +2.5 +3 +1.5 +2 +2.5 +3 +a/ IIrj +a / IIrj ll +2:5 +1:4 +2:3 +1:3 +1:5(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), +(1, −1), (2, −1), (3, −1), (1, −2), (1, −3), +thus including all harmonics associated with the mean-motion resonances detected in the FLI +cartography of Fig. 7 for a > 1.5aJ. Through the closed-form normalization (Eqs.(70) and (86)) +we then obtain small divisors in the series at every value of the semi-major axis a∗ for which one +of the resonant combinations s1n∗ − s2nJ, nJ = n1, takes a value near zero. All these incidences +lead to Arnold tongue-like spikes pointing downwards in the curve (iii), marking the failure of +the approximation of the orbits based on a non-resonant normal form construction. On the +other hand, we observe that, for any value of a∗ there is a threshold value of the eccentricity +e∗,s, such that, for e∗ < e∗,s no visible effects of the harmonics associated with mean-motion +resonances are visible in the FLI cartography. This implies that the secular models constructed +by eliminating all harmonics involving the fast angles of the problem describe with good precision +the dynamics in this domain, called, for this reason, the domain of secular motions. In physical +terms, the domain of secular motions corresponds to initial conditions for which the gravitational +perturbation of Jupiter is only felt in the ‘Laplacian’ meaning, i.e., as a mass distributed along a +ring coinciding with Jupiter’s orbit. The curve (iii) then yields the limit of this domain, which, +as found by the FLI cartography, is well distinct from the limit of the Hill domain. +The overall situation can therefore be summarized with the identification of four regimes of +motion (specified in the FLI chart): +• the ‘crossing orbit regime’ (above curve (i)); +• the ‘close encounter regime’ (between curves (i) and (ii)); +• the ‘resonant regime’ (between curves (ii) and (iii)); +• the ‘secular regime’ (below curve (iii)). +4 +Conclusions +In summary, in the present paper we have proposed a closed-form method for the derivation +of secular Hamiltonian models (normal forms) with a small (albeit finite minimum) remainder +applicable to the R3BP in the case when the particle’s trajectory is exterior to the trajectory of +the primary perturber. Also, using this method we were led to the definition of a new heuristic +limit separating the motions whose character is ‘secular’, i.e., not affected by short-period effects, +from the rest of motions in the R3BP. In particular: +1. Section 2 develops the formal aspects of the method, which heavily relies on the use of a +book-keeping parameter to simultaneously account for all small quantities of the problem +as they appear not only in the Hamiltonian and Lie generating functions, but also in the +closed-form version of all formulas involved in the Poisson algebra between the Delaunay +canonical variables of the problem. A rigorous demonstration of the consistency of the +method is then given through Propositions 1, 2 and 3, which also estabilish the explicit +formulas for the implementation of one iterative step of the closed-form normalization +algorithm. +2. Section 3 gives numerical examples of the implementation and precision of the algorithm +in the spatial elliptic, as well as in the planar circular R3BP, examining, also numerically, +the method’s convergence properties. The effect of choosing different truncation orders (in +powers of the mass parameter µ or in the multipole expansion) is discussed, along with +32 + +several simplifications to the normalization procedure which hold in the circular case. The +essentially asymptotic character of the series is established through numerical examples, +showing the existence of an optimal number of normalization steps, after which the size of +the remainder becomes the minimum possible. +3. A key aspect of the above presented method lies in the possibility to exploit the behavior of +the size of the remainder as a function of the number of normalizing steps in order to obtain +a clear separation of two well-distinct domains, as also identified by purely numerical (FLI +cartography) means: one, called the domain of secular motions corresponds to the domain +where the harmonics in the Hamiltonian associated with resonant combinations of the fast +angles (anomalies) of the problem produce no dynamical effect on the orbits visible at the +level of the FLI cartography. From the semi-analytical point of view, this turns to be the +domain where a non-resonant construction as the one proposed in section 2 produces no +(nearly-)resonant divisors up to the optimal normalization step. As a consequence, only +the angles associated with the motions of the perihelion and of the line of nodes survive in +the final normal form. We show numerically how to use the information on the size of the +normal form remainder in order to determine semi-analytically the border of the domain +of secular motions in the case of the Sun-Jupiter system. We finally give evidence that this +border is well distinct from the border of the domains defined either by the Hill stability +or by the perihelion crossing criterion. +Appendix +A +Computation of Poisson bracket’s intermediate derivatives +Derivatives (31)–(43) are computed combining adequately definitions (3), the polar relationship +(15), including its alternative expression involving the eccentric anomaly E +∥R∥ = a(1 − e cos E) , +(109) +∥r1∥ via (2) (analogous to (109)), Kepler’s equations +ℓ = E − e sin E , +M1 = E1 − e1 sin E1 , +(110) +and the trigonometric equalities +cos f = cos E − e +1 − e cos E , +sin f = +η sin E +1 − e cos E . +(111) +Eq.(31) comes from (109) and (15) by total differentiation with respect to ℓ: +d +dℓ ∥R∥ +(109) += +∂ ∥R∥ +∂E +∂E +∂ℓ = +ae sin E +1 − e cos E +(15) += ∂ ∥R∥ +∂f +∂f +∂ℓ = +aη2e sin f +(1 + e cos f)2 +∂f +∂ℓ , +since a, e do not depend on ℓ, where ∂E/∂ℓ is deduced from the first of (110) making use of the +derivative of inverse functions (∂ℓ/∂E ̸= 0 is ensured). Thus the result by (111). +Eqs.(32), (33) are straightforwardly yielded taking respectively ordinary differentiation and the +inverse derivative once again of dM1/dE1 ̸= 0 from the second of (110): +dE1 +dM1 += +1 +1 − e1 cos E1 += +a1 +∥r1∥ . +33 + +Now solving for e in (3) and partially differentiating, we immediately have Eqs.(35) and (38), +from which Eqs.(36), (39) as +∂η +∂δL = −e +η +∂e +∂δL = − η +L , +∂η +∂G = −e +η +∂e +∂G = 1 +L . +The true anomaly derivatives with respect to the actions are slightly more elaborated. Employing +(111), +− sin f ∂f +∂δL = +∂ +∂δL cos f = ∂ +∂e +� cos E − e +1 − e cos E +� ∂e +∂δL + ∂ +∂E +� cos E − e +1 − e cos E +� ∂E +∂δL , +that leads upon simplifications to +∂f +∂δL = sin f +eL + 1 + e cos f +η +∂E +∂δL ; +finally we explicit ∂E/∂δL exploiting the corresponding Kepler equation (110) and the inter- +independence ℓ, δL by conjugacy: +0 = +d +dδL(E − e sin E) = ∂E +∂δL − ∂e +∂δL sin E − e cos E ∂E +∂δL +=⇒ +∂E +∂δL = η sin f +eL +, +thereby Eq.(34). +The relation for ∂f/∂G is achieved precisely in the same manner, so one finds out +∂f +∂G = −sin f +ηeL + 1 + e cos f +η +∂E +∂G , +∂E +∂G = −sin f +eL +, +that is Eq.(37). +Finally, derivatives (40), (42) involving ιc = cos i easily follow again by partial differentiation in +(3) with respect to G and H respectively; while for those containing ιs = sin i we can rely, for +example, to the identity sin2 i + cos2 i = 1: +0 = 2 sin i∂ιs +∂G + 2 cos i∂ιc +∂G +and consequently Eq.(41) provided sin i ̸= 0, as well as Eq.(43) repeating the same argument +with the variable H. +B +Example of normalization for a µ2 quadrupolar expansion +Consider the following toy model Hamiltonian with kµ = kmp = ν = 2, ν1 = 1, according to +conventions introduced in §2.4.1: +H (0) = Z0 + R(0) +2,2 + R(0) +2,3 + R(0) +2,4 , +34 + +where +R(0) +2,2 = σ2 +� +− 3a3 +1G4µm4 +0ι2 +c cos (2 (E1 − f − g − h)) +16L6∗ ∥r1∥ +− 3a3 +1G4µm4 +0ι2 +c cos (2 (E1 + f + g − h)) +16L6∗ ∥r1∥ +− 3a3 +1G4µm4 +0ιc cos (2 (E1 − f − g − h)) +8L6∗ ∥r1∥ ++ 3a3 +1G4µm4 +0ιc cos (2 (E1 + f + g − h)) +8L6∗ ∥r1∥ ++ 3a3 +1G4µm4 +0ι2 +c cos (2 (E1 − h)) +8L6∗ ∥r1∥ ++ 3a3 +1G4µm4 +0ι2 +c cos(2(f + g)) +8L6∗ ∥r1∥ +− 3a3 +1G4µm4 +0ι2 +c +8L6∗ ∥r1∥ +− 3a3 +1G4µm4 +0 cos (2 (E1 − f − g − h)) +16L6∗ ∥r1∥ +− 3a3 +1G4µm4 +0 cos (2 (E1 + f + g − h)) +16L6∗ ∥r1∥ +− 3a3 +1G4µm4 +0 cos (2 (E1 − h)) +8L6∗ ∥r1∥ +− 3a3 +1G4µm4 +0 cos(2(f + g)) +8L6∗ ∥r1∥ ++ a3 +1G4µm4 +0 +8L6∗ ∥r1∥ − 3a1δL2G2m2 +0 +2L4∗ ∥r1∥ +− a1G2µm2 +0 +L2∗ ∥r1∥ +� +. +35 + +The first step j = 1 of the method aims precisely at normalizing R(0) +2,2 via (71) solved by +χ(1) +2 += σ3 +� +3G4µa2 +1ι2 +cφ1n2 +∗m4 +0 +8n1L6∗ +� +n2 +1 − n2∗ +� − +G4µa2 +1φ1n2 +∗m4 +0 +8n1L6∗ +� +n2 +1 − n2∗ +� +− 3G4µa2 +1n1ι2 +cφ1m4 +0 +8L6∗ +� +n2 +1 − n2∗ +� ++ G4µa2 +1n1φ1m4 +0 +8L6∗ +� +n2 +1 − n2∗ +� + +G2µφ1n2 +∗m2 +0 +n1L2∗ +� +n2 +1 − n2∗ +� ++ 3G2δL2φ1n2 +∗m2 +0 +2n1L4∗ +� +n2 +1 − n2∗ +� − G2µn1φ1m2 +0 +L2∗ +� +n2 +1 − n2∗ +� − 3G2δL2n1φ1m2 +0 +2L4∗ +� +n2 +1 − n2∗ +� +� ++ σ2 +� +− 3G4µ sin (2 (E1 − h)) a2 +1ι2 +cn2 +∗m4 +0 +16n1L6∗ +� +n2 +1 − n2∗ +� ++ 3G4µ sin (2 (E1 − h)) a2 +1n2 +∗m4 +0 +16n1L6∗ +� +n2 +1 − n2∗ +� ++ 3G4µ sin(2(f + g))a2 +1n∗m4 +0 +16L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (−f − g − h + E1)) a2 +1n∗m4 +0 +32L6∗ +� +n2 +1 − n2∗ +� ++ 3G4µ sin (2 (f + g − h + E1)) a2 +1n∗m4 +0 +32L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin(2(f + g))a2 +1ι2 +cn∗m4 +0 +16L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (−f − g − h + E1)) a2 +1ι2 +cn∗m4 +0 +32L6∗ +� +n2 +1 − n2∗ +� ++ 3G4µ sin (2 (f + g − h + E1)) a2 +1ι2 +cn∗m4 +0 +32L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (−f − g − h + E1)) a2 +1ιcn∗m4 +0 +16L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (f + g − h + E1)) a2 +1ιcn∗m4 +0 +16L6∗ +� +n2 +1 − n2∗ +� ++ 3G4µ sin (2 (E1 − h)) a2 +1n1ι2 +cm4 +0 +16L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (−f − g − h + E1)) a2 +1n1ι2 +cm4 +0 +32L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (f + g − h + E1)) a2 +1n1ι2 +cm4 +0 +32L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (E1 − h)) a2 +1n1m4 +0 +16L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (−f − g − h + E1)) a2 +1n1m4 +0 +32L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (f + g − h + E1)) a2 +1n1m4 +0 +32L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin (2 (−f − g − h + E1)) a2 +1n1ιcm4 +0 +16L6∗ +� +n2 +1 − n2∗ +� ++ 3G4µ sin (2 (f + g − h + E1)) a2 +1n1ιcm4 +0 +16L6∗ +� +n2 +1 − n2∗ +� +− 3G4µ sin(2(f + g))a2 +1n2 +1m4 +0 +16L6∗n∗ +� +n2 +1 − n2∗ +� ++ 3G4µ sin(2(f + g))a2 +1n2 +1ι2 +cm4 +0 +16L6∗n∗ +� +n2 +1 − n2∗ +� +� +, +so that the new truncated Hamiltonian becomes +H (1) = Z0 + Z (1) +2 ++ R(1) +3,3 + R(1) +3,4 , +with +Z (1) +2 += σ2 +� +−3a2 +1G4µm4 +0ι2 +c +8L6∗ ++ a2 +1G4µm4 +0 +8L6∗ +− 3δL2G2m2 +0 +2L4∗ +− G2µm2 +0 +L2∗ +� +36 + +and +R(1) +3,3 = σ3 +� +− 3eG4µ cos (f + 2g + 2h − 2E1) a3 +1ι2 +cn∗m4 +0 +8η3 ∥r1∥ L6∗ (2n1 − 2n∗) +− 3eG4µ cos (3f + 2g + 2h − 2E1) a3 +1ι2 +cn∗m4 +0 +8η3 ∥r1∥ L6∗ (2n1 − 2n∗) ++ 3G4µ cos (2f + 2g + 2h − 3E1) a3 +1e1ι2 +cn∗m4 +0 +16 ∥r1∥ L6∗ (2n1 − 2n∗) ++ 3G4µ cos (2f + 2g + 2h − E1) a3 +1e1ι2 +cn∗m4 +0 +16 ∥r1∥ L6∗ (2n1 − 2n∗) +− 3eG4µ cos (f + 2g + 2h − 2E1) a3 +1ιcn∗m4 +0 +4η3 ∥r1∥ L6∗ (2n1 − 2n∗) +− 3eG4µ cos (3f + 2g + 2h − 2E1) a3 +1ιcn∗m4 +0 +4η3 ∥r1∥ L6∗ (2n1 − 2n∗) ++ 3G4µ cos (2f + 2g + 2h − 3E1) a3 +1e1ιcn∗m4 +0 +8 ∥r1∥ L6∗ (2n1 − 2n∗) ++ 3G4µ cos (2f + 2g + 2h − E1) a3 +1e1ιcn∗m4 +0 +8 ∥r1∥ L6∗ (2n1 − 2n∗) +− 3eG4µ cos (f + 2g + 2h − 2E1) a3 +1n∗m4 +0 +8η3 ∥r1∥ L6∗ (2n1 − 2n∗) +− 3eG4µ cos (3f + 2g + 2h − 2E1) a3 +1n∗m4 +0 +8η3 ∥r1∥ L6∗ (2n1 − 2n∗) ++ 3G4µ cos (2f + 2g + 2h − 3E1) a3 +1e1n∗m4 +0 +16 ∥r1∥ L6∗ (2n1 − 2n∗) ++ 3G4µ cos (2f + 2g + 2h − E1) a3 +1e1n∗m4 +0 +16 ∥r1∥ L6∗ (2n1 − 2n∗) ++ 3eG4µ cos (f + 2g − 2h + 2E1) a3 +1ι2 +cn∗m4 +0 +8η3 ∥r1∥ L6∗ (2n1 + 2n∗) ++ 3eG4µ cos (3f + 2g − 2h + 2E1) a3 +1ι2 +cn∗m4 +0 +8η3 ∥r1∥ L6∗ (2n1 + 2n∗) +− 3G4µ cos (2f + 2g − 2h + E1) a3 +1e1ι2 +cn∗m4 +0 +16 ∥r1∥ L6∗ (2n1 + 2n∗) +− 3G4µ cos (2f + 2g − 2h + 3E1) a3 +1e1ι2 +cn∗m4 +0 +16 ∥r1∥ L6∗ (2n1 + 2n∗) +− 3eG4µ cos (f + 2g − 2h + 2E1) a3 +1ιcn∗m4 +0 +4η3 ∥r1∥ L6∗ (2n1 + 2n∗) +− 3eG4µ cos (3f + 2g − 2h + 2E1) a3 +1ιcn∗m4 +0 +4η3 ∥r1∥ L6∗ (2n1 + 2n∗) ++ 3G4µ cos (2f + 2g − 2h + E1) a3 +1e1ιcn∗m4 +0 +8 ∥r1∥ L6∗ (2n1 + 2n∗) ++ 3G4µ cos (2f + 2g − 2h + 3E1) a3 +1e1ιcn∗m4 +0 +8 ∥r1∥ L6∗ (2n1 + 2n∗) ++ 3eG4µ cos (f + 2g − 2h + 2E1) a3 +1n∗m4 +0 +8η3 ∥r1∥ L6∗ (2n1 + 2n∗) ++ 3eG4µ cos (3f + 2g − 2h + 2E1) a3 +1n∗m4 +0 +8η3 ∥r1∥ L6∗ (2n1 + 2n∗) +− 3G4µ cos (2f + 2g − 2h + E1) a3 +1e1n∗m4 +0 +16 ∥r1∥ L6∗ (2n1 + 2n∗) +− 3G4µ cos (2f + 2g − 2h + 3E1) a3 +1e1n∗m4 +0 +16 ∥r1∥ L6∗ (2n1 + 2n∗) +− 9eG4µ cos(f)a3 +1ι2 +cm4 +0 +8 ∥r1∥ L6∗ ++ 9eG4µ cos(f + 2g)a3 +1ι2 +cm4 +0 +16 ∥r1∥ L6∗ +− 3eG4µ cos(f + 2g)a3 +1ι2 +cm4 +0 +8η3 ∥r1∥ L6∗ ++ 9eG4µ cos(3f + 2g)a3 +1ι2 +cm4 +0 +16 ∥r1∥ L6∗ +− 3eG4µ cos(3f + 2g)a3 +1ι2 +cm4 +0 +8η3 ∥r1∥ L6∗ ++ 9eG4µ cos (f + 2h − 2E1) a3 +1ι2 +cm4 +0 +16 ∥r1∥ L6∗ +− 9eG4µ cos (f + 2g + 2h − 2E1) a3 +1ι2 +cm4 +0 +32 ∥r1∥ L6∗ +− 9eG4µ cos (3f + 2g + 2h − 2E1) a3 +1ι2 +cm4 +0 +32 ∥r1∥ L6∗ ++ 9eG4µ cos (f − 2h + 2E1) a3 +1ι2 +cm4 +0 +16 ∥r1∥ L6∗ +− 9eG4µ cos (f + 2g − 2h + 2E1) a3 +1ι2 +cm4 +0 +32 ∥r1∥ L6∗ +− 9eG4µ cos (3f + 2g − 2h + 2E1) a3 +1ι2 +cm4 +0 +32 ∥r1∥ L6∗ +− 3G4µ cos (2h − 3E1) a3 +1e1ι2 +cm4 +0 +16 ∥r1∥ L6∗ ++ 3G4µ cos (2f + 2g + 2h − 3E1) a3 +1e1ι2 +cm4 +0 +32 ∥r1∥ L6∗ +37 + +− 3G4µ cos (2f + 2g − E1) a3 +1e1ι2 +cm4 +0 +8 ∥r1∥ L6∗ +− 15G4µ cos (2h − E1) a3 +1e1ι2 +cm4 +0 +16 ∥r1∥ L6∗ ++ 15G4µ cos (2f + 2g + 2h − E1) a3 +1e1ι2 +cm4 +0 +32 ∥r1∥ L6∗ ++ 9G4µ cos (E1) a3 +1e1ι2 +cm4 +0 +8 ∥r1∥ L6∗ +− 3G4µ cos (2f + 2g + E1) a3 +1e1ι2 +cm4 +0 +8 ∥r1∥ L6∗ ++ 15G4µ cos (2f + 2g − 2h + E1) a3 +1e1ι2 +cm4 +0 +32 ∥r1∥ L6∗ ++ 3G4µ cos (2f + 2g − 2h + 3E1) a3 +1e1ι2 +cm4 +0 +32 ∥r1∥ L6∗ +− 9eG4µ cos (f + 2g + 2h − 2E1) a3 +1ιcm4 +0 +16 ∥r1∥ L6∗ +− 9eG4µ cos (3f + 2g + 2h − 2E1) a3 +1ιcm4 +0 +16 ∥r1∥ L6∗ ++ 9eG4µ cos (f + 2g − 2h + 2E1) a3 +1ιcm4 +0 +16 ∥r1∥ L6∗ ++ 9eG4µ cos (3f + 2g − 2h + 2E1) a3 +1ιcm4 +0 +16 ∥r1∥ L6∗ ++ 3G4µ cos (2f + 2g + 2h − 3E1) a3 +1e1ιcm4 +0 +16 ∥r1∥ L6∗ ++ 15G4µ cos (2f + 2g + 2h − E1) a3 +1e1ιcm4 +0 +16 ∥r1∥ L6∗ +− 15G4µ cos (2f + 2g − 2h + E1) a3 +1e1ιcm4 +0 +16 ∥r1∥ L6∗ +− 3G4µ cos (2f + 2g − 2h + 3E1) a3 +1e1ιcm4 +0 +16 ∥r1∥ L6∗ ++ 3eG4µ cos(f)a3 +1m4 +0 +8 ∥r1∥ L6∗ +− 9eG4µ cos(f + 2g)a3 +1m4 +0 +16 ∥r1∥ L6∗ ++ 3eG4µ cos(f + 2g)a3 +1m4 +0 +8η3 ∥r1∥ L6∗ +− 9eG4µ cos(3f + 2g)a3 +1m4 +0 +16 ∥r1∥ L6∗ ++ 3eG4µ cos(3f + 2g)a3 +1m4 +0 +8η3 ∥r1∥ L6∗ +− 9eG4µ cos (f + 2h − 2E1) a3 +1m4 +0 +16 ∥r1∥ L6∗ +− 9eG4µ cos (f + 2g + 2h − 2E1) a3 +1m4 +0 +32 ∥r1∥ L6∗ +− 9eG4µ cos (3f + 2g + 2h − 2E1) a3 +1m4 +0 +32 ∥r1∥ L6∗ +− 9eG4µ cos (f − 2h + 2E1) a3 +1m4 +0 +16 ∥r1∥ L6∗ +− 9eG4µ cos (f + 2g − 2h + 2E1) a3 +1m4 +0 +32 ∥r1∥ L6∗ +− 9eG4µ cos (3f + 2g − 2h + 2E1) a3 +1m4 +0 +32 ∥r1∥ L6∗ ++ 3G4µ cos (2h − 3E1) a3 +1e1m4 +0 +16 ∥r1∥ L6∗ ++ 3G4µ cos (2f + 2g + 2h − 3E1) a3 +1e1m4 +0 +32 ∥r1∥ L6∗ ++ 3G4µ cos (2f + 2g − E1) a3 +1e1m4 +0 +8 ∥r1∥ L6∗ ++ 15G4µ cos (2h − E1) a3 +1e1m4 +0 +16 ∥r1∥ L6∗ ++ 15G4µ cos (2f + 2g + 2h − E1) a3 +1e1m4 +0 +32 ∥r1∥ L6∗ +− 3G4µ cos (E1) a3 +1e1m4 +0 +8 ∥r1∥ L6∗ ++ 3G4µ cos (2f + 2g + E1) a3 +1e1m4 +0 +8 ∥r1∥ L6∗ ++ 15G4µ cos (2f + 2g − 2h + E1) a3 +1e1m4 +0 +32 ∥r1∥ L6∗ ++ 3G4µ cos (2f + 2g − 2h + 3E1) a3 +1e1m4 +0 +32 ∥r1∥ L6∗ +− eG2µ cos(f)a1m2 +0 +∥r1∥ L2∗ ++ G2µ cos (E1) a1e1m2 +0 +∥r1∥ L2∗ ++ 3G2δL2 cos (E1) a1e1m2 +0 +2 ∥r1∥ L4∗ +� +. +Next, we move on with the second and last iteration j = 2 targeted to R(1) +3,3: +H (2) = Z0 + Z (1) +2 ++ Z (2) +3 ++ R(2) +4,4 , +in which χ(2) +3 +is omitted for brevity and +Z (2) +3 += 0 +38 + +as expected, being R(1) +3,3 solely made up of harmonics containing fast angles. +Acknowledgements. +C.E. was partially supported by the MIUR-PRIN 20178CJA2B New +Frontiers of Celestial Mechanics: Theory and Applications. +References +[1] +I. 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Universidad de Zaragoza, 1992. +39 + diff --git a/49E1T4oBgHgl3EQfSwOL/content/tmp_files/load_file.txt b/49E1T4oBgHgl3EQfSwOL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67abbd90efaef795d147e006ccec929639b27f3b --- /dev/null +++ b/49E1T4oBgHgl3EQfSwOL/content/tmp_files/load_file.txt @@ -0,0 +1,1821 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf,len=1820 +page_content='Relegation-free closed-form perturbation theory and the domain of secular motions in the Restricted 3-Body Problem Mattia Rossi and Christos Efthymiopoulos Università degli Studi di Padova Dipartimento di Matematica “Tullio Levi-Civita” Via Trieste, 63 - 35121 Padova, Italy mrossi@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='it, cefthym@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='it January 10, 2023 Abstract We propose a closed-form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' without expansion in the orbital eccentricities) scheme for computations in perturbation theory in the restricted three-body problem (R3BP) when the massless particle is in an orbit exterior to the one of the primary perturber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Starting with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamilto- nian model without use of relegation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' To this end, we introduce a book-keeping analogous to the one proposed in [1] for test particle orbits interior to the one of the primary perturber, but here adapted, instead, to the case of exterior orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We give numerical examples of the performance of the method in both the planar circular and the spatial elliptic restricted three-body problem, for parameters pertinent to the Sun-Jupiter system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In particular, we demonstrate the method’s accuracy in terms of reproducibility of the orbital elements’ vari- ations far from mean-motion resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As a basic outcome of the method, we show how, using as criterion the size of the series’ remainder, we reach to obtain an accurate semi- analytical estimate of the boundary (in the space of orbital elements) where the secular Hamiltonian model arrived at after eliminating the particle’s fast degree of freedom provides a valid approximation of the true dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Keywords: Celestial mechanics – Astrodynamics – R3BP – Closed-form – No relegation – Secular motion 1 Introduction As opposed to the usual (Laplace-Lagrange) theory, closed-form perturbation theory [11] provides a framework for series calculations in perturbed Keplerian problems without expansions in powers of the bodies’ orbital eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is mainly motivated by the necessity to construct secular models for sufficiently eccentric orbits, like those of many asteroids, in our solar system, or the planets in extrasolar planetary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The efficiency of the usual series methods of expansion in the orbital eccentricities is limited by the fact that the inversion of Kepler’s equation in powers of the eccentricity converges only up to the so-called Laplace limit eL ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='66274 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Generally, such convergence slows down way before this value (around e ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 in many applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In order to address this issue, closed form perturbation theory aims at solving in ‘closed-form’ the homological equation by which the Lie generating function is computed at every perturbative step (see for example [3, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='03070v1 [math-ph] 8 Jan 2023 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The process is far from being priceless: a major obstruction appears when the kernel of the homological equations contains addenda beyond the Keplerian terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The most common such addendum ([11]) is the centrifugal term −νH, where ν is the angular frequency in a frame co-rotating with the primary perturber, and H is the Delaunay action equal to the particle’s angular momentum in the direction of the axis of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In the case of a planet’s orbiter, ν is equal to the planet’s rotation frequency, and the problem appears for all non-axisymmetric terms (tesseral harmonics) of the planet’s multipole potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In the R3BP, instead, ν represents the mean motion of the primary perturber (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Jupiter in the Sun-Jupiter system), while the problem appears in a similar way after introducing a multipole expansion of the disturbing function in the particle’s Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' An algorithm to overcome the above issue, called the relegation algorithm, has been proposed in works by Deprit, Palaciań and collaborators [2, 4, 7, 13, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Briefly, given a quasi-integrable Hamiltonian H = H0 + εH1, where ε is a small parameter, suppose that H0 = H′ 0 + H′′ 0 , where, in a domain in phase space we have that H′ 0 yields the dominant contribution to the Hamiltonian flow of H0 versus the H′′ 0 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In usual perturbation theory, we seek to partly normalize the perturbation H1 via a sequence of canonical transformations defined by generating functions χ(r), r = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' satisfying a homological equation of the form {H0, χ(r)} + h(r) 1 = 0, where {·, ·} denotes the Poisson bracket between two functions of the canonical variables and h(r) 1 is a term in the Hamiltonian to be normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In the relegation technique, we use instead the equation {H′ 0, χ(r)}+h(r) 1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', letting only the dominant function H′ 0 in the kernel of the homological equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Such a choice stems mostly from motives of algorithmic convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For example, identifying H′ 0 with the Keplerian term (when ν is small) leads to a homological equation that can be solved in closed form (we set, instead, H′ 0 = −νH when ν is large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' However, all Poisson brackets of χ(r) with the part H′′ 0 left out of the kernel lead to terms which need to be ‘relegated’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', pushed to normalization in subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For reasons explained in detail in [14], only a finite number or relegation steps can be performed before reaching a point beyond which the scheme generates divergent sequences of terms (see also [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This implies that the process necessarily stops after some steps, leading to a finite, albeit possibly quite small remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Relegation is a technique particularly suitable to the limiting situation of a strongly hierarchi- cal problem, when the integrable part H0 depends on a frequency vector involving n frequencies ω = (ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , ωn) out of which one, say ωi for some i with 1 ≤ i ≤ n is significantly larger in absolute value than the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In particular, the harmonics cos(k · ϕ) in the Hamiltonian whose normalization can be ‘relegated’ should satisfy |kiωi| ≫ |kjωj|, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , n, j ̸= i, for every integer ki, kj ∈ Z \\ {0} (assuming also the non-resonant condition k · ω ̸= 0, k = (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , kn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For example, as explained in [14] in the simple case with n = 2 and ω2 ≫ ω1, the generating function χ(N) produced after N relegation steps contains terms with coefficients growing as a geometric sequence with ratio k1ω1/k2ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Thus, relagation is limited to those terms for which the above ratio is smaller than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This includes most harmonics of low Fourier order in the Hamiltonian perturbation when ω2 ≫ ω1, but only few when the two frequencies become comparable in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Hence, by construction, relegation has limited applicability in this latter, non-hierarchical, case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Variants of the relegation technique have been discussed in literature to address perturbed Keplerian problems in which the gravitational potential is due to an extended body expanded in spherical harmonics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' [7, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' To address the non-hierarchical case, a techique similar to the one of the present paper is discussed in [7], referring to the averaging of the tesseral harmonics in the case of the Earth’s artificial satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In the case of the R3BP, instead, Cavallari and Efthymiopoulos [1] discuss a relegation-free algorithm for the elimination of short-period terms in the particle’s Hamiltonian, when the orbit of the particle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' an asteroid) is totally interior to the orbit of the primary perturber (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Jupiter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We are aware of no relegation-free algorithm 2 proposed in literature which addresses, instead, the case when the particle’s orbit is exterior to the orbit of the primary perturber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Providing such an algorithm, discussing some of its im- portant differences with past-proposed algorithms, as well as checking its limits of applicability, constitutes the primary goal of our present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The R3BP is defined by the motion of a body P of negligible mass in the gravitational field of two massive bodies P0 (the primary or central body) and P1 (the secondary or primary perturber), which perform a motion r1(t) either elliptic in the more general version (ER3BP) or circular (CR3BP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The starting point for our analysis in the sequel is the Hamiltonian of the model, obtained after reduction via Jacobi coordinates (R, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Expressing time through the secondary’s mean anomaly M1 = n1t, where n1 is the mean motion of the secondary, and canonically conjugating M1 with a dummy action variable J1 allows to express the Hamiltonian as H(R, M1, P, J1) = ∥P∥2 2 − Gm0 ∥R + µr1(M1)∥ − Gm1 ∥R − (1 − µ)r1(M1)∥ + n1J1 , (1) where G is the gravitational constant and µ = m1 m0 + m1 ∈ (0, 1/2] is the mass parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' r1(M1) = a1 � cos E1(M1) − e1, � 1 − e2 1 sin E1(M1), 0 � (2) is the elliptic revolution of P0 − P1 around their barycenter with eccentricity e1 and semi- major axis a1, in which the dependence of the system’s eccentric anomaly E1 ∈ T = R/(2πZ) on the mean anomaly M1 ∈ T is given through Kepler’s equation according to standard two- body problem setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (R = (X, Y, Z), P = (PX, PY , PZ)) ∈ T ∗(R3 \\ {−µr1, (1 − µ)r1}) is the position-momentum couple of P and the phase space is endowed with standard symplectic form dPX ∧ dX + dPY ∧ dY + dPZ ∧ dZ + dJ1 ∧ dM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We make use then of Delaunay elements (ℓ, g, h, L, G, H), defined by L = � Gm0a , ℓ = M , G = L � 1 − e2 , g = ω , (3) H = G cos i , h = Ω , where a, e, i, M, Ω, ω stand for the semi-major axis, the eccentricity, the inclination, the mean anomaly, the longitude of the ascending node, the argument of pericenter of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' A key ingredient of the method proposed below is the following: similarly as in [1], we introduce a book-keeping symbol σ with numerical value equal to 1, whose role is to organize the perturbative scheme so as to successively normalize terms of similar order of smallness, treating together all small quantities of the problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', – the eccentricities e, e1 (when e1 ̸= 0), – the mass ratio µ, – the semi-major axis fluctuation δL around the mean L∗ for a particular particle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 1In the R3BP problem the Jacobi transformation is implemented when ∥R∥ > ∥r1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 3 The book-keeping symbol acts by assigning powers σ1 and σν1, σν, σν respectively, for non-zero natural numbers ν, ν1 defined below, to all the terms in the original Hamiltonian as well as in the Hamiltonian produced after every normalization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Given this baseline,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' we arrive (in Section 2) to the following result: we demonstrate that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' for kµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' kmp ∈ N \\ {0} with kµ > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' the combination of expansions of (1) up to µkµ and (∥r1∥ / ∥R∥)kmp is canonically conjugate by ν(kµ − 1) near-identity transformations to a secular model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' obtained as a normal form with respect to the fast angles ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' M1 H (ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' J1) = H0(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' J1) + R(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' H) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (4) with H0 = n∗δL + n1J1 + νkµ−1 � l=ν � p∈Z2 cl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' µ, L∗, a1, e1) cos(p1g + p2h)σl , (5) R = � s∈Z4 dνkµ,s(E1, δL, e, i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' µ, L∗, a1, e1) cos(s1f + s2g + s3h + s4E1)σνkµ + O � σνkµ+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' �∥r1∥ ∥R∥ �kmp+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (6) The dependencies f = f(ℓ, δL, G) for the true anomaly, e = e(δL, G) and i = i(G, H) are implied in all the above expressions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' cl,p, dνkµ,s are real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' A crucial point is the way by which the positive integers ν = ν(e∗, µ) ≥ 1, ν1 = ν1(e∗, e1) ≥ 1 are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As detailed below, these integers, which regulate the book-keeping scheme, are suitably tuned on the basis of a selected reference value e∗ ∈ (0, 1): ν = � log10 µ log10 e∗ � , ν1 = �log10 e1 log10 e∗ � , (7) where ⌈·⌉ is the ceiling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The normalizing scheme leading to (4) is local: knowing that the semi-major axis is preserved under the flow of the (secular) normal form, we introduce the splitting L = L∗ + δL, where L∗ = √Gm0a∗ ≫ δL, n∗ = √Gm0a−3/2 ∗ is a targeted reference value for the semi-major axis a∗, and expand the Hamiltonian in powers of δL, rendering δL the new action variable canonically conjugated to the particle’s mean anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Given the above, the normalization algorithm provides a sequence of Lie generating functions χ(j) ν+j−1 = O(σν+j−1), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , ν(kµ−1), which yields the Lie canonical transformation allowing to recursively normalize all terms depending on the angles f and E1 in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The normalizing trasformations are possible to define for values of the frequencies n∗ (mean motion of the particle at the semi-major axis a∗) and n1 far from mean-motion resonances (see Remark 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Furthermore, the generating functions are computed as solutions of a homological equation of the form {Z0, χ(j) ν+j−1} + R(j−1) ν+j−1,ν+j−1 = O(σν+j−1) , (8) where Z0 = n∗δL + n1J1 and R(j−1) ν+j−1,ν+j−1 ∼ σν+j−1 collects the trigonometric monomials of O(σν+j−1) depending on at least one of the two anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The key to obtaining a closed- form solution for (8) is, precisely, the appropriate choice of a O(σν+j−1) remainder left in the second hand of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In words, we do not seek for an exact cancellation of the terms R(j−1) ν+j−1,ν+j−1, but only for an approximate cancellation, leading to a remainder, which, however, is of higher order in book-keeping, and, hence, possible to reduce at subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 4 As discussed in Section 3, a relevant outcome of the analysis of the behavior of the re- mainder obtained by the above method stems from an estimation of the optimal number of normalization steps jopt, where the remainder becomes of order ν + jopt − 1 in the book-keeping parameter, with jopt ≤ ν(kµ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The value of jopt is defined as the one where the error bound E (j)(a∗, e∗) = � ν+j≤l≤νkµ,s |d(j) l,s | ≥ ∥R(j) ν+j∥∞ = sup |R(j) ν+j| becomes minimum, with R(j) ν+j = O(σν+j) and d(j) l,s as in (6) after j normalization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As typical in perturbation the- ory, the value of jopt depends on the chosen reference values (a∗, e∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' With the present method one can then obtain a map of the size of the optimal remainder as a function of (a∗, e∗) in the semi-plane a > a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Using this information, we compute the limiting locus uniting all points in (a∗, e∗) such that the normal form computation yields no improvement with increasing number of normalization steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', where jopt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Comparing with numerical stability maps obtained with the Fast Lyapunov Indicator (FLI) [9], one sees that, the limiting locus found semi-analytically essentially coincides with the numerical (FLI map) limit where no harmonic in the Hamiltonian associated with one of the exterior mean-motion resonances affects the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As a conse- quence, all motions in the sub-domain of the plane (a∗, e∗) below the limiting locus are stable in the secular sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', protected against instabilities caused by short-period resonant effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For this reason, we identify this locus as the border of the domain of secular motions, and sub- stantiate the fact that its semi-analytical computation (through the normal forms) yields results in precise agreement with those found by the heuristic definition of the same border via the fully numerical (FLI) computation of stability maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Section 2 presents step-by-step the algorithm that gives rise to (5) and (6), supplemented with the formulas for the Poisson algebra in Keplerian elements used in all closed-form computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Section 3 is devoted to a numerical investigation of the method’s accuracy for an asteroid in the Sun-Jupiter system, first in the spatial ER3BP, and then in the planar CR3BP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' in the latter case, the computations are short enough to allow for a speci- fication of the optimal normalization order in a grid of values in the (a∗, e∗) plane, leading to the semi-analytical determination of the border of the domain of secular motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Section 4 summa- rizes the basic conclusions of the present study and gives some relevant comments for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2 The closed-form method for the outermost R3BP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 Multipole expansion of the perturbation Referring to section 1, let H be given in barycentric Cartesian coordinates as in (1): H = ∥P∥2 2 + n1J1 − Gm0R , (9) 5 X Y Z O r1 P0 P1 −µr1 (1 − µ)r1 R P Figure 1: Representation of the R3BP in the barycentric frame (or equivalently in Jacobi variables) with ∥R∥ > ∥r1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Assuming ∥r1∥ / ∥R∥ < 1, we carry out a multipole expansion of the function R(R, M1) in powers of the ratio ∥r1∥ / ∥R∥ < 1: R = 1 ∥R + µr1∥ + µ 1 − µ 1 ∥R + (1 − µ)r1∥ = 1 ∥R∥ � ∞ � l=0 �−1/2 l � � 2µr1 · R ∥R∥2 + µ2 �∥r1∥ ∥R∥ �2�l + µ 1 − µ ∞ � l=0 �−1/2 l � � −2(1 − µ)r1 · R ∥R∥2 + (1 − µ)2 �∥r1∥ ∥R∥ �2�l � = 1 1 − µ 1 ∥R∥ + O ��∥r1∥ ∥R∥ �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (10) where, for β ∈ R �β l � = β(β − 1) · · · (β − l + 1) l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' indicates the generalized binomial coefficient (equal to 1 for l = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For l = 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (10) the coefficients of the dipole term (r1 · R)/ ∥R∥3 in the two sums in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' of the equation cancel each other exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Thus, no dipole term appears in the disturbing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is a consequence of the choice of Jacobi coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 Canonical form of the Hamiltonian Performing an extra series expansion in powers of µ < 1 yields the standard nearly-integrable form H = H0 + µH1 , (11) 6 where the Keplerian part reads H0 = ∥P∥2 2 − Gm0 ∥R∥ + n1J1 (12) and the disturbing function becomes H1 = −Gm0 ∥R∥ � ∞ � l=0 µl + ∞ � l=1 µl−1 �−1/2 l � � 2r1 · R ∥R∥2 + µ �∥r1∥ ∥R∥ �2�l + ∞ � l=1 (1 − µ)l−1 �−1/2 l � � −2r1 · R ∥R∥2 + (1 − µ) �∥r1∥ ∥R∥ �2�l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (13) We now move to Delaunay action-angle variables (3) by replacing into (11) the relationships H0 = −Gm0 2a + n1J1 , (14) ∥R∥ = a(1 − e2) 1 + e cos f , (15) r1 · R = a1 ∥R∥ � (cos E1 − e1) (cos h cos(g + f) − sin h sin(g + f) cos i) + � 1 − e2 1 sin E1 (sin h cos(g + f) + cos h sin(g + f) cos i) � (16) as well as (2) for the vector r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We get H = −Gm0 2a + n1J1 + µH1(f, g, h, E1, a, e, i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' µ, a1, e1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (17) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Only the square of the norm ∥r1∥2 = r1 · r1 is required in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (13), while the norm ∥R∥ appears only in the denominator of the above equation, in powers equal to or higher than quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Then equations (15) and (2), respectively dependent on f and E1, lead to a represen- tation of the disturbing function as a sum of trigonometric polynomials depending on harmonics of the form cos(s1f + s2g + s3h + s4E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is a key ingredient of the closed-form method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', working with the angles f and E1, instead of the mean anomalies M, M1, no series reversion of Kepler’s equation is used throughout the whole perturbative scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In order to avoid relegation, our method discussed below works locally, by constructing a model for the secular Hamiltonian valid for a particle’s semi-major axis varying as a = a∗+δa(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', by a small quantity δL around some reference value a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' By standard secular theory, we have the estimate δa = O(µ) far from mean-motion resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Formally, introducing the new canonical variable δL as L = L∗ + δL = � Gm0a∗ + 1 2 � Gm0 a∗ δa + O(δa2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (18) and expanding the Hamiltonian in powers of the quantity δL around L∗, we obtain H = −G2m2 0 2L2∗ ∞ � l=0 �−2 l � �δL L∗ �l + n1J1 + µ ∞ � l=0 1 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ∂lH1 ∂Ll ���� L=L∗ δLl = n∗δL + n1J1 + µ � H1|δL=0, µ=0 + ∂H1 ∂δL ���� δL=0, µ=0 δL � + O(µ2, δL2) , (19) 7 where a constant term −G2m2 0/(2L2 ∗) was dropped from the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The constant n∗ = G2m2 0/L3 ∗ is equal to the particle’s mean motion under Keplerian orbit at the semi-major axis a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The choice of the reference value a∗ determines the kind of divisors appearing in the normalization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In the present paper, we deal only with the ‘non-resonant’ case, in which the frequencies n∗ and n1 satisfy no-commensurability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For example, to be far from any resonance we may require that n∗ and n1 satisfy a diophantine condition |k∗n∗ + k1n1| > γ |k|τ , ∀k = (k∗, k1) ∈ Z2 \\ {0} (20) with |k| = |k∗| + |k1| and some suitable γ > 0, τ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' However, the algorithm presented below can be readily extended to cases of mean-motion res- onance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We leave the details for a future work, noting only that in resonant cases we have the estimate δL = O(µ1/2), instead of O(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The effect of approaching close to a mean-motion resonance with the present series is seen, instead, as a rise in the value of the series’ remainder, caused by (non-zero) small divisors in the series (as visible, for example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 7 discussed in section 3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 Poisson structure and book-keeping 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 Poisson bracket formulas All steps of closed-form perturbation theory involve Poisson brackets between differentiable functions of the form F(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' J1) ∈ C∞(T4 × D),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' D ⊂ R4 being an open set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' whose dependence on the variables ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G and H is given in implicit form through the func- tions f(ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' E1(M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' e(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' e(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιc(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' H) = cos i(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιs(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' H) = sin i(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' η(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G) = � 1 − e(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ∥r1∥ (M1) = a1(1−e1 cos E1(M1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' and φ1(M1) = E1(M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' e(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G))− M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The Poisson bracket between two functions F1, F2 of the above form is computed by the formulas {F1, F2} = dF1 dℓ dF2 dδL + dF1 dg dF2 dG + dF1 dh dF2 dH + dF1 dM1 dF2 dJ1 − dF1 dδL dF2 dℓ − dF1 dG dF2 dg − dF1 dh dF2 dH − dF1 dJ1 dF2 dM1 (21) implemented to the closed-form version of the functions F1, F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The closed-form version of a function F is defined as: F = F(f, g, h, E1, δL, e, η, ιc, ιs, J1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (22) The derivatives in the canonical variables of a function F as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (21) are computed by the chain rule formulas dF dℓ = ∂F ∂f ∂f ∂ℓ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (23) dF dg = ∂F ∂g ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (24) dF dh = ∂F ∂h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (25) dF dM1 = � ∂F ∂E1 + ∂F ∂ ∥r1∥ d ∥r1∥ dE1 + ∂F ∂φ1 � dE1 dM1 − ∂F ∂φ1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (26) 8 dF dδL = ∂F ∂f ∂f ∂δL + ∂F ∂δL + ∂F ∂e ∂e ∂δL + ∂F ∂η ∂η ∂δL ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (27) dF dG = ∂F ∂f ∂f ∂G + ∂F ∂e ∂e ∂G + ∂F ∂η ∂η ∂G + ∂F ∂ιc ∂ιc ∂G + ∂F ∂ιs ∂ιs ∂G ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (28) dF dH = ∂F ∂ιc ∂ιc ∂H + ∂F ∂ιs ∂ιs ∂H ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (29) dF dJ1 = ∂F ∂J1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (30) where ∂f ∂ℓ = (1 + e cos f)2 η3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (31) d ∥r1∥ dE1 = a1e1 sin E1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (32) dE1 dM1 = a1 ∥r1∥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (33) ∂f ∂δL = 1 L �2 sin f e + sin(2f) 2 � = 1 L∗ �2 sin f e + sin(2f) 2 � � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (34) ∂e ∂δL = η2 eL = η2 eL∗ � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (35) ∂η ∂δL = − η L = − η L∗ � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (36) ∂f ∂G = − 1 ηL �2 sin f e + sin(2f) 2 � = − 1 ηL∗ �2 sin f e + sin(2f) 2 � � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (37) ∂e ∂G = − η eL = − η eL∗ � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (38) ∂η ∂G = 1 L = 1 L∗ � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (39) ∂ιc ∂G = − ιc ηL = − ιc ηL∗ � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (40) ∂ιs ∂G = −1 − ι2 s ηLιs = −1 − ι2 s ηL∗ιs � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (41) ∂ιc ∂H = 1 ηL = 1 ηL∗ � 1 − δL L∗ � + O(δL2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (42) ∂ιs ∂H = − ιc ηLιs = − ιc ηL∗ιs � 1 − δL L∗ � + O(δL2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (43) A sketch of the derivation of the above formulas can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' They are strictly valid with e ∈ (0, 1), i ∈ (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' However, several cancellations lead to no singular behavior of the Poisson bracket formulas arising throughout the various perturbative steps also when e = 0 or i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 Book-keeping: Hamiltonian We introduce in the series a book-keeping symbol σ (see [5] for an introduction to the book- keeping technique), with numerical value σ = 1, whose role is to provide a grouping of all the various terms in the series according to their ‘order of smallness’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Hence, a group of terms with common factor σl, l ∈ Z, indicates a term considered as of the ‘l-th order of smallness’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Since in our series there are several small quantities, we introduce a book-keeping scheme allowing to simultaneously deal with all small quantities while maintaining the closed-form char- acter of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' To this end, we make the following substitutions, called ‘book-keeping rules’, within the initial Hamiltonian: BK-Rule 1: e � σ1e = σe (not applicable to the quantity e2 within η = √ 1 − e2), BK-Rule 2: η � σ0η = η, BK-Rule 3: µ � σνµ, with ν as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (7), BK-Rule 4: e1 � σν1e1, with ν1 as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (7) (not applicable to the quantity e2 1 within η1 := � 1 − e2 1), BK-Rule 5: 1 η2 � � 1 η2 − 1 � σ2 + 1, BK-Rule 6: η1 � (η1 − 1)σ2ν1 + 1, BK-Rule 7: δLλ � σlνδLλ with l = � λ , if δLλ comes from H1 , λ − 1 , if δLλ comes from H0 , λ ∈ N \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Since σ = 1, the above substitutions affect the structure of the series only at the formal level, and can be substituted directly into the original Hamiltomian, whereby they propagate at sub- sequent normalization steps once these steps are organized in successive powers σ, σ2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', of the book-keeping symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The BK-Rules 1 to 7 above are justified on physical ground as well as on motives of algorithmic convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In particular: BK-Rule 1 implies that, despite the use of closed-form formulas, the basic small quantity in powers of which the series are organized is the eccentricity of the test particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' BK-Rule 3 implies that a factor µ in front of a series term should be treated as of compa- rable order of smallness as a term of order eν, with ν given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Similarly, BK-Rule 4 implies that a term containing a factor e1 raised to some power should be treated as of compa- rable order of smallness with a term eν 1 raised to the same power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Note that the eccentricity e is a quantity variable in time, so that to compute the exponents ν, ν1 we need to use, for any examined trajectory, a reference value e∗ yielding an estimate of the overall level of eccentricity all along the orbital evolution for that trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Note that, by standard secular theory we have e(t) = e∗ + O(µ) if e∗ is close to the mean eccentricity (see also discussion at the introduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Note finally that we obtain exponents ν, ν1 ≥ 1 in the typical case in which e > µ and e ≥ e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' These inequalities arise naturally in the case of small bodies in highly eccentric orbits perturbed by some planet of, say, our solar system, which are the cases of main interest in applying the present method (see, nevertheless, Remark 4 on the treatment of cases where the above condi- tions are not met).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' BK-Rule 7 stems from the estimate δL = O(δa) = O(µ) holding for the oscillations in semi- major axis of trajectories far from mean-motion resonances (as already pointed outin the latter 10 case, instead, we have in general δL = O(δa) = O(µ1/2) and the corresponding rule has to be adapted accordingly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The lowering of the book-keeping power by one for within H0 is intro- duced for reasons of algorithmic convenience, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', in order to maintain n∗δL in the kernel of the homological equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' BK-Rules 5 and 6 imply just a partition of the unity aiming at keeping the perturbative scheme in closed-form while splitting the corresponding expressions (involving η and η1 respec- tively) in two parts, of orders O(1) and O(e2), or O(e2 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 Book-keeping: Poisson structure Some of the formulas in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 imply differentiation with respect to e through the corresponding partial derivatives in (27), (28), thus yielding a lowering of the power of the eccentricity in some terms arising through Poisson brackets at consecutive steps of perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' To account for this fact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' similarly as in [1] we introduce the use of the book-keeping symbol σ in the formulas of the Poisson algebra as follows: first,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' we re-write the derivatives with respect to the angles ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' M1 as dF dℓ = ∂F ∂f ∂f ∂ℓ a1(1 − e1σν1 cos E1) ∥r1∥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (44) dF dg = ∂F ∂g a1(1 − e1σν1 cos E1) ∥r1∥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (45) dF dh = ∂F ∂h a1(1 − e1σν1 cos E1) ∥r1∥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (46) dF dM1 = � ∂F ∂E1 + ∂F ∂ ∥r1∥ d ∥r1∥ dE1 + ∂F ∂φ1 σ−ν1 � dE1 dM1 − ∂F ∂φ1 σ−ν1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (47) and with respect to the actions δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' G as dF dδL = ∂F ∂f ∂f ∂δL + ∂F ∂δL + ∂F ∂e ∂e ∂δLσ−1 + ∂F ∂η ∂η ∂δL ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (48) dF dG = ∂F ∂f ∂f ∂G + ∂F ∂e ∂e ∂Gσ−1 + ∂F ∂η ∂η ∂G + ∂F ∂ιc ∂ιc ∂G + ∂F ∂ιs ∂ιs ∂G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (49) Note that in (47) use was made of the identity φ1 = e1 sin E1 (Kepler’s equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' we revise formulas (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (34)–(43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' attributing a book-keeping to all factors involving the eccentricity function η as ∂f ∂ℓ = 1 + 2e cos f η3 σ + � 1 η3 − 1 + e2 cos2 f η3 � σ2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (50) d ∥r1∥ dE1 = a1e1σν1 sin E1 (51) ∂f ∂δL = 1 L∗ �2 sin f e σ−1 + sin(2f) 2 � + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (52) ∂e ∂δL = 1 L∗ �1 eσ−1 + η2 − 1 e σ � + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (53) ∂η ∂δL = − 1 L∗ � 1 + (η − 1)σ2� + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (54) 11 ∂f ∂G = − 1 L∗ � 2 sin f e σ−1 + sin(2f) 2 + 2 sin f e �1 η − 1 � σ + sin 2f 2 �1 η − 1 � σ2 � + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (55) ∂e ∂G = − 1 L∗ �1 eσ−1 + η − 1 e σ � + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (56) ∂η ∂G = 1 L∗ + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (57) ∂ιc ∂G = − ιc L∗ � 1 + �1 η − 1 � σ2 � + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (58) ∂ιs ∂G = −1 − ι2 s L∗ιs � 1 + �1 η − 1 � σ2 � + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (59) ∂ιc ∂H = 1 L∗ � 1 + �1 η − 1 � σ2 � + O(δLσν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (60) ∂ιs ∂H = − ιc L∗ιs � 1 + �1 η − 1 � σ2 � + O(δLσν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (61) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The small eccentricity problem consists of the fact that the above-proposed book- keeping rules are not applicable in the case 0 < e∗ ≲ µ < e1, since, by (7), the exponents ν, ν1 would be smaller than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The simple solution of rounding these exponents to 1, while maintaining the same book-keeping rules as above, fails, since, at any given normalization order r, the presence of σ−1, σ−ν1 terms in the formulas of the Poisson algebra leads to the generation of terms of order lower than r in the normal form’s remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Notwithstanding our focus on a method dealing with large eccentricity orbits (for which the problem does not appear), we discuss below a variant of the main algorithm that deals with trajectories in the case ν = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', when e∗ ≲ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 Iterative normalization algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 Preliminary step: Hamiltonian preparation After implementing BK-Rules 1 to 7 the Hamiltonian (19) resumes the form: H = n∗δL + n1J1 + � s∈Z4 qs(δL, e, η, ιc, ιs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' µ, L∗, a1, e1, η1) cos(s1f + s2g + s3h + s4E1)σs (62) where σs ∈ {σν, σν+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='} and, by D’Alembert rules, only cosines and real coefficients qs appear (invariance under simultaneous change of sign of all angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Setting Z0 = n∗δL + n1J1, for ob- taining a closed-form normalization algorithm it turns convenient to re-express the Hamiltonian according to H = Z0 + (H − Z0)a1(1 − e1σν1 cos E1) ∥r1∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (63) 12 The Hamiltonian (63) resumes the form: H = H (0) = Z0 + R(0) ν , (64) where R(0) ν = � l≥ν R(0) ν,l = � l≥ν a1 ∥r1∥ � � � � � p∈Z2 q′ l,p cos(p1g + p2h) + � s∈Z4 (s1,s4)̸=(0,0) q′′ l,s cos(s1f + s2g + s3h + s4E1) � � � � σl ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (65) We call R(0) ν the remainder at the zero-th normalization step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' in the original Hamiltonian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The terms R(0) ν,l contain terms of book-keeping order σl, with l ≥ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 Step 1: normalization of the σν-terms For a suitable generating function χ(1) ν to be determined in a while, we introduce the Lie series operator as exp � Lχ(1) ν � : Cω(T4 × D) −→ Cω(T4 × D) exp � Lχ(1) ν � = � n≥0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='Ln χ(1) ν = I + Lχ(1) ν + 1 2Lχ(1) ν Lχ(1) ν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , (66) where Cω(T4 × D) denotes the set of real analytic functions in the phase space and Lχ(1) ν · = {·, χ(1) ν } (67) is the time derivative along the Hamiltonian vector field generated by χ(1) ν (Lie derivative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Applying (66) to (63) we get the transformed Hamiltonian H (1) = Z0 + R(0) ν + {Z0, χ(1) ν } + {R(0) ν , χ(1) ν } + 1 2{{H, χ(1) ν }, χ(1) ν } + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , (68) in which, with the usual abuse of notation, we still indicate with ℓ, g, h, M1, δL, G, H, J1 the new canonical variables given by the inverse transformation exp � Lχ(1) ν �−1 = exp � L−χ(1) ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (69) Our scope will be to define the Lie generating function χ(1) ν in such a way that, after imple- menting the transformation (68), H (1) contains no terms depending on the angles f and E1 at order σν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The required generating function χ(1) ν is computed as an outcome of the following: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Define χ(1) ν as χ(1) ν = φ1 n1 σν+ν1 � p∈Z2 q′ ν,p cos(p1g + p2h) + σν � s∈Z4 (s1,s4)̸=(0,0) q′′ ν,s s1n∗ + s4n1 sin(s1f + s2g + s3h + s4E1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (70) 13 Then, it holds that {Z0, χ(1) ν } + R(0) ν,ν = Z (1) ν + O � σν+1� , (71) where Z (1) ν = σν � p q′ ν,p cos(p1g + p2h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (72) Furthermore, the function H (1) as computed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (68) takes the form H (1) = Z0 + Z (1) ν + R(1) , (73) where the remainder R(1) is O(σν+1) ∀ν ≥ 1 independently of the value of ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Setting χ(1) ν (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιs) = σν � � � �φ1σν1 � p∈Z2 ˆq′ ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιs) cos(p1g + p2h) + � s∈Z4 (s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s4)̸=(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0) ˆq′′ ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιs) sin(s1f + s2g + s3h + s4E1) � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' and recalling the chain rules (44),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (47) and (50),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (51),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (33),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' we find {Z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' χ(1) ν } + R(0) ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='ν = −n∗ � 1 + 2e cos f η3 σ + � 1 η3 − 1 + e2 cos2 f η3 � σ2 � a1(1 − e1σν1 cos E1) ∥r1∥ σν � (s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s4)̸=(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0) s1ˆq′′ ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s cos(s1f + s2g + s3h + s4E1) − n1 a1 ∥r1∥σν � � � (s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s4)̸=(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0) s4ˆq′′ ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s cos(s1f + s2g + s3h + s4E1) + � p ˆq′ ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p cos(p1g + p2h) � � + n1σν � p ˆq′ ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p cos(p1g + p2h) + σν a1 ∥r1∥ �� p q′ ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p cos(p1g + p2h) + � (s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s4)̸=(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0) q′′ ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s cos(s1f + s2g + s3h + s4E1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Requiring that no trigonometric terms depending on f, E1 be present at order σν then leads to ˆq′′ ν,s = q′′ ν,s s1n∗ + s4n1 , s ∈ Z4 : (s1, s4) ̸= (0, 0) , ˆq′ ν,p = q′ ν,p n1 , p ∈ Z2 , which implies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='(70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' At order σν we then obtain immediately the formula Z (1) ν = σν � p q′ ν,p cos(p1g + p2h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 14 We now consider the function H (1) computed by replacing (70) into (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The function H (1) can be decomposed as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='(73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We shall demonstrate that the remainder R(1) contains no terms of order lower than σν+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' To this end, it suffices to show that {R(0) ν , χ(1) ν } = O(σ2ν) , 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{H, χ(1) ν }, χ(1) ν }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(1) ν � �� � n≥2 } = O(σn(ν−1)+2) , (74) since n(ν − 1) + 2 > ν, for all n ≥ 2, ν ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The term R(0) ν contains terms of order equal to or larger than σν, while χ(1) ν contains only terms of order σν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Thus, except for the Poisson bracket {Z0, χ(1) ν }, which only contributes to the secular terms Z (1) ν due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (71), the first Poisson bracket in (74) contains prefactors of order σ2ν or higher, while the second contains prefactors σnν or higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' However, the exponent of σ in these brackets can be lowered due to the negative powers introduced in the book-keeping formulas in the following three classes of factors: (i) partial derivatives with respect to the eccentricity in (48), (49) (carrying σ−1) multiplied by corresponding formulae (53), (56) (another σ−1), hence a total of σ−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (ii) differentiations (52), (55) involving f (weighting σ−1) again in (48), (49), thus a pre-factor σ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (iii) partial derivatives with respect to φ1 in (47) (σ−ν1, ν1 ≥ 1), thus a prefactor at least σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As regards (iii) φ1 shows up in the numerator of χ(1) ν accompanied by a prefactor σν+ν1 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (70)), thus the negative powers σ−ν1 are cancelled by the positive powers σν1, implying no dependence of the minimum order of the remainder on ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As regards (i), we first note that χ(1) ν has no explicit dependence on e, but only an implicit dependence through η, which in the closed-form context is treated as an independent symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This follows from the fact that χ(1) stems from balancing the coefficients of R(0) ν,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The latter term contains a pre-factor µ, which is already O(σν), thus it cannot contain any further factors produced by any explicit power of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In view of the above, setting ∂χ(1)/∂e = 0, we find that for any F ∈ C∞(T4 × D) the expression in {F, χ(1) ν } pertaining (i) can be factored out as {F, χ(1) ν }(i) = −∂F ∂e σ−1 � ∂f ∂ℓ ∂e ∂δL ∂χ(1) ν ∂f + ∂e ∂G ∂χ(1) ν ∂g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (75) We now have the following lemma: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For every term in the Hamiltonian (63) of the form qs(∥r1∥ , δL, η, ιc, ιs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' µ, L∗, a1, e1, η1) cos(s1f + s2g + s3h + s4E1)σs , (76) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', explicitly independent on e, we have s1 = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is a consequence of D’Alembert rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Using modified Delaunay angular elements ˜λ = ℓ + g + h , ˜p = −g − h , (77) ˜q = −h , as well as the formulas f = ℓ + 2e sin ℓ + O(e2), eη(e)−2λ = e + λe3 + O(e5), λ ∈ N, we find that, after expanding in the eccentricity e, (76) should give the terms qs cos(s1(˜λ + ˜p) + s2(˜q − ˜p) − s3˜q + s4E1)σs + O(e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (78) 15 However, according to the D’Alembert rules, in a generic trigonometric monomial of the form bw(∥r1∥ , δL, η, ιc, ιs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' µ, L∗, a1, e1, η1)elσl cos(w1˜λ + w2˜p + w3˜q + w4E1)σw , l ∈ N , (79) appearing after expanding H in the eccentricities e, e1, we necessarily have that l − |w2| must be non-negative and even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Since for any closed-form term in the Hamiltonian, explicitly independent of e, the lowermost term in e produced after the expansion satisfies l = 0, we necessarily have w2 = 0, that is s1 = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In view, now, of (70), the relation s1 = s2 implies ∂χ(1) ν /∂f = ∂χ(1) ν /∂g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Therefore, making use of (50), (53) and (56), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (75) translates into {F, χ(1) ν }(i) = −∂F ∂e σ−1 ∂χ(1) ν ∂f �σ−1 L∗e − σ−1 L∗e + O(σ0) � = −∂F ∂e σ−1 ∂χ(1) ν ∂f O(σ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' It follows that for any of the functions F = R(0) ν , {H, χ(1) ν }, {{H, χ(1) ν }, χ(1) ν }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', terms produced by derivatives of the type (i) in (68) are subject to a lowering of the exponent of σ per Poisson bracket only by a factor σ−1, instead of σ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In particular, in the case F = R(0) ν,ν (as well as for any other closed-form function explicitly independent on the eccentricity) we have that (75) is identically vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As regards (ii), we find that for any F1, F2 ∈ C∞(T4 × D), the derivative ∂f/∂δL (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (52)) participates in the Poisson bracket {F1, F2} only through the combination ∂f ∂ℓ ∂f ∂δL �∂F1 ∂f ∂F2 ∂f − ∂F1 ∂f ∂F2 ∂f � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (80) On the other hand, the derivative ∂f/∂G (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (55)) participates in the same Poisson bracket through the combination ∂f ∂G �∂F1 ∂g ∂F2 ∂f − ∂F1 ∂f ∂F2 ∂g � (81) which, by Lemma 1, is also equal to zero for F1 = R(0) ν,ν (or any other term O(σν+1) in H not depending explicitly on e), and F2 = χ(1) ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In conclusion, returning to (74), and taking all the above deductions into account, we arrive at the expressions {R(0) ν , χ(1) ν } = {R(0) ν,ν, χ(1) ν } + � � � � l≥ν+1 R(0) ν,l , χ(1) ν � � � = O(σν+ν) + O(σν+1+ν−1) = O(σ2ν) and similarly, 1 2{{H, χ(1) ν }, χ(1) ν } = 1 2{{Z0, χ(1) ν }, χ(1) ν } + 1 2{{R(0) ν , χ(1) ν }, χ(1) ν } = O(σ2ν) + O(σ3ν−1) = O(σ2ν) , since {Z0, χ(1) ν } satisfies Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We then have {Z0, χ(1) ν } = Z (1) ν − R(0) ν,ν + O(σν+1), with Z (1) ν independent on f, g, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Proceeding by induction 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{Z0 + R(0) ν , χ(1) ν }, χ(1) ν }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(1) ν � �� � n≥3 } = O(σmin{nν−(n−2), (n+1)ν−(n−1)}) = O(σn(ν−1)+2) which concludes the proof of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 16 By Proposition 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' computing all Poisson brackets in (68),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' substituting φ1 = e1 sin E1 where appropriate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' and multiplying all terms missing a factor 1/ ∥r1∥ with the factor a1(1 − σν1e1 cos(E1))/ ∥r1∥ (equal to 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' the remainder R(1) ν+1 resumes the standard form R(1) ν+1 = � l≥ν+1 R(1) ν+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='l = � l≥ν+1 � λ≥1 a1 ∥r1∥λ � s∈Z4 d(1) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s cos(s1f + s2g + s3h + s4E1)σl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (82) where the coefficients d(1) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s satisfy the relations d(1) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s = d(1) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s(δL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ιs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' µ, L∗, a1, e1, η1) = � � � d′(1) l,λ,p , s1 = s4 = 0, (s2, s3) = p , d′′(1) l,λ,s , (s1, s4) ̸= (0, 0) , ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' These last algebraic operations conclude the first normalization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 Loop: normalization of the σν+j−1-terms The procedure followed in the first step can be repeated iteratively in order to normalize consec- utively terms of order σν+j−1, with each time an O(σν+j) remainder, for ν, j > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As anticipated in Remark 4, the iterative procedure described below fails in the case ν = 1 at step j = 2, so an adjustment (involving one more iteration) is required, as discussed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The j-th normalization step is carried out as follows from the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Assume ν ≥ 2, ν1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Assume that the Hamiltonian before the j-th normal- ization step has the form: H (j−1) = Z0 + j−1 � l=1 Z (l) ν+l−1 + R(j−1) ν+j−1 (83) where Z (l) ν+l−1 = σν+l−1 � λ≥1 � p∈Z2 ζ(l) ν+l−1,λ,p cos(p1g + p2h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (84) R(j−1) ν+j−1 = � l≥ν+j−1 R(j−1) ν+j−1,l = � l≥ν+j−1 � λ≥1 a1 ∥r1∥λ � � � � � p∈Z2 d′(j−1) l,λ,p cos(p1g + p2h) + � s∈Z4 (s1,s4)̸=(0,0) d′′(j−1) l,λ,s cos(s1f + s2g + s3h + s4E1 � � � � σl , (85) for some real coefficients ζ(l) ν+l−1,λ,p, d′(j−1) l,λ,p , d′′(j−1) l,λ,s specified at previous steps, where ζ(1) ν,λ,p = � q′ ν,p , λ = 1 0 , λ > 1 by (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 17 Define the j-th step Lie generating function χ(j) ν+j−1 as χ(j) ν+j−1 = φ1 n1 σν+j−1+ν1 � λ≥1 λ � ψ=1 1 aψ−1 1 ∥r1∥λ−ψ � p∈Z2 d′(j−1) ν+j−1,λ,p cos(p1g + p2h) + σν+j−1 � λ≥1 1 ∥r1∥λ−1 � s∈Z4 (s1,s4)̸=(0,0) d′′(j−1) ν+j−1,λ,s s1n∗ + s4n1 sin(s1f + s2g + s3h + s4E1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (86) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' the Hamiltonian H (j) produced by the Lie operation H (j) = exp � Lχ(j) ν+j−1 � H (j−1) has the form H (j) = exp � Lχ(j) ν+j−1 � H (j−1) = Z0 + j � l=1 Z (l) ν+l−1 + R(j) ν+j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (87) where Z (j) ν+j−1 = σν+j−1 � λ≥1 � p∈Z2 ζ(j) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p cos(p1g + p2h) (88) with ζ(j) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p = 1 aλ−1 1 d′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (89) and R(j) ν+j = � l≥ν+j R(j) ν+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='l = � l≥ν+j � λ≥1 a1 ∥r1∥λ � � � � � p∈Z2 d′(j) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p cos(p1g + p2h) + � s∈Z4 (s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s4)̸=(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0) d′′(j) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s cos(s1f + s2g + s3h + s4E1 � � � � σl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (90) with real coefficients d′(j) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' d′′(j) l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s computed from the known coefficients ζ(l) ν+l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p (l = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , j−1), d′(j−1) l,λ,p , d′′(j−1) l,λ,s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We repeat the strategy of Proposition 1 and look for a generating Hamiltonian this time dependent on ∥r1∥: χ(j) ν+j−1(f, g, h, E1, φ1, ∥r1∥ , δL, e, η, ιc, ιs) = σν+j−1 � � � �φ1σν1 � λ≥1 � p∈Z2 ˆd′(j−1) ν+j−1,λ,p(∥r1∥ , δL, e, η, ιc, ιs) cos(p1g + p2h) + � λ≥1 � s∈Z4 (s1,s4)̸=(0,0) ˆd′′(j−1) ν+j−1,λ,s sin(s1 + s2g + s3h + s4E1) � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 18 Requiring {Z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' χ(j) ν+j−1} + R(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='ν+j−1 to be O(σν+j) in fast angles we come up with −n∗ ˆd′′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='ss1 − n1 ˆd′′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='ss4 + 1 ∥r1∥λ−1 d′′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' −n1 a1 ∥r1∥ ˆd′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p + n1 ˆd′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p + a1 ∥r1∥λ d′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p = 1 aλ−1 1 d′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' for λ ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ˆd′′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s = 1 ∥r1∥λ−1 d′′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='s s1n∗ + s4n1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' s ∈ Z4 : (s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' s4) ̸= (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ˆd′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p = 1 aλ−1 1 d′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p n1 λ−1 � ψ=0 � a1 ∥r1∥ �ψ = d′(j−1) ν+j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='p n1 λ � ψ=1 1 aψ−1 1 ∥r1∥λ−ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' p ∈ Z2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' which proves Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (86), and new accumulated addenda in normal form Z (j) ν+j−1 = σν+j−1 � λ≥1 1 aλ−1 1 � p d′(j−1) ν+j−1,λ,p cos(p1g + p2h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' which proves Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='(89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' It remains to demonstrate that the expression (90) is O(σν+j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The proof is done by induction: for j = 2 we get H (2) = Z0 + Z (1) ν + Z (2) ν+1 + O(σν+2) + � l≥ν+2 R(1) ν+1,l + {Z (1) ν , χ(2) ν+1} + {R(1) ν+1, χ(2) ν+1} + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' + � n≥2 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{H (1), χ(2) ν+1}, χ(2) ν+1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(2) ν+1 � �� � n } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (91) Similarly as in Proposition 1, a lowering of the book-keeping exponents in a Poisson bracket of the form {F, χ(2) ν+1} can occur through derivatives of the form (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' However, this time the latter can only appear in a Poisson bracket via the combination σ−1 � ∂f ∂ℓ ∂e ∂δL � ∂F ∂f ∂χ(2) ν+1 ∂e − ∂F ∂e ∂χ(2) ν+1 ∂f � + ∂e ∂G � ∂F ∂g ∂χ(2) ν+1 ∂e − ∂F ∂e ∂χ(j) ν+1 ∂g �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (92) so we can infer that {Z (1) ν , χ(2) ν+1} = O(σ2ν+1) , {R(1) ν+1, χ(2) ν+1} = O(σ2ν) , 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{H (1), χ(2) ν+1}, χ(2) ν+1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(2) ν+1 � �� � n≥2 } = O(σmin{n(ν+1)−2(n−1), n(ν+1)+ν−2(n−1), (n+1)(ν+1)−2n}) = O(σn(ν−1)+2) because (80), (81), (92) vanish when F = F1 = Z (1) ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Now, for all ν ≥ 2, n(ν − 1) + 2 > ν + 1, n ≥ 2, hence, the proposition is valid for j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For j ≥ 3, we have H (j) = Z0 + Z (1) ν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' + Z (j−1) ν+j−2 + Z (j) ν+j−1 + O(σν+j) + � l≥ν+j R(j−1) ν+j−1,l + {Z (1) ν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' + Z (j−1) ν+j−2, χ(j) ν+j−1} + {R(j−1) ν+j−1, χ(j) ν+j−1} + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' + � n≥2 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{H (j−1), χ(j) ν+j−1}, χ(j) ν+j−1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(j) ν+j−1 � �� � n } , (93) 19 and analogously {Z (1) ν , χ(j) ν+j−1} = O(σ2ν+j−1) , {Z (j−1) ν+j−2, χ(j) ν+j−1} = O(σ2ν+2j−5) , {R(j−1) ν+j−1, χ(j) ν+j−1} = O(σ2ν+2j−4) , 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{H (j−1), χ(j) ν+j−1}, χ(j) ν+j−1}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(j) ν+j−1 � �� � n≥2 } = O(σmin{n(ν+j−1)−2(n−1), n(ν+j−1)+ν−2(n−1), n(ν+j−1)+ν+j−2−2n, (n+1)(ν+j−1)−2n}) = O(σn(ν+j−3)+2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' However, since ν > 1, n ≥ 2, we readily find n(ν + j − 3) + 2 > ν + j − 1, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 The case ν = 1 Coming to ν = 1, one realizes that (91) produces same order σ2 non-normalized terms via {R(1) 2 , χ(2) 2 } and {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{Z0 + R(1) 2 , χ(2) 2 }, χ(2) 2 }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(2) 2 }, namely the resulting remainder is R(2) 2 , so the scheme in Proposition 2 is not directly applicable beyond j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Despite this, it is worth noticing that if we manage to get rid of these spurious terms, by performing, for instance, an extra normalization II, such that the new outcome returns R(II) = R(II) 3 , then the algorithm (87) will work for j ≥ 3 upon restarting the recursion from iteration II in place of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is precisely the claim we are about to show to complete the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Let us write (91) as H (2) = Z0 + Z (1 1 + Z (2) 2 + R(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Introduce the extra second normalization II based on Proposition 2 targeted to R(2) 2,2 with generating function χ(II) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For ν = 1 and any ν1 ≥ 1, H (II) = exp � Lχ(II) 2 � H (2) = Z0 + Z (1) 1 + Z (2) 2 + Z (II) 2 + R(II) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (94) Moreover the loop composed by (87)–(90) in Proposition 2 holds true for any j ≥ 4 under the modifications H (3) = exp � Lχ(3) 3 � H (II) = Z0 + Z (1) 1 + Z (2) 2 + Z (II) 2 + Z (3) 3 + R(3) 4 , (95) H (j) = exp � Lχ(j) j � H (j−1) = Z0 + j � l=1 Z (l) l + Z (II) 2 + R(j) j+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (96) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We begin with a necessary generalization of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Given F1, F2 ∈ Cω(T×D) trigonometric monomials of the form (76), or equivalently in terms of the sine, fulfilling the property of Lemma 1, addenda of the same type in the Lie series transformation applied to F1 with respect to F2 preserve such property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Since exp (LF2) F1 involves the computation of Poisson brackets of functions explicitly independent on e, we have that (92), with F1, F2 in place of F, χ(2) ν+1, is identically null, as well as (81) because ∂F1/∂f = ∂F1/∂g, ∂F2/∂f = ∂F2/∂g by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Thus, the bracket {F1, F2} 20 in the Lie series either does not introduce any eccentricity dependence at all, or only at numerator through (50) multiplied by cos f or cos2 f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' therefore its derivatives contain products of cosines (sines) whose coefficients are independent on e like G1(s1f + s2g + s3h + s4E1)G2(u1f + u2g + u3h + u4E1) , G1, G2 = cos, sin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The arguments are now either summed or subtracted, hence they clearly satisfy the property concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' By cascade reasoning for further nested brackets we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' A straightforward use of the lemma in conjunction with formulae (80), (81), (92) (χ(2) ν+1 replaced by generic differentiable function) reveal that any transformed Hamiltonian H (j) and corresponding generating function χ(j) ν+j−1 encountered are regular at e = 0 in agreement with D’Alembert rules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' they never depend on negative powers of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Furthermore, every time one of the two entries of {·, ·} does not depend on e, the upshot due to item (i) in the proof of Proposition 1, as soon as non-zero, is diminished by σ−1 instead of σ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We consider step II: H (II) = Z0 + Z (1) 1 + Z (2) 2 + Z (II) 2 + O(σ3) + � l≥3 R(2) 2,l + {Z (1) 1 , χ(II) 2 } + {Z (2) 2 , χ(II) 2 } + {R(2) 2 , χ(II) 2 } + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' + � n≥2 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{H (2), χ(II) 2 }, χ(II) 2 }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(II) 2 � �� � n } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (97) The analysis of the contributions reports these deductions, by which (94) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {Z (1) 1 , χ(II) 2 } = O(σ3) because Z (1) 1 is independent on f, g, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {Z (2) 2 , χ(II) 2 } = O(σ4) because Z (2) and χ(II) 2 fulfil Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Indeed, R(1) 2,2 depends on e at most linearly by book-keeping rules, so it does χ(2) 2 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' At this point we show that for eccentricity dependent terms stemming from R(1) 2,2 (or equivalently χ(2) 2 ) d′(1) 2,λ,p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Every trigonometric monomial in R(1) 2,2 explicitly dependent on e carries the dependence on at least one of the two fast anomalies f, E1 as well, namely corresponding coefficients in (82) are d(1) 2,λ,s = d′′(1) 2,λ,s, (s1, s4) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' By Proposition 1, Lemma 1 and 2, the substitution φ1 = e1 sin E1 and the formulas listed in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' we take out of (68) the order σ2 remainder and it is not restrictive to assume ν1 = 1 in order to include also the e1 cos E1 dependent term in (71): R(1) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 = R(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 + a1 ∥r1∥ � n∗ � e1 cos E1 − 2e cos f η3 � σ∂χ(1) 1 ∂f + ∂R(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 ∂f ∂χ(1) 1 ∂δL − ∂R(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 ∂δL ∂χ(1) 1 ∂f − 1 L∗ ∂χ(1) 1 ∂ιc � ιc ∂R(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 ∂f − ∂R(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 ∂h � + 1 L∗ ∂R(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 ∂ιc � ιc ∂χ(1) 1 ∂f − ∂χ(1) 1 ∂h � − 2 sin f L∗e σ−1 ∂χ(1) 1 ∂f � ∂R(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 ∂g − ∂R(0) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 ∂f � � − a1 2 � 1 ∥r1∥ � n∗ � 1 + 2e cos f η3 σ � ∂χ(1) 1 ∂f + n1 ∂χ(1) 1 ∂E1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' χ(1) 1 � 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 21 where {·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ·}2 indicates that we retain only σ2 quantities after the operation (in virtue of Lemma 2 and Remark 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' inductions derived to demonstrate Proposition 1 are a coarser bound and no other parts of order σ2 come out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Plugging in (70) and (65) for l = 1, 2 and taking into account Lemma 1, upon simplifications the contributions involving e result R(0) 1,2e − a1en∗ η3 ∥r1∥σ2 � (s1,s4)̸=(0,0) s1q′′ 1,s s1n∗ + s4n1 (cos((1 − s1)f − s1g − s3h − s4E1) + cos((1 + s1)f + s1g + s3h + s4E1)) , (98) where R(0) 1,2e = a1 ∥r1∥σ2 � s∈Z4 q2,s cos(s1f + s2g + s3h + s4E1) , q2,s = e¯q2,s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (99) We employ now all D’Alembert rules to show that only the harmonics of interest can exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Following the same argument as in Lemma 1, let us write the cosine input of (99) using modified Delaunay angles (77) also for P1 in relation to corresponding orbital elements (3) (subscript ‘1’): s1˜λ + (s1 − s2)˜p + (s2 − s3)˜q + s4˜λ1 + (s4 − s5)˜p1 + (s5 − s6)˜q1 , sl ∈ Z , in which ˜p1 = ˜q1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For the elimination of the apparent singularity at e = 0, we must have 1 − |s1 + s2| ≥ 0 and even, hence s2 = s1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Then, since R(0) 1,2e is independent on e1 by book-keeping setting, analogously we must end up with s4 = s5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Regarding instead the regularity at i1 = 0, because of the absence of i1 we must conclude that 0 − |s5 − s6| ∈ 2N, namely s5 = s6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' At this stage, we invoke the invariance under rotation around the Z axis, which prescribes s1 − s1 + s2 − s2 + s3 + s4 − s4 + s5 − s5 + s6 = s3 + s6 = 0 , and summing up this implies s3 = −s4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Ultimately, concerning the inclination, we must ensure that l − |s2 − s3| ∈ 2N, with l even as well again being i1 not involved, thus s2 = s3 ± 2n, n ≤ l/2 natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Putting all together we arrive at s1f + s2g + s3h + s4E1 =⇒ s1f + (s1 ± 1)g + (s1 ∓ 2n ± 1)h + (±2n ∓ 1 − s1)E1 , which always depends on at least one among f, E1 since the coefficients s1, ±2n ∓ 1 − s1 never vanish simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' By means of an identical reasoning and given the preservation of D’Alembert rules under exp � Lχ(1) 1 � , we achieve the same outcome for the remaining part of (98) after replacing s1 �→ 1 ± s1, indeed we find (1 ± s1) + (1 ± s1 ± 1)g + (1 ± s1 ∓ 2n ± 1)h + (±2n ∓ 1 − 1 ∓ s1)E1 , and no solutions to 1 ± s1 = 0, ±2n ∓ 1 − 1 ∓ s1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Given that the order 2 normal form is sourced from the part of R(1) 2,2 explicitly indepen- dent on fast angles, it turns out that it is free of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Finally, R(2) 2,2 is free of e too, being generated by terms in {R(1) 2,2, χ(2) 2 } and {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{Z0 + R(1) 2,2, χ(2) 2 }, χ(2) 2 }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(2) 2 } subjected to computation (i) of Proposition 1 (Remark 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Again by construction, the same applies to χ(II) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 22 {R(2) 2 , χ(II) 2 } = O(σ4) by Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' {{H (2), χ(II) 2 }, χ(II) 2 }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , χ(II) 2 � �� � n≥2 } = O(σ4) consequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In order to conclude, we just need to check that the next step gives rise to an O(σ4) perturbation and the cycle of normalizations can restart for j ≥ 4 in light of the bounds on σ from (93) at the end of the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Upon repeating the usual argument, it is easy to see that the only bracket worth investigating is {Z (II) 2 , χ(3) 3 }, that is, nevertheless, O(σ4) because Z (II) 2 is made out of R(2) 2,2 independent on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' By the above argument it is immediate to realize that even p2 ≡ 0 in (70) and (65) for l = ν, so q′ ν,p = 0 for all p ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Serving as an example, a detailed demonstration of the normalization procedure exposed in the present section for a simple model, containing just few terms of the disturbing function, is presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 3 Numerical tests 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 Computer-algebraic implementation of the normalization algorithm Implementing the above normalization procedure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' by use of a Computer Algebra System (CAS), requires working with a finite truncation of the initial Hamiltonian model (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' To this end, the disturbing function (13) multiplied by µ can be re-arranged as µH1 = −Gm0µ ∥R∥ ∞ � κ1=0 ∞ � κ2=0 κ2̸=1 ∞ � κ3=0 ˜hκ1,κ2,κ3µκ1 �2r1 · R ∥R∥2 �κ2 �∥r1∥ ∥R∥ �2κ3 , (100) where ˜hκ1,κ2,κ3 are real coefficients derived from the coefficients of (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' A convenient truncation of (100) stems from defining two separate truncation orders in powers of µ (truncation order kµ), and in powers of ∥r1∥ / ∥R∥ (multipole truncation order kmp), through the formula H≤kµ,kmp 1 = −Gm0µ ∥R∥ kµ−1 � κ1=0 kmp � κ2=0,κ̸=1 ⌊kmp/2⌋ � κ3=0 ˜hκ1,κ2,κ3µκ1 �2r1 · R ∥R∥2 �κ2 �∥r1∥ ∥R∥ �2κ3 , (101) where ⌊·⌋ is the integer part function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Working with the truncated Hamiltonian H≤kµ,kmp = H0 + H≤kµ,kmp 1 , we then obtain a sequence of secular models Z (j), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', where j denotes the normalization step, computed via the formula Z (j) = Z0 + j � l=1 Z (l) ν+l−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (102) In particular, we implement the following steps of the CAS algorithm: (i) for a fixed value of µ, choose values for kµ, kmp, perform the corresponding expansions of the Hamiltonian as in (100) and compute the truncated model H≤kµ,kmp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 23 (ii) choose the reference values of a∗ and e∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (iii) pass to variables (f, g, h, E1, δL, e, η, ιc, ιs, J1) and parameters L∗, e1, a1, η1 on the basis of the selected a∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (iv) compute ν and ν1 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (7));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (v) set the appropriate book-keeping weights following the rules in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 and expand correspondingly the Hamiltonian in δL up to σνkµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (vi) drop constants, perform the identity operation (63), discard book-keeping powers larger than νkµ and introduce n∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (vii) if ν > 1, compute the generating function (70) as well as the first-normalized Hamiltonian H (1) by the Lie series operation (66) truncated at the maximum book-keeping order Nbk = νkµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' if ν = 1, compute H (1) (always truncated to the book-keeping order Nbk) via the procedure of Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (viii) compute the successive normalizations H (j), truncated at book-keeping order Nbk via the procedure of Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3, up to a maximum normalization order ν + jmax − 1 < Nbk, jmax ≤ ν(kµ − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' this allows to obtain truncated Hamiltonian models containing a finite number of normal form terms as well as a finite number of terms provided by the truncated remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In the CAS implementation of the above algorithm we work with numerical coefficients, substituting all constants with their corresponding numerical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Several types of numerical tests of the precision and overall performance of the method can be carried out as exemplified in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 Numerical examples in the Sun-Jupiter ER3BP: semi-analytic orbit prop- agation For all numerical tests below we refer to the Sun-Jupiter one (µ = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5364 · 10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We employ Earth-orbit based units, such that Gm0 = 4π2AU3/y2, a1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2044AU, so that Jupiter’s period is T1 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='86 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Jupiter’s mean motion is n1 = 2π/T1, and eccentricity e1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0489, used throughout all computations in the framework of the ER3BP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In all tests below, a particle’s orbit is defined by providing the initial conditions a(0), e(0), i(0), complemented by f(0) = g(0) = h(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Our basic probe of the efficiency of the normalization method in the framework of the ER3BP is given by comparing the short-period oscillations of the orbital elements a(t), e(t), i(t), g(t), h(t), as found by two different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Direct Cartesian propagation: the initial conditions z(0) := (a(0), e(0), i(0), f(0), g(0), h(0)) are mapped into initial conditions for the Cartesian canonical positions and conjugate momenta (X(0), Y (0), Z(0), PX(0), PY (0), PZ(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Using Hamilton’s equations with the full Hamiltonian (1) (setting also J1(0) = 0, M1(0) = 0), we obtain the numerical evolution (X(t), Y (t), Z(t), PX(t), PY (t), PZ(t)), which can be transformed to element evolution z(t) = (a(t), e(t), i(t), f(t), g(t), h(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Semi-analytical propagation: following the implementation of the normalization algorithm as de- scribed in the previous subsection, the initial osculating element state vector z(0) is transformed 24 0 500 1000 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='98 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='99 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='00 0 500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='100 0 500 1000 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='99992 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='99996 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='00000 0 500 1000 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='98 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='99 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='00 0 500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='150 0 500 1000 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='99974 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='99987 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='00000 Figure 2: First and second example (ER3BP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Data: a∗ = 50AU, e∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 (ν = 3), i(0) = 10◦, kµ = kmp = 2 (top panels);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' a∗ = 30AU, e∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='15 (ν = 4), i(0) = 10◦, kµ = kmp = 2 (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Black curves represent semi-analytic time variations (our method), while red curves stand for Cartesian series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' into an initial condition for the corresponding ‘mean element’ state vector ξ(j)(z(0)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', the element vector corresponding to the new canonical variables conjugated to the original ones af- ter j near-identity normalizing transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is computed by the Lie series composition formula truncated at book-keeping order Nbk: ξ(j)(z) = � exp � L−χ(1) ν � exp � L−χ(2) ν+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ◦ exp � L−χ(j) ν+j−1 � z �≤Nbk , (103) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (69) for the inverse series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We then obtain the evolution of the mean element vector ξ(j)(t) through numerical integration of the secular equations of motion ˙ξ(j) = J∇Z (j)(ξ(j)) (104) (J standard symplectic unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This can be back-transformed to yield the evolution of the oscu- lating element vector z(t) using the truncated Lie series composition formula z(ξ(j)) = � exp � Lχ(j) ν+j−1 � exp � Lχ(j−1) ν+j−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' ◦ exp � Lχ(1) ν � ξ(j) �≤Nbk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (105) Note that both the direct and inverse transformations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (103) and (105)), as well as Hamilton’s secular equations (104), can be computed in closed form, using the Poisson algebra rules of Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We then call semi-analytic the evolution of the element vector z(t) obtained via the formula z(t) = z(ξ(j)(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (106) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2 shows the comparison between the Cartesian and the semi-analytical propagation of the elements in ‘easy’ cases, where the particle departs from initial conditions a(0) = 50AU (top left 25 0 200 400 600 800 1000 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='70 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='75 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='80 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='85 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='90 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='95 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='00 0 200 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='20 Figure 3: Third example (ER3BP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Data: a∗ = 50AU, e∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='7 (ν = 20), i(0) = 20◦, kµ = 2, kmp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' On the left, the black curve represents the semi-analytic time variation of the semi- major axis (our method) versus the one found by propagation of the Cartesian equations of motion (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The right panel shows the evolution of the corresponding percent relative error E% a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' panel) or a(0) = 30AU (bottom left panel), with a relatively low value of the eccentricity e(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 or e(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='15 respectively (middle panels) and inclination i(0) = 10◦ (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In these cases, the distance ratio ∥r1∥ / ∥R∥ is small (about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2), a fact implying that the quadrupolar expansion (kmp = 2) suffices to have obtained a relative error of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1% in the representation of the Hamiltonian perturbation H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Going to higher multipoles is straightforward, albeit with a significant computational cost as the number of terms in the Hamiltonian grows significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' On the other hand, even with low-order truncations of the Hamiltonian we achieve to have an accurate semi-analytical representation of the O(µ) short-period oscillations in all three ‘action- like’ elements (semi-major axis, eccentricity, inclination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Most notably, keeping a(0) = 50AU but changing the eccentricity to e(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='7, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', beyond the Laplace value, yields an orbit whose pericenter is at ∥Rp∥ = 15AU, implying a distance ratio ∥r1∥ / ∥R∥ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This time, an octupole truncation (kmp = 3) is required to produce an approximation of the Hamiltonian model at the level of a relative error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Still, however, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 3 the semi-analytical propagation of the orbit is able to track the fully numerical one with an error which does not exceed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2% even close to the orbit’s pericentric passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In the above examples, the maximum number of normalization steps at which the secular Hamiltonian is computed was set equal to jmax = 3, jmax = 4 and jmax = 4 respectively, which corresponds to the best match in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As discussed in the next subsection, an estimate of the minimum possible error in the semi-analytic propagation of the trajectories requires computing first the so-called optimal number of normalizations jopt (or equivalently optimal normalization order ν + jopt − 1) as a function of the reference values (a∗, e∗) within a model given by a preset fixed multipole truncation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Owing to the fact that the same divisors appear in the ER3BP and in the CR3BP, we verify with numerical examples that the error analysis yields essentially identical results in either case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' However, the computation of the optimal normalization is easier to perform in the CR3BP, owing to the considerably smaller number of terms produced in the CAS computation of the normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Hence, we now turn our attention to this latter computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 Numerical examples in the Sun-Jupiter planar CR3BP: order and size of the optimal remainder 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 Trajectory propagation: optimal remainder A considerable reduction of the computational cost occurs in the case of the planar and circular R3BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is due, in particular, to the following: the dependence on M1 becomes explicit (M1 = E1 in (2)), while a1 = ∥r1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As a conse- quence, φ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' no terms involving (h, H) appear in the disturbing function, thus ιc, ιs are discarded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' no terms requiring a book-keeping in terms of the exponent ν1 appear, hence, only ν is defined, as in (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' d′(j) l,λ,p = 0 for every j, l, λ, p in (90), (86), and consequently p1 = p2 ≡ 0 in (88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is due to the fact that the expression (16) reduces to r1 · R = ∥r1∥ ∥R∥ cos(f + g − M1) , (107) which always depends on the difference g − M1 by D’Alembert rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This implies that, unlike the ER3BP, the action G (and the corresponding eccentricity e) are integrals of the secular Hamiltonian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' as a consequence no lower or equal book-keeping order terms appear in any Poisson bracket of the first normalization step in the case ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Hence Proposition 3 is redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Owing to the above, in the planar CR3BP we are able to make normal form computations in a grid of points in the plane (a∗, e∗) up to a sufficiently high normalization order so that the asymptotic character of the series computed by the algorithm of Section 2 can show up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' To this end, we introduce an estimate of the size of the series’ remainder after j normalization steps via the upper norm bound E (j) = νkµ � l=ν+j � s∈Z3 |d(j) l,s | ≥ ���R(j) ν+j ��� ∞ , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' , ν(kµ − 1) , (108) where ∥·∥∞ denotes the sup norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Plotting E (j) against the number of normalization steps j allows then to estimate the error committed at any step (size of the remainder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Figure 4 yields an example of such computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The relevant fact is that there is an optimal number of normalization steps (j = jopt = 6) where the estimate E (j) of the remainder size yields a global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Although a systematic investigation of the dependence of the optimal number of normalization steps jopt on the parameters (a∗, e∗) is beyond our present scope, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 5 and 6 allow to gain some insight into the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The most relevant remark concerns the dependence of the behavior of the curve E (j) (versus j) on how close to the ‘hierarchical’ regime the trajectory with reference values (a∗, e∗) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As a measure of the hierarchical character of an orbit we adopt either the ratio of the semi-major axes a1/a∗, or of the pericentric distances ∥r1∥ / ∥Rp∥ = a1(1 − e1)/(a∗(1 − e∗)) = a1/(a∗(1 − e∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 5 (a∗ = 30AU, e∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5) implies a pericentric distance ratio ∥r1∥ / ∥Rp∥ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 smaller than the one of the example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 4 (∥r1∥ / ∥Rp∥ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We observe that the optimal number of normalization steps in the former case satisfies jopt = 10, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', it is larger than in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 6 shows, instead, an example of orbit far from the hierarchical limit, satisfying the estimate ∥r1∥ / ∥Rp∥ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In this case a higher order multipole expansion 27 1 2 3 4 5 6 7 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 0 150 300 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='97 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='99 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='01 0 150 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4005 0 150 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='004 Figure 4: Fourth example (planar CR3BP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Data: a∗ = 20AU, e∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 (ν = 8), kµ = 2, kmp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The estimate E (j) is depicted in semi-logarithmic scale on top left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The direct comparison of the semi-analytic (black) evolution vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' the fully numerical (red) one for the osculating elements a(t), e(t), g(t) are shown in the top right and bottom panels respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The semi-analytic curves are obtained for j = jopt = 6, where E (j) is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 28 1 2 3 4 5 6 7 8 9 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 0 150 300 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='94 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='97 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='00 0 150 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='500 0 150 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='012 Figure 5: Fifth example (planar CR3BP): a∗ = 30AU, e∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 (ν = 10), kµ = 2, kmp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Plot types and color conventions are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The semi-analytic curves are obtained for j = jopt = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (kmp = 5) is required to obtain a precise truncated Hamiltonian model for this orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We note, however, that the normalization procedure performs well, producing a decreasing remainder as a function of j up to the point where it is arrested, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' j = 6 = ν(kµ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We find numerically that this performance is deteriorated as we gradually approach the condition ∥r1∥ / ∥R∥ = 1, beyond which the multipole expansion of the Hamiltonian is no longer convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 Semi-analytical determination of the domain of secular motions The results shown in the two previous subsections refer to isolated examples of orbits treated within various multipole truncation orders as well as different choices of the number of nor- malization steps, searching each time to arrive at the best approximating secular model given computational restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In the present subsection, we aim to investigate the behavior of the remainder in a closed-form normalization with uniform choice of all truncation orders of the problem, but performed, instead, in a fine grid (100 × 20) of reference values in the plane (a∗, e∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' To this end, we set kµ = 2 (second order in the mass parameter), and fix kmp = 3 (octupole approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The latter choice, imposed by computational restrictions, yields an initial model whose error with respect to the full Hamiltonian becomes of the order of 1% only for a∗ > 2a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' However, for reasons explained below, a computation within the framework of the octupole approximation becomes relevant to the problem addressed in the sequel also in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5a1 < a∗ < 2a1, while higher multipoles are required to address still smaller values of a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 29 1 2 3 4 5 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 0 150 300 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='98 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='04 0 150 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='102 0 150 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='18 Figure 6: Sixth example (planar CR3BP): a∗ = 8, e∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 (ν = 3), kµ = 3, kmp = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Plot types and color conventions are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The semi-analytic curves are obtained for j = jopt = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The result of the above computation is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 7: the left panel shows in logarithmic color scale the size of the remainder, estimated by the value of E (n)(a∗, e∗) computed as in (108), corresponding to each point in the plane (a∗, e∗), where the number of normalization steps is set as n = min{ν(kµ − 1), 7} = min{ν, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The maximum value n = 7 is, again, imposed by computational restrictions, and it implies that n varies with e∗ up to about e∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The relevant information in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 7 is provided by the black curve, which corresponds to the isocontour E (n)(a∗, e∗) = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Since in the original Hamiltonian we have the estimate E (0)(a, e) := H≤kµ,kmp 1 = O(10−2), the black curve provides a rough estimate of the limiting border dividing the plane (a∗, e∗) in two domains: in the one below the black curve the progressive elimination of the fast angles by the iterative normalization steps leads to a secular model whose remainder decreases with the number of normalization steps j at least up to j = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' A physical interpretation of the border approximated through the isocontour E (n)(a∗, e∗) = 10−2 can be given through a comparison with a numerical stability map obtained, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', as in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For each trajectory in a 300 × 900 grid in (a, e), the plot shows in color scale the value of the Fast Lyapunov Indicator (FLI, see [9] for a review) obtained after integrating the variational equations of motion together with the equations of motion of the full Hamiltonian model for a time equal to 50 periods of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Thus, deep blue colors indicate the most regular, and light yellow the most chaotic orbits as identified by the value of the FLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Superposed to the FLI cartography are three curves: 30 Figure 7: Left panel: computation of log10(E (n)), n = min{ν, 7}, kmp = 3, over a 100 × 20 (a, e) grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' For every e = e∗, n different normalizations are executed and then evaluated for each a = a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Right panel: short-period FLI map over a 300 × 900 (a, e) grid of initial data integrated for 50T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As indicated, the three curves represent, respectively, the line of constant pericenter of the particle’s trajectory equal to the radius of Jupiter’s orbit ∥r1∥ = ∥rJ∥ (red), Hill’s stability criterion (brown) and the isolevel E (n) = 1% (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Each region enclosed by two consecutive above curves is labeled with the corresponding regime of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The main mean-motion resonances are reported below the pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (i) the ‘perihelion crossing curve’ (red) yields the locus of values satisfying the condition a(1 − e) = ∥rJ∥ = aJ (in the circular case), that is the points where the pericenter of the test particle’s orbit comes at distance equal to the radius of Jupiter’s orbit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (ii) the Hill limit [12] (brown) is based on the relationship CJac(a, e) = CJac(L1), where CJac is the particle’s Jacobi constant as function of the orbital elements and CJac(L1) its value at the Lagrangian point L1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (iii) the isocontour E (n)(a, e) = 10−2 (black, same as in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Of the above three curves, the perihelion crossing curve is analogous, in the R3BP, of the so-called Angular Momentum Deficit criterion (AMD, [8]) used to separate systems protected from perihelia crossings in the case of the full planetary three-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As indicated by the FLI cartography data, Hill’s curve gives an overall better approximation separating the domain of strong chaos (yellow) from the domain of regular or weakly-chaotic orbits (all blue nuances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This is expected, since the Hill’s curve separates orbits for which Jupiter’s gravitational effect becomes (at least temporarily) dominant from those for which it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Nevertheless, through the FLI cartography we note the presence of a large domain between the curves (ii) and (iii), where the trajectories, while protected from close encounters, are subject to the long term effects on dynamics produced by resonant multiplets associated with the mean-motion resonances of the problem (the most important of which are marked in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Note that in the octupole approximation, the Hamiltonian contains harmonics including all combinations of the fast angles of the form cos(s1f + s2(g − M1)), with (s1, s2) = (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (7, 3), 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='55 Apsidal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='55 Hill Apsidal Close encounter regime Method Hill 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 Method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='45 Crdssing orbit 4 Resonant regime 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 regime 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 e e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 Secular regime 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5 3 a/ IIrj a / IIrj ll 2:5 1:4 2:3 1:3 1:5(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (1, −1), (2, −1), (3, −1), (1, −2), (1, −3), thus including all harmonics associated with the mean-motion resonances detected in the FLI cartography of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 7 for a > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='5aJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Through the closed-form normalization (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (70) and (86)) we then obtain small divisors in the series at every value of the semi-major axis a∗ for which one of the resonant combinations s1n∗ − s2nJ, nJ = n1, takes a value near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' All these incidences lead to Arnold tongue-like spikes pointing downwards in the curve (iii), marking the failure of the approximation of the orbits based on a non-resonant normal form construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' On the other hand, we observe that, for any value of a∗ there is a threshold value of the eccentricity e∗,s, such that, for e∗ < e∗,s no visible effects of the harmonics associated with mean-motion resonances are visible in the FLI cartography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' This implies that the secular models constructed by eliminating all harmonics involving the fast angles of the problem describe with good precision the dynamics in this domain, called, for this reason, the domain of secular motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In physical terms, the domain of secular motions corresponds to initial conditions for which the gravitational perturbation of Jupiter is only felt in the ‘Laplacian’ meaning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', as a mass distributed along a ring coinciding with Jupiter’s orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The curve (iii) then yields the limit of this domain, which, as found by the FLI cartography, is well distinct from the limit of the Hill domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The overall situation can therefore be summarized with the identification of four regimes of motion (specified in the FLI chart): the ‘crossing orbit regime’ (above curve (i));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' the ‘close encounter regime’ (between curves (i) and (ii));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' the ‘resonant regime’ (between curves (ii) and (iii));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' the ‘secular regime’ (below curve (iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 4 Conclusions In summary, in the present paper we have proposed a closed-form method for the derivation of secular Hamiltonian models (normal forms) with a small (albeit finite minimum) remainder applicable to the R3BP in the case when the particle’s trajectory is exterior to the trajectory of the primary perturber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Also, using this method we were led to the definition of a new heuristic limit separating the motions whose character is ‘secular’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=', not affected by short-period effects, from the rest of motions in the R3BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In particular: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Section 2 develops the formal aspects of the method, which heavily relies on the use of a book-keeping parameter to simultaneously account for all small quantities of the problem as they appear not only in the Hamiltonian and Lie generating functions, but also in the closed-form version of all formulas involved in the Poisson algebra between the Delaunay canonical variables of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' A rigorous demonstration of the consistency of the method is then given through Propositions 1, 2 and 3, which also estabilish the explicit formulas for the implementation of one iterative step of the closed-form normalization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Section 3 gives numerical examples of the implementation and precision of the algorithm in the spatial elliptic, as well as in the planar circular R3BP, examining, also numerically, the method’s convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The effect of choosing different truncation orders (in powers of the mass parameter µ or in the multipole expansion) is discussed, along with 32 several simplifications to the normalization procedure which hold in the circular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The essentially asymptotic character of the series is established through numerical examples, showing the existence of an optimal number of normalization steps, after which the size of the remainder becomes the minimum possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' A key aspect of the above presented method lies in the possibility to exploit the behavior of the size of the remainder as a function of the number of normalizing steps in order to obtain a clear separation of two well-distinct domains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' as also identified by purely numerical (FLI cartography) means: one,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' called the domain of secular motions corresponds to the domain where the harmonics in the Hamiltonian associated with resonant combinations of the fast angles (anomalies) of the problem produce no dynamical effect on the orbits visible at the level of the FLI cartography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' From the semi-analytical point of view, this turns to be the domain where a non-resonant construction as the one proposed in section 2 produces no (nearly-)resonant divisors up to the optimal normalization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' As a consequence, only the angles associated with the motions of the perihelion and of the line of nodes survive in the final normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We show numerically how to use the information on the size of the normal form remainder in order to determine semi-analytically the border of the domain of secular motions in the case of the Sun-Jupiter system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' We finally give evidence that this border is well distinct from the border of the domains defined either by the Hill stability or by the perihelion crossing criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Appendix A Computation of Poisson bracket’s intermediate derivatives Derivatives (31)–(43) are computed combining adequately definitions (3), the polar relationship (15), including its alternative expression involving the eccentric anomaly E ∥R∥ = a(1 − e cos E) , (109) ∥r1∥ via (2) (analogous to (109)), Kepler’s equations ℓ = E − e sin E , M1 = E1 − e1 sin E1 , (110) and the trigonometric equalities cos f = cos E − e 1 − e cos E , sin f = η sin E 1 − e cos E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (111) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (31) comes from (109) and (15) by total differentiation with respect to ℓ: d dℓ ∥R∥ (109) = ∂ ∥R∥ ∂E ∂E ∂ℓ = ae sin E 1 − e cos E (15) = ∂ ∥R∥ ∂f ∂f ∂ℓ = aη2e sin f (1 + e cos f)2 ∂f ∂ℓ , since a, e do not depend on ℓ, where ∂E/∂ℓ is deduced from the first of (110) making use of the derivative of inverse functions (∂ℓ/∂E ̸= 0 is ensured).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Thus the result by (111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (32), (33) are straightforwardly yielded taking respectively ordinary differentiation and the inverse derivative once again of dM1/dE1 ̸= 0 from the second of (110): dE1 dM1 = 1 1 − e1 cos E1 = a1 ∥r1∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 33 Now solving for e in (3) and partially differentiating, we immediately have Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (35) and (38), from which Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (36), (39) as ∂η ∂δL = −e η ∂e ∂δL = − η L , ∂η ∂G = −e η ∂e ∂G = 1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The true anomaly derivatives with respect to the actions are slightly more elaborated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Employing (111), − sin f ∂f ∂δL = ∂ ∂δL cos f = ∂ ∂e � cos E − e 1 − e cos E � ∂e ∂δL + ∂ ∂E � cos E − e 1 − e cos E � ∂E ∂δL , that leads upon simplifications to ∂f ∂δL = sin f eL + 1 + e cos f η ∂E ∂δL ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' finally we explicit ∂E/∂δL exploiting the corresponding Kepler equation (110) and the inter- independence ℓ, δL by conjugacy: 0 = d dδL(E − e sin E) = ∂E ∂δL − ∂e ∂δL sin E − e cos E ∂E ∂δL =⇒ ∂E ∂δL = η sin f eL , thereby Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' The relation for ∂f/∂G is achieved precisely in the same manner, so one finds out ∂f ∂G = −sin f ηeL + 1 + e cos f η ∂E ∂G , ∂E ∂G = −sin f eL , that is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Finally, derivatives (40), (42) involving ιc = cos i easily follow again by partial differentiation in (3) with respect to G and H respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' while for those containing ιs = sin i we can rely, for example, to the identity sin2 i + cos2 i = 1: 0 = 2 sin i∂ιs ∂G + 2 cos i∂ιc ∂G and consequently Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (41) provided sin i ̸= 0, as well as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' (43) repeating the same argument with the variable H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' B Example of normalization for a µ2 quadrupolar expansion Consider the following toy model Hamiltonian with kµ = kmp = ν = 2, ν1 = 1, according to conventions introduced in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1: H (0) = Z0 + R(0) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 + R(0) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 + R(0) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 34 where R(0) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 = σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='c cos (2 (E1 − f − g − h)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='c cos (2 (E1 + f + g − h)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0ιc cos (2 (E1 − f − g − h)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0ιc cos (2 (E1 + f + g − h)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='c cos (2 (E1 − h)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='c cos(2(f + g)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 cos (2 (E1 − f − g − h)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 cos (2 (E1 + f + g − h)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 cos (2 (E1 − h)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 cos(2(f + g)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1G4µm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ∥r1∥ − 3a1δL2G2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2L4∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− a1G2µm2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='L2∗ ∥r1∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 35 The first step j = 1 of the method aims precisely at normalizing R(0) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 via (71) solved by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='χ(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='= σ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3G4µa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cφ1n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8n1L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='G4µa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1φ1n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8n1L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cφ1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin (2 (−f − g − h + E1)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin (2 (f + g − h + E1)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ sin (2 (E1 − h)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin (2 (−f − g − h + E1)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin (2 (f + g − h + E1)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin (2 (E1 − h)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin (2 (−f − g − h + E1)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin (2 (f + g − h + E1)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin (2 (−f − g − h + E1)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ sin (2 (f + g − h + E1)) a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ sin(2(f + g))a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗n∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ sin(2(f + g))a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16L6∗n∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='n2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1 − n2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' so that the new truncated Hamiltonian becomes H (1) = Z0 + Z (1) 2 + R(1) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 + R(1) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' with Z (1) 2 = σ2 � −3a2 1G4µm4 0ι2 c 8L6∗ + a2 1G4µm4 0 8L6∗ − 3δL2G2m2 0 2L4∗ − G2µm2 0 L2∗ � 36 and R(1) 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 = σ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos (f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos (3f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos (f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4η3 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos (3f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4η3 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos (f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos (3f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1n∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1n∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ (2n1 − 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3eG4µ cos (f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3eG4µ cos (3f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (2f + 2g − 2h + E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (2f + 2g − 2h + 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos (f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4η3 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos (3f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='4η3 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g − 2h + E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g − 2h + 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ιcn∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3eG4µ cos (f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3eG4µ cos (3f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1n∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (2f + 2g − 2h + E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1n∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (2f + 2g − 2h + 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1n∗m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ (2n1 + 2n∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos(f)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 9eG4µ cos(f + 2g)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos(f + 2g)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 9eG4µ cos(3f + 2g)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3eG4µ cos(3f + 2g)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 9eG4µ cos (f + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (3f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 9eG4µ cos (f − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (3f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (2h − 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (2f + 2g − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 15G4µ cos (2h − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 15G4µ cos (2f + 2g + 2h − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 9G4µ cos (E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (2f + 2g + E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 15G4µ cos (2f + 2g − 2h + E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g − 2h + 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ι2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='cm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (3f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 9eG4µ cos (f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 9eG4µ cos (3f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 15G4µ cos (2f + 2g + 2h − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 15G4µ cos (2f + 2g − 2h + E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (2f + 2g − 2h + 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1ιcm4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3eG4µ cos(f)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos(f + 2g)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3eG4µ cos(f + 2g)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos(3f + 2g)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3eG4µ cos(3f + 2g)a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8η3 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (f + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (3f + 2g + 2h − 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (f − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 9eG4µ cos (3f + 2g − 2h + 2E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2h − 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + 2h − 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 15G4µ cos (2h − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='16 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 15G4µ cos (2f + 2g + 2h − E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− 3G4µ cos (E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g + E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='8 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 15G4µ cos (2f + 2g − 2h + E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G4µ cos (2f + 2g − 2h + 3E1) a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='1e1m4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='32 ∥r1∥ L6∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='− eG2µ cos(f)a1m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='∥r1∥ L2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ G2µ cos (E1) a1e1m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='∥r1∥ L2∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='+ 3G2δL2 cos (E1) a1e1m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='2 ∥r1∥ L4∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Next, we move on with the second and last iteration j = 2 targeted to R(1) 3,3: H (2) = Z0 + Z (1) 2 + Z (2) 3 + R(2) 4,4 , in which χ(2) 3 is omitted for brevity and Z (2) 3 = 0 38 as expected, being R(1) 3,3 solely made up of harmonics containing fast angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' was partially supported by the MIUR-PRIN 20178CJA2B New Frontiers of Celestial Mechanics: Theory and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' References [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Cavallari and C.' metadata={'source': 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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Coffey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' “An analytical theory for tesseral gravitational harmon- ics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' In: Celestial Mechanics and Dynamical Astronomy 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content='3 (2000), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' 139–156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Subiela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' “Teoría del satélite artificial: armónicos teserales y su relegación mediante simplificaciones algebraicas”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E1T4oBgHgl3EQfSwOL/content/2301.03070v1.pdf'} +page_content=' Universidad de Zaragoza, 1992.' metadata={'source': 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0000000000000000000000000000000000000000..38c5353accd0e03b0f4496f92967abf2511d28ee --- /dev/null +++ b/7NAzT4oBgHgl3EQfEvqd/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d2332ad92bd6c6b822fe8476f67f53b4e72f6432497a728050d24b71cd245fa5 +size 5898285 diff --git a/7NAzT4oBgHgl3EQfgPyo/content/tmp_files/2301.01466v1.pdf.txt b/7NAzT4oBgHgl3EQfgPyo/content/tmp_files/2301.01466v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7e0943228beb4b0556d8c4cd8fdb66fe90d0ba71 --- /dev/null +++ b/7NAzT4oBgHgl3EQfgPyo/content/tmp_files/2301.01466v1.pdf.txt @@ -0,0 +1,1361 @@ +arXiv:2301.01466v1 [math.PR] 4 Jan 2023 +A Bayesian Perspective on Feller, Pollard and the +Complete Monotonicity of the Mittag-Leffler Function +Nomvelo Karabo Sibisi +sbsnom005@myuct.ac.za +January 5, 2023 +Abstract +Pollard used contour integration to show that the Mittag-Leffler function is the +Laplace transform of a positive function, thereby proving that it is completely +monotone. He also cited personal communication by Feller of a discovery of the +result by “methods of probability theory”. Feller used the two-dimensional Laplace +transform of a bivariate distribution to derive the result. We prove the result by a +Bayesian approach. We proceed to prove the complete monotonicity of the multi- +parameter Mittag-Leffler function, thereby generalising the Pollard result by meth- +ods of Bayesian probability theory. +Keywords— +Bayesian reasoning; complete monotonicity; stable & gamma distributions; +Mittag-Leffler function; Prabhakar function. +1 +Introduction +The problem of interest in this paper is the study of the complete monotonicity of the Mittag- +Leffler function. +Complete monotonicity is an analytic property of functions. +Accordingly, +Pollard [18] used analytic methods to prove the property in the instance of the Mittag-Leffler +function. Pollard also cited personal communication by Feller of a discovery of the result by +“methods of probability theory”. However, Pollard’s comment notwithstanding, the published +proof by Feller [7] (XIII.8) is also analytic rather than probabilistic (we discuss both approaches +later in this section). This prompted us to ask the following: +1. What might constitute a “method of probability theory” in proving an analytic property of +a function, at least in the context of proving that the Mittag-Leffler function is completely +monotone? +2. What additional or complementary insight, if any, might the method of probability theory +offer relative to an analytic method? + +The approach of this paper is to assign appropriate probability distributions and use the sum +and product rules of probability theory to explore analytic attributes of associated functions. +This is an instance of Bayesian reasoning for analytic purposes, without the Bayesian inference +step associated with data analysis. We do not have cause to invoke Bayes’ rule to generate a +posterior distribution from an assigned prior distribution and a prescribed likelihood. Beyond +reproducing known analytic results due to Pollard and Feller, we discuss the generalisation that +flows from adopting such Bayesian reasoning. We start with definitions of complete monotonicity +and the Mittag-Leffler function. +1.1 +Definitions +An infinitely differentiable function ϕ(x) on x > 0 is completely monotone if its derivatives +ϕ(n)(x) satisfy (−1)nϕ(n)(x) ≥ 0, n ≥ 0. Bernstein’s theorem states that ϕ(x) is completely +monotone iff it may be expressed as +ϕ(x) = +� ∞ +0 +e−xt dF(t) = +� ∞ +0 +e−xtf(t)dt +(1) +for a non-decreasing distribution function F(t) with density f(t), i.e. F(t) = +� t +0 f(u)du. The +first integral in (1) is formally called the Laplace-Stieltjes transform of F and the latter the +(ordinary) Laplace transform of f. For bounded F(t), ϕ(x) is defined on x ≥ 0. Integrating (1) +by parts in this case gives ϕ(x) in terms of the ordinary Laplace transform of F: +ϕ(x) = x +� ∞ +0 +e−xtF(t) dt = +� ∞ +0 +e−tF(t/x) dt +(2) +The Mittag-Leffler function Eα(x) is defined by the infinite series +Eα(x) = +∞ +� +k=0 +xk +Γ(αk + 1) +α ≥ 0 +(3) +For later reference, the Laplace transform of Eα(−λxα) (λ > 0) is +� ∞ +0 +e−sxEα(−λxα) dx = sα−1 +λ + sα +Re(s) ≥ 0 +(4) +We may turn to the problem of proving the complete monotonicity of Eα(−x). We discuss the +approaches due to Pollard and Feller in turn before turning to the Bayesian perspective. +1.2 +Pollard’s Method +In a 1948 paper, Pollard [18] led with the opening remark: +“W. Feller communicated to me his discovery – by the methods of probability theory +– that if 0 ≤ α ≤ 1 the function Eα(−x) is completely monotonic for x ≥ 0. This +means that it can be written in the form +Eα(−x) = +� ∞ +0 +e−xtdPα(t) +where Pα(t) is nondecreasing and bounded. In this note we shall prove this fact +directly and determine the function Pα(t) explicitly.” +[we use Pα where Pollard used Fα, which we reserve for another purpose] +2 + +Having dispensed with E0(−x) = 1/(1 + x) and E1(−x) = e−x since “there is nothing to be +proved in these cases”, Pollard used a contour integral representation of Eα(−x): +Eα(−x) = +1 +2πi +� +C +sα−1es +x + sα ds = +1 +2πiα +� +C′ +ez +1 +α +x + z dz +(5) +to prove that +pα(t) ≡ P ′ +α(t) = 1 +α fα(t−1/α) t−1/α−1 +0 < α < 1 +(6) +where fα(t) is defined by +e−sα = +� ∞ +0 +e−stfα(t) dt +0 < α < 1 +(7) +Pollard [17] had earlier proved that fα(t) > 0, so that pα(t) ≥ 0, thereby completing his proof +that Eα(−x) is completely monotone for 0 ≤ α ≤ 1. Pollard stopped at the point of deriving (6), +the density pα(t) ≡ P ′ +α(t). As per initial task, we proceed to discuss Pα(t) explicitly. We first +recognise fα(t) as the density of the stable distribution Fα on [0, ∞) +Fα(t) = +� t +0 +fα(u) du +0 < α < 1 +(8) +with normalisation Fα(∞) = 1. In turn, the Pollard distribution Pα(t) is +Pα(t) = +� t +0 +pα(u) du = 1 +α +� t +0 +fα(u−1/α) u−1/α−1 du +(9) +Janson [13] derived Pα(t) as a limiting distribution of a P´olya urn scheme. Pα(t) is known as +the Mittag-Leffler distribution in the probabilistic literature (one of two distributions bearing +the same name as discussed later). +Setting y = u−1/α in (9) gives a simple relation between Pα and Fα: +Pα(t) = +� ∞ +t−1/α fα(y) dy = 1 − +� t−1/α +0 +fα(y) dy ≡ 1 − Fα(t−1/α) +(10) +This ‘duality’ between the Mittag-Leffler and stable distributions is key to the discussion that +follows. The Pollard result may accordingly be written in several equivalent forms: +Eα(−x) = +� ∞ +0 +e−xtdPα(t) = +� ∞ +0 +e−tPα(t/x) dt +or +Eα(−xα) = +� ∞ +0 +e−tPα(x−αt) dt = +� ∞ +0 +e−t(1 − Fα(xt−1/α)) dt +(11) +Another representation arising from change of variable in Pollard’s original result is +αEα(−xα) = +� ∞ +0 +e−xαu fα(u−1/α) u−1/α−1 du += x +� ∞ +0 +e−t fα(xt−1/α) t−1/α−1 dt +(12) +Setting aside Pollard’s contour integral proof, it is hard to evaluate directly any of the equivalent +integral representations above to demonstrate that they do indeed generate Eα(−x), Eα(−xα). +A method that may be convenient to prove one representation effectively proves all other rep- +resentations because they are interchangeable ways of stating the Pollard result. In particular, +Feller followed an indirect route to prove the representation (11), discussed next. +3 + +1.3 +Feller’s Method +In an illustration of the use of the two-dimensional Laplace transform, Feller [7](p453) considered +1 − Fα(xt−1/α) as a bivariate distribution over x > 0, t > 0. The Laplace transform over x, +followed by that over t gives +� ∞ +0 +e−sx(1 − Fα(xt−1/α)) dx = 1 +s − e−tsα +s +(13) +1 +s +� ∞ +0 +e−λt � +1 − e−tsα� +dt = 1 +λ +sα−1 +λ + sα +(14) +By reference to (4), the right hand side of (14) is the Laplace transform of Eα(−λxα)/λ. Since +the two-dimensional Laplace transform equivalently can be evaluated first over t then over x, +Feller concluded that +Eα(−λxα) = λ +� ∞ +0 +e−λt(1 − Fα(xt−1/α)) dt +(15) +which, for λ = 1, is the Pollard result in the form (11). Feller’s proof is based on the interchange +of the order of integration (Fubini’s theorem) and the uniqueness of Laplace transforms. We +represent it by the commutative diagram below, where Ls|t denotes the one-dimensional Laplace +transform of a bivariate source function at fixed t, to give a bivariate function of (s, t) where s +is the Laplace variable. +1 − Fα(xt−1/α) +1 +s − e−tsα +s +1 +λEα(−λxα) +1 +λ +sα−1 +λ + sα +Ls|t +easy +Lλ|s +easy +Lλ|x +hard +L −1 +x|λ +easy +(16) +The desired proof is the “hard” direct path, which is equivalent to the “easy” indirect path. +We will return to commutative diagram representation in a different context later in the paper +when we discuss infinite divisibility. +Feller’s concise proof uses “methods of probability theory”, as cited by Pollard, only to the extent +of choosing the bivariate distribution as input to the two-dimensional Laplace transform. Short +of any further insight, the methods by both Pollard and Feller might be described as analytic +rather than probabilistic. We may now turn to an approach that may indeed be described as a +“method of probability theory” in the context of the Pollard problem. +1.4 +Purpose and Scope of Paper +As stated earlier, the approach is this paper is that of Bayesian reasoning, involving strict use of +the sum and product rules of probability theory. The assignment of appropriate distribution in +our context is guided by the task of proving that Eα(−x) is completely monotone. We first cast +Feller’s argument in such terms before proceeding to a more general probabilistic discussion. +The Mittag-Leffler function is of growing interest in probability theory and physics, with a +diversity of applications, notably fractional calculus. A comprehensive study of the properties +4 + +and applications of the Mittag-Leffler function and its numerous generalisations is beyond the +scope of this paper. We consciously restrict the scope to the theme of complete monotonicity +and Mittag-Leffler functions, underpinned by Bayesian reasoning. +Other studies that explicitly discuss complete monotonicity and Mittag-Leffler functions build +upon complex analytic approaches similar to Pollard’s rather than the probabilistic underpin- +ning discussed here. For example, de Oliviera et al. [5] and Mainardi and Garrappa [14] studied +the complete monotonicity of xβ−1Eγ +α,β(−xα), whereas G´orska [10] explored the complete mono- +tonicity of Eγ +α,β(−x). Eγ +α,β(x) is the three-parameter variant of the Mittag-Leffler function, also +known as the Prabhakar function. These papers comment on the fundamental importance of the +complete monotonicity of Mittag-Leffler functions used in the modelling of physical phenomena, +such as anomalous dielectric relaxation and viscoelasticity. +Finally, we are keenly aware that there are other views on the interpretation of “methods of +probability theory”. We comment on this before discussing the Bayesian approach in detail. +1.5 +Probabilistic Perspectives +The phrase ‘methods of probability theory’ used by Pollard may suggest an experiment with +random outcomes as a fundamental metaphor. As noted earlier, Pα is derived as a limiting +distribution of a P´olya urn scheme in the probabilistic literature. +Diversity of approach is commonplace in probability theory and mathematics more generally. +For example, in a context of nonparametric Bayesian analysis, Ferguson [8] constructed the +Dirichlet process based on the gamma distribution as the fundamental probabilistic concept, +without invoking a random experiment. Blackwell and MacQueen [3] observed that the Ferguson +approach “involves a rather deep study of the gamma process” as they proceeded to give an +alternate construction based on the metaphor of a generalised P´olya urn scheme. Adopting the +one approach is not to deny or diminish the other, but to bring attention to the diversity of +thinking in probability theory, even when the end result is the same mathematical object. We +look upon this as healthy complementarity rather than undesirable contestation. +We discuss complete monotonicity by methods of probability theory in the sense of Bayesian +reasoning. +For the purpose at hand, we have no need to invoke an underlying random ex- +periment or indeed an explicit random variable, while not denying the latter as an alternative +probabilistic approach. Hence, for example, we shall continue to express the Laplace transform +of a distribution as an explicit integral rather than as an expectation E +� +e−sX� +for a random +variable X. +2 +A Bayesian Method +First, we note that the scale change s → t1/αs (t > 0) in (7) gives +e−tsα = +� ∞ +0 +e−sxfα(x t−1/α)t−1/α dx ≡ +� ∞ +0 +e−sxfα(x|t) dx +(17) +5 + +where fα(x|t) ≡ fα(x t−1/α)t−1/α is the stable density conditioned on the scale parameter t, with +fα(x) ≡ fα(x|1). Correspondingly, the stable distribution conditioned on t is +Fα(x|t) = +� x +0 +fα(u|t) du = +� xt−1/α +0 +fα(u) du ≡ Fα(xt−1/α) +(18) +with Laplace transform e−tsα/s. +We then assign a distribution G(t) to the scale parameter t of Fα(x|t). Then, by the sum and +product rules of probability theory, the unconditional or marginal distribution Mα(x) over x is +Mα(x) = +� ∞ +0 +Fα(x|t)dG(t) +(19) +with Laplace transform +� ∞ +0 +e−sxMα(x) dx = 1 +s +� ∞ +0 +e−tsα dG(t) +(20) +Mα is also referred to as a mixture distribution, arising from randomising or mixing the parame- +ter t in Fα(x|t) with G(t). This has the same import as saying that we assign a prior distribution +G(t) on t and we shall continue to use the latter language. +G may depend on one or more parameters. A notable example is the gamma distribution G(µ, λ) +with shape and scale parameters µ > 0, λ > 0 respectively: +dG(t|µ, λ) = +λµ +Γ(µ) tµ−1e−λt dt +(21) +λ is not fundamental and may be set to λ = 1 by change of scale t → λt, while µ controls the +shape of G(t|µ, λ). The marginal (19) becomes Mα(x|µ, λ), with Laplace transform +� ∞ +0 +e−sxMα(x|µ, λ) dx = 1 +s +� +λ +λ + sα +�µ += 1 +s +� +1 − +sα +λ + sα +�µ +(22) +We may now state Feller’s approach from a Bayesian perspective. +2.1 +A Bayesian View of Feller’s Approach +The case µ = 1 in (21) gives the exponential distribution dG(t|λ) = λe−λtdt. Then Mα(x|λ) ≡ +Mα(x|µ = 1, λ) is +Mα(x|λ) = +� ∞ +0 +Fα(x|t)dG(t|λ) = λ +� ∞ +0 +Fα(x|t)e−λt dt +(23) +The Laplace transform of Mα(x|λ), read from (22) with µ = 1, is +� ∞ +0 +e−sxMα(x|λ) dx = 1 +s − sα−1 +λ + sα +(24) +=⇒ +Mα(x|λ) = 1 − Eα(−λxα) +(25) +=⇒ Eα(−λxα) = 1 − Mα(x|λ) = λ +� ∞ +0 +(1 − Fα(x|t))e−λt dt +(26) +This reproduces Feller’s result (15) from a Bayesian perspective. +The difference is purely a +matter of conceptual outlook: +6 + +Feller: Study the two-dimensional Laplace transform of the bivariate distribution 1−Fα(xt−1/α), +where Fα is the stable distribution. Deduce that Eα(−λxα)/λ is the Laplace transform +of 1 − Fα(xt−1/α) over t at fixed x, where λ is the Laplace variable. +Bayes: Assign an exponential prior distribution G(t|1, λ) to the scale factor t of Fα(x|t) ≡ +Fα(xt−1/α), where G(t|µ, λ) is the gamma distribution. Marginalise over t to generate the +Feller result directly. +Feller himself might also have established the result by the latter reasoning. Under subordination +of processes, Feller [7](p451) discussed mixture distributions but he did not specifically discuss +the Mittag-Leffler function in this context in his published work. The task fell on Pillai [15] to +study Mα(x|µ) ≡ Mα(x|µ, λ = 1), including its infinite divisibility and the corresponding Mittag- +Leffler stochastic process. He also proved that Mα(x|1) = 1 − Eα(−xα) (as discussed above), +which he referred to as the Mittag-Leffler distribution. There are thus two distributions bearing +the name “Mittag-Leffler distribution”: Mα(x) = 1 − Eα(−xα) and Pα(t) = 1 − Fα(t−1/α). +The natural question arising from the Bayesian approach is whether there might be other choices +of µ in G(µ, λ) (or indeed other choices of G altogether) that yield the Pollard result and, if so, +what insight they might offer. At face value, there would appear to be nothing further to be +said since other choices of µ can be expected to lead to different results, beyond the study of +the Mittag-Leffler function. The main contribution of this paper is that, in fact, there is a limit +relationship that generates the Pollard result for any µ > 0, as discussed next. +We first note, given the definition of the conditional stable density +fα(x|t) ≡ fα(x t−1/α)t−1/α =⇒ fα(1|t) ≡ fα(t−1/α)t−1/α +that we may write Pα(t) of (9) and the representation (12) of the Pollard result as +Pα(t) = +� t +0 +pα(u) du = 1 +α +� t +0 +fα(1|u) u−1 du +(27) +αEα(−λxα) = x +� ∞ +0 +fα(x|t) t−1e−λt dt +0 < α < 1 +(28) +u = x−αt : +Eα(−λxα) = +� ∞ +0 +e−λxαu dPα(u) +(29) +The intent is to generate this representation using the general G(µ, λ) prior distribution, i.e. +without reference to Pollard’s analytic method and without explicit restriction to the G(µ = 1, λ) +case that is equivalent to Feller’s approach, as demonstrated above. +3 +Main Contribution +We first state Theorem 1, which warrants dedicated discussion, even though it is actually a +special case of the more general Theorem 3 stated later. We note first that the density of the +marginal distribution Mα(x|µ, λ) of Section 2 is +mα(x|µ, λ) = +� ∞ +0 +fα(x|t) dG(t|µ, λ) +µ > 0, λ > 0 += +λµ +Γ(µ) +� ∞ +0 +fα(x|t) tµ−1e−λt dt += +µλµ +Γ(µ + 1) +� ∞ +0 +fα(x|t) tµ−1e−λt dt +(30) +7 + +where the latter expression follows from the identity µΓ(µ) = Γ(µ + 1). +Theorem 1. The limit +lim +n→∞ +n +µ x mα(x|µ +n, λ) = lim +n→∞ +n +µ x +� ∞ +0 +fα(x|t) dG(t|µ +n, λ) +(31) +is finite and independent of µ for any µ > 0. This limit yields the following integral representa- +tion of the Mittag-Leffler function Eα(−λxα) +αEα(−λxα) = x +� ∞ +0 +fα(x|t) t−1e−λt dt +(32) +u = x−αt : +Eα(−λxα) = +� ∞ +0 +e−λxαu dPα(u) +(33) +where Pα(t) is the (one-parameter) Pollard distribution +Pα(t) = 1 +α +� t +0 +fα(1|u) u−1 du += 1 +α +� t +0 +fα(u−1/α) u−1/α−1 du +Hence Eα(−x) is completely monotone. +Proof of Theorem 1. The Laplace transform of x mα(x|µ, λ) is +� ∞ +0 +e−sxx mα(x|µ, λ) dx = +� ∞ +0 +e−sxx +� ∞ +0 +fα(x|t) dG(t|µ, λ) dx += − d +ds +� ∞ +0 +� ∞ +0 +e−sxfα(x|t) dx dG(t|µ, λ) += − d +ds +� ∞ +0 +e−tsα dG(t|µ, λ) += αsα−1 +� ∞ +0 +t e−tsα dG(t|µ, λ) += αsα−1 λµ +Γ(µ) +� ∞ +0 +tµ e−(λ+sα)t dt += αsα−1 λµ +Γ(µ) +Γ(µ + 1) +(λ + sα)µ+1 += λµµα +sα−1 +(λ + sα)µ+1 +=⇒ +lim +n→∞ +n +µ +� ∞ +0 +e−sxx mα(x|µ +n, λ) dx = α sα−1 +λ + sα +which is the Laplace transform of αEα(−λxα). With the aid of (30), it also readily follows that +the limit (31) is +lim +n→∞ +n +µ x mα(x|µ +n, λ) = x +� ∞ +0 +fα(x|t) t−1e−λt dt +The integral representations (32) and (33) of Eα(−λxα) follow, hence the conclusion that Eα(−x) +is completely monotone. +Pursuing the Bayesian theme, we turn next to Laplace convolution to demonstrate the complete +monotonicity of the two and three parameter Mittag-Leffler functions. +8 + +4 +A Convolution Representation +Toward a more general discussion, we first present an alternative representation of xfα(x|t) using +Laplace convolution. The convolution {ρ ⋆ f}(x) of ρ(x), f(x) is given by +{ρ ⋆ f}(x) = +� x +0 +ρ(x − u)f(u) du +(34) +The convolution theorem states that the Laplace transform of {ρ⋆f} is a product of the Laplace +transforms of ρ, f. +4.1 +One Parameter Case +Proposition 1. Let ρα(x) = x−α/Γ(1 − α), 0 < α < 1 with Laplace transform sα−1. +Let +{ρα ⋆ fα(·|t)}(x) be the convolution of ρα(x) and fα(x|t) with Laplace transform sα−1e−tsα. +Then +x fα(x|t) = α t{ρα ⋆ fα(·|t)}(x) = α {ρα ⋆ fα}(xt−1/α) +(35) +where {ρα ⋆ fα}(x) is the convolution of ρα(x) and fα(x) ≡ fα(x|1). For compatibility with later +discussion, we also use the name wα(x|t) defined by αwα(x|t) ≡ x fα(x|t). +Proof of Proposition 1. By the convolution theorem, {ρα ⋆ fα(·|t)}(x) has Laplace transform +sα−1e−tsα = − 1 +αt +d +dse−tsα = 1 +αt +� ∞ +0 +e−sxxfα(x|t) dx +=⇒ +α t {ρα ⋆ fα(·|t)}(x) = xfα(x|t) +The convolution {ρα ⋆ fα(·|t)}(x) takes the explicit form: +{ρα ⋆ fα(·|t)}(x) = +� x +0 +ρα(x − u)fα(u|t) du += +� x +0 +ρα(x − u)fα(ut−1/α)t−1/α du +y = ut−1/α : += +� xt−1/α +0 +ρα(x − yt1/α)fα(y) dy += +� xt−1/α +0 +ρα(t1/α(xt−1/α − y))fα(y) dy += t−1 +� xt−1/α +0 +ρα(xt−1/α − y)fα(y) dy += t−1{ρα ⋆ fα}(xt−1/α) +so that αwα(x|t) ≡ x fα(x|t) = α t{ρα ⋆ fα(·|t)}(x) = α {ρα ⋆ fα}(xt−1/α). +Hence the following are equivalent representations of the Pollard distribution Pα(t): +Pα(t) = +� t +0 +wα(1|t) u−1 du ≡ 1 +α +� t +0 +fα(1|u) u−1 du += +� t +0 +{ρα ⋆ fα(·|u)}(1) du += +� t +0 +{ρα ⋆ fα}(u−1/α) u−1 du +(36) +9 + +The motivation for the convolution representation is to facilitate generalisation. Specifically, +the Laplace transform αtsα−1e−tsα of xfα(x|t) is the derivative of −e−tsα. However, a more +general term like tsα−βe−tsα cannot arise from simple derivatives of e−tsα for non-integer β. It +might be interpreted as a fractional derivative, as can be represented instead by convolutions. +Accordingly, we proceed to consider more general convolutions than the convolution form (35) +for xfα(x|t). +4.2 +Two Parameter Case +First, we introduce the two-parameter Mittag-Leffler function +Eα,β(x) = +∞ +� +k=0 +xk +Γ(αk + β) +(37) +The Laplace transform of xβ−1Eα,β(−λxα) is +� ∞ +0 +e−sxxβ−1Eα,β(−λxα) dx = sα−β +λ + sα +(38) +We may now proceed to prove that Eα,β(−x) is completely monotone by showing that it is the +Laplace transform of a two-parameter variant Pα,β(t) of the Pollard distribution. We follow +a corresponding two-parameter variant of the convolution argument presented above for the +one-parameter case. +Proposition 2. Let ρα,β(x) = xβ−α−1/Γ(β − α) β > α, with Laplace transform sα−β. Let +{ρα,β ⋆ fα(·|t)}(x) be the convolution of ρα,β(x) and fα(x|t). Then +wα,β(x|t) ≡ t {ρα,β ⋆ fα(·|t)}(x) = t(β−1)/α {ρα,β ⋆ fα}(xt−1/α) +(39) +(the name wα,β(x|t) is a shorthand adopted for convenience). +Proof of Proposition 2. +{ρα,β ⋆ fα(·|t)}(x) = +� x +0 +ρα,β(x − u)fα(u|t) du += +� xt−1/α +0 +ρα,β(t1/α(xt−1/α − u))fα(u) du += t(β−1)/α−1 +� xt−1/α +0 +ρα,β(xt−1/α − u)fα(u) du += t(β−1)/α−1{ρα,β ⋆ fα}(xt−1/α) +Thus wα,β(x|t) ≡ t {ρα,β ⋆ fα(·|t)}(x) = t(β−1)/α{ρα,β ⋆ fα}(xt−1/α). +Theorem 2. The two-parameter Mittag-Leffler function Eα,β(−λxα) has the integral represen- +tation +Eα,β(−λxα) = +� ∞ +0 +e−λxαt dPα,β(t) +(40) +10 + +where Pα,β(t), which we refer to as the two-parameter Pollard distribution, is +Pα,β(t) = +� t +0 +wα,β(1|u) u−1 du +≡ +� t +0 +{ρα,β ⋆ fα(·|u)}(1) du += +� t +0 +{ρα,β ⋆ fα}(u−1/α) u(β−1)/α−1 du +(41) +Hence Eα,β(−x) is completely monotone. +Proof of Theorem 2. The theorem is a particular case of the more general Theorem 3 below, +hence the current proof is deferred to that of the latter theorem. +4.3 +Three Parameter Case +The three-parameter Mittag-Leffler function, also known as the Prabhakar function, is given by +Eγ +α,β(x) = +1 +Γ(γ) +∞ +� +k=0 +Γ(γ + k) +k! Γ(αk + β) xk +(42) +The Laplace transform of xβ−1Eγ +α,β(−λxα) is +� ∞ +0 +e−sxxβ−1Eγ +α,β(−λxα) dx = +sαγ−β +(λ + sα)γ +(43) +We may now proceed to prove that Eγ +α,β(−x) is completely monotone by showing that it is the +Laplace transform of a three-parameter variant P γ +α,β(t) of the Pollard distribution. In principle, +we need only have discussed the three-parameter case from the outset because the two and one- +parameter instances are the special cases γ = 1 and γ = β = 1 respectively. We chose instead +to present in sequential order for clarity of exposition. +We devote a separate section to the three-parameter case, which subsumes all prior discussion, +by restating Theorem 1 in the three-parameter context. +5 +Main Theorem +We start with a proposition required for the general theorem that follows: +Proposition 3. Let ργ +α,β(x) = xβ−αγ−1/Γ(β − αγ) (0 < α < 1, γ > 0, β > αγ) and let {ργ +α,β ⋆ +fα(·|t)}(x) be the convolution of ργ +α,β(x) and the stable density fα(x|t). Then +wγ +α,β(x|t) ≡ tγ {ργ +α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ +α,β ⋆ fα}(xt−1/α) +(44) +11 + +Proof of Proposition 3. +{ργ +α,β ⋆ fα(·|t)}(x) = +� x +0 +ργ +α,β(x − u)fα(u|t) du += +� xt−1/α +0 +ργ +α,β(t1/α(xt−1/α − u))fα(u) du += t(β−1)/α−γ +� xt−1/α +0 +ργ +α,β(xt−1/α − u)fα(u) du += t(β−1)/α−γ{ργ +α,β ⋆ fα}(xt−1/α) +Thus wγ +α,β(x|t) ≡ tγ {ργ +α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ +α,β ⋆ fα}(xt−1/α). +Theorem 3. Let ργ +α,β(x), wγ +α,β(x|t) (0 < α < 1, γ > 0, β > αγ) be as defined in Proposition 3 and +let G(µ, λ) be the gamma distribution with shape and scale parameters µ > 0, λ > 0 respectively. +Let the distribution Mγ +α,β(x|µ, λ) have density +mγ +α,β(x|µ, λ) = +� ∞ +0 +wγ +α,β(x|t) dG(t|µ, λ) += +λµ +Γ(µ) +� ∞ +0 +wγ +α,β(x|t) tµ−1e−λt dt +(45) +≡ +λµ +Γ(µ) +� ∞ +0 +{ργ +α,β ⋆ fα(·|t)}(x) tγ+µ−1e−λt dt +(46) += +λµ +Γ(µ) +� ∞ +0 +{ργ +α,β ⋆ fα}(xt−1/α) t(β−1)/α+µ−1e−λt dt +(47) +where the latter two forms follow from Proposition 3. Then the following limit is finite and +independent of µ for any µ > 0 +lim +n→∞ +n +µ mγ +α,β(x|µ +n, λ) +(48) +This limit yields the following integral representation of the three-parameter Mittag-Leffler or +Prabhakar function Eγ +α,β(−λxα) +Eγ +α,β(−λxα) = +� ∞ +0 +wγ +α,β(x|t) t−1e−λt dt = +� ∞ +0 +e−λxαt dP γ +α,β(t) +(49) +where P γ +α,β(t), which we refer to as the three-parameter Pollard distribution, is +P γ +α,β(t) = +� t +0 +wγ +α,β(1|u) u−1 du +≡ +1 +Γ(γ) +� t +0 +{ργ +α,β ⋆ fα(·|u)}(1) uγ−1 du += +1 +Γ(γ) +� t +0 +{ργ +α,β ⋆ fα}(u−1/α) u(β−1)/α−1 du +(50) +Hence Eγ +α,β(−x) is completely monotone. +12 + +Proof of Theorem 3. The Laplace transform �mγ +α,β(s|µ, λ) of (45) is +�mγ +α,β(s|µ, λ) ≡ +� ∞ +0 +e−sx mγ +α,β(x|µ, λ) dx += sαγ−β λµ +Γ(µ) +� ∞ +0 +tγ+µ−1e−(λ+sα)t dt += λµ Γ(γ + µ) +Γ(µ) +sαγ−β +(λ + sα)γ+µ +(51) +=⇒ +lim +n→∞ +n +µ +� ∞ +0 +e−sxmγ +α,β(x|µ +n, λ) dx = Γ(γ) +sαγ−β +(λ + sα)γ +(52) +By (43), the right hand side is the Laplace transform of Γ(γ) xβ−1Eγ +α,β(−λxα). Given (46) and +(47), it also readily follows that the limit (48) is +� ∞ +0 +tγ{ργ +α,β ⋆ fα(·|t)}(x)t−1e−λtdt = +� ∞ +0 +{ργ +α,β ⋆ fα}(xt−1/α) t(β−1)/α−1e−λtdt +=⇒ +Eγ +α,β(−λxα) = x1−β +Γ(γ) +� ∞ +0 +{ργ +α,β ⋆ fα}(xt−1/α) t(β−1)/α−1e−λt dt +u = x−αt : = +1 +Γ(γ) +� ∞ +0 +e−λxαu {ργ +α,β ⋆ fα}(u−1/α) u(β−1)/α−1 du += +� ∞ +0 +e−λxαu dP γ +α,β(u) +Hence Eγ +α,β(−x) is completely monotone. +Theorem 3 may be visually represented by the following commutative diagram, where mγ +α,β(x|µ, λ) +and its Laplace transform �mγ +α,β(s|µ, λ) are given by (45) and (51) respectively. The equivalence +of the two routes from the top left node to the bottom left node induces the integral represen- +tation of the Mittag-Leffler function. +mγ +α,β(x|µ, λ) +�mγ +α,β(s|µ, λ) +Γ(γ)xβ−1Eγ +α,β(−λxα) +Γ(γ) +sαγ−β +(λ + sα)γ +L +lim +n→∞ +n +µ �mγ +α,β(s|µ +n, λ) +lim +n→∞ +n +µ mγ +α,β(x|µ +n, λ) +L −1 +(53) +The representation (49) of Eγ +α,β(x), with P γ +α,β(t) given by (50), is equivalent to equation (2.4) +in G´orska et al. [10]. The difference is one of approach. +This paper offers a fundamentally +probabilistic argument, while G´orska et al. [10] follows a complex analytic route inspired by +Pollard [18]. The balance of G´orska et al. [10] is devoted to finding an explicit formula for a +function f γ +α,β(x) featuring in the paper in terms of the Meijer G function and associated confluent +Wright function. In turns out that f γ +α,β(x) in G´orska et al. [10] is identical to {ργ +α,β ⋆ fα}(x) in +13 + +this paper. We are content to leave it in the conceptually simple convolution form: +{ργ +α,β ⋆ fα}(x) = +� x +0 +ργ +α,β(x − u)fα(u) du += +1 +Γ(β − αγ) +� x +0 +(x − u)β−αγ−1fα(u) du +(54) +rather than express it in terms of special functions. In our context, we have actually worked +with the conditional density +wγ +α,β(x|t) ≡ tγ {ργ +α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ +α,β ⋆ fα}(xt−1/α) +where we assigned a gamma prior distribution to the scale parameter t. The density wγ +α,β(x|t) +reduces to (54) for the particular choice t = 1. +We have completed the task of proving that the three-parameter Mittag-Leffler function Eγ +α,β(−x) +is completely monotone by methods of probability theory, using Bayesian reasoning to derive an +explicit form for P γ +α,β(t), whose Laplace transform is Eγ +α,β(−x). Beyond that, we draw conclu- +sions on the complete monotonicity of related functions, notably xβ−1Eγ +α,β(−xα) and Eγ +α,β(−xα) +in isolation. First, we discuss xβ−1Eγ +α,β(−xα), the bottom left node of the commutative dia- +gram (53), in the Bayesian context of Theorem 3. The discussion involves an alternative repre- +sentation of the fundamental probabilistic object – the convolution density {ργ +α,β ⋆ fα(·|t)}(x). +6 +An Alternative Representation +For xβ−1Eγ +α,β(−λxα) to be completely monotone, there must exist a distribution Rγ +α,β(u|λ) +defined by the Laplace transform +xβ−1Eγ +α,β(−λxα) = +� ∞ +0 +e−xu dRγ +α,β(u|λ) +(55) +In turn, the Laplace transform of (55) is the Stieltjes transform (or iterated Laplace transform) +of Rγ +α,β(u|λ): +sαγ−β +(λ + sα)γ = +� ∞ +0 +1 +s + u dRγ +α,β(u|λ) +(56) +Then, as de Oliviera et al. [5], Mainardi and Garrappa [14] show, the Stieltjes inversion formula +(Titchmarsh [22](11.8, p318), Widder [23](VIII.7, p342)) gives +dRγ +α,β(u|λ) = 1 +π Im +� +(e−iπu)αγ−β +(λ + (e−iπu)α)γ +� +du +(57) +The expression in braces on the RHS of (57) is (56) at s = e−iπu. In particular, for γ = β = 1, +(57) reduces to +dRα(u|λ) = 1 +π +λ uα−1 sin πα +λ2 + 2λ uα cos πα + u2α du +(58) +which has been discussed in various contexts in the fractional calculus and probabilistic literature +(e.g. James [12] in the latter context). +14 + +We have mentioned (55) for completeness but it was not the core of our probabilistic discussion, +whose focus was to determine P γ +α,β(t), with Laplace transform Eγ +α,β(−x). That said, we can +offer a ‘hybrid’ derivation of (55) that combines the core of the probabilistic argument in the +form of the convolution density {ργ +α,β ⋆ fα(·|t)}(x) with the complex analytic Stieltjes inversion +argument presented above. +Assume {ργ +α,β ⋆ fα(·|t)}(x) to be the Laplace transform of a distribution Sγ +α,β(u|t): +{ργ +α,β ⋆ fα(·|t)}(x) = +� ∞ +0 +e−xu dSγ +α,β(u|t) +(59) +In turn, the Laplace transform of (59) is the Stieltjes transform of Sγ +α,β(u|t): +sαγ−βe−tsα = +� ∞ +0 +1 +s + u dSγ +α,β(u|t) +(60) +By the Stieltjes inversion formula: +dSγ +α,β(u|t) = 1 +π Im +� +(ue−iπ)αγ−βe−t(ue−iπ)α� +du +(61) +Hence, using the representation (59) in the proof of Theorem 3: +Γ(γ) xβ−1Eγ +α,β(−λxα) = +� ∞ +0 +tγ{ργ +α,β ⋆ fα(·|t)}(x) t−1e−λt dt += +� ∞ +0 +dt tγ−1e−λt +� ∞ +0 +e−xu dSγ +α,β(u|t) += 1 +π Im +� ∞ +0 +du e−xu(ue−iπ)αγ−β +� ∞ +0 +tγ−1e−(λ+(ue−iπ)α)t dt += Γ(γ) +π +Im +� ∞ +0 +e−xu +(e−iπu)αγ−β +(λ + (e−iπu)α)γ du += Γ(γ) +� ∞ +0 +e−xu dRγ +α,β(u|λ) +(62) +thereby reproducing (55). +The Stieltjes transform and its complex analytic inverse are not unfamiliar in probability theory. +In his study of a family of distributions known as generalised gamma convolutions, Bondesson [4] +used the concept under the guise of Pick functions (also known as Nevanlinna functions). +We turn next to the complete monotonicity of Eγ +α,β(−λxα). +7 +A Further Consequence +There is a well-known property of completely monotone functions (e.g. Schilling et al. [20]) that +we state without proof in Proposition 4. We start with a definition: +Definition 1. A Bernstein function is a nonnegative function η(x), x ≥ 0 with a completely +monotone derivative, i.e. η(x) ≥ 0 and (−1)k−1η(k)(x) ≥ 0, k ≥ 1. For example, η(x|λ) = λxα +(0 ≤ α ≤ 1, λ > 0) is a Bernstein function. +15 + +Proposition 4. If ϕ(x) is completely monotone and η is a Bernstein function, ϕ(η) is completely +monotone. +Theorem 4. Given a Bernstein function η, the Mittag-Leffler function Eγ +α,β(−η) is completely +monotone. For example, Eγ +α,β(−λxα) is completely monotone. +Proof of Theorem 4. We have already shown that Eγ +α,β(−x) is completely monotone. Hence, +by Proposition 4, Eγ +α,β(−η) is completely monotone for a Bernstein function η. Specifically, +η(x|λ) = λxα (0 ≤ α ≤ 1, λ > 0) is a Bernstein function, hence Eγ +α,β(−λxα) is completely +monotone. +The complete monotonicity of Eγ +α,β(−λxα) implies that there exists a distribution Qγ +α,β(t|λ) +whose Laplace transform is Eγ +α,β(−λxα): +Eγ +α,β(−λxα) = +� ∞ +0 +e−xt dQγ +α,β(t|λ) +(63) +Qγ +α,β(t|λ) is to Eγ +α,β(−λxα) what P γ +α,β(t) is to Eγ +α,β(−x). However, determining Qγ +α,β(t|λ) appears +to be a challenging problem, whether the approach is analytic or probabilistic. +Clearly, (63) and (57) are identical for β = 1, i.e. Qγ +α,1(t|λ) ≡ Rγ +α,1(t|λ). But, to our awareness, +determining Qγ +α,β(t|λ) for β ̸= 1 is an open problem. We shall not pursue it further here. Our +primary purpose in this section was to bring attention to Theorem 4 and hence the existence of +a distribution Qγ +α,β(t|λ) defined by (63). +8 +A Different Generalisation +As mentioned in Section 1.5, the Pollard distribution Pα is known as the Mittag-Leffler distri- +bution in probabilistic literature. For completeness, we briefly discuss a different generalisation +of Pα that features extensively in such literature. It is known as the generalised Mittag-Leffler +distribution Pα,θ (Pitman [16], p70 (3.27)), also denoted by ML(α, θ) (Goldschmidt and Haas [9], +Ho et al. [11]). +Despite its name, Pα,θ(t) is different from the two-parameter Pollard distribution Pα,β(t) dis- +cussed above, whose Laplace transform is the Mittag-Leffler function Eα,β(−x). Janson [13] +showed that Pα,θ may be constructed as a limiting distribution of a P´olya urn scheme. +It +is also intimately linked to a concept known as ‘polynomial tilting’. +For some parameter θ, +fα,θ(x) ∝ x−θfα(x) is said to be a polynomially tilted variant of fα(x) (e.g. Arbel et al. [1], De- +vroye [6], James [12]). Here, we consider the polynomially tilted density fα,θ(x|t) ∝ x−θfα(x|t) +conditioned on a scale factor t > 0. Normalisation gives +fα,θ(x|t) = +Γ(θ + 1) +Γ(θ/α + 1)tθ/α x−θfα(x|t) +(64) +so that fα,θ(x|t) is defined for θ/α + 1 > 0, or θ > −α. We then consider a two-parameter +16 + +function hα,θ(x|λ) defined by: +α hα,θ(x|λ) = x +� ∞ +0 +fα,θ(x|t) t−1e−λt dt +(65) += +Γ(θ + 1) +Γ(θ/α + 1) x1−θ +� ∞ +0 +fα(x|t) tθ/α−1 e−λt dt +u = x−αt : +hα,θ(x|λ) = +� ∞ +0 +e−λxαu dPα,θ(u) +(66) +where +Pα,θ(t) = +Γ(θ + 1) +Γ(θ/α + 1) +1 +α +� t +0 +fα(u−1/α) u(θ−1)/α−1 du +(67) +or +dPα,θ(t) = +Γ(θ + 1) +Γ(θ/α + 1) tθ/α dPα(t) +(68) +It is clear from (66) that hα,θ(x|λ) may be written as hα,θ(λxα). It follows that: +1. hα,θ(x) is completely monotone +2. θ = 0: Pα,0(t) = Pα(t) =⇒ hα,0(x) = Eα(−x), as directly apparent from comparing (32) +and (65). +3. hα,θ(η) is completely monotone where η is a Bernstein function as discussed in Section 7. +In particular, hα,θ(λxα) is completely monotone and thus expressible as the Laplace trans- +form of a corresponding distribution Qα,θ(t|λ) (distinct from Qα,β(t|λ) discussed in Sec- +tion 7). +We are not aware of a representation of hα,θ other than that generated by Pα,θ in (66). By +comparison, the two-parameter Mittag-Leffler function Eα,β has a well-established infinite se- +ries representation (37), in addition to the representation (40) generated by the two-parameter +Pollard distribution Pα,β. +9 +Discussion +The integral representation (49) of Eγ +α,β(−λxα) in Theorem 3, arising from the limit (48), con- +tains the L´evy measure t−1e−λtdt of the infinitely divisible gamma distribution. There is indeed +an intimate relationship between completely monotone functions and the theory of infinitely di- +visible distributions on the nonnegative half-line R+ = [0, ∞) (Feller [7] (XIII.4, XIII.7), Steutel +and van Harn [21] (III)). Sato [19] considers infinitely divisible distributions on Rd, but the de- +liberate restriction to R+ makes for simpler discussion and relates directly to the core concept of +complete monotonicity that is of interest here. There is also an intimate link to the generalised +gamma convolutions studied by Bondesson [4]. +The limit (48) of Theorem 3 is an instance of a limit rule to generate the L´evy measure of +an infinitely divisible distribution given in Steutel and van Harn [21] (III(4.7)) and Sato [19] +(Corollary 8.9 restricted to R+ rather than Rd). Barndorff-Nielsen and Hubalek [2] also cite +Sato’s Corollary. +Further exploration using the probabilistic machinery of this paper possibly includes the ex- +plicit determination of the three-parameter distribution Qγ +α,β(t|λ), whose Laplace transform is +Eγ +α,β(−λxα), as per (63). +17 + +10 +Conclusion +We have presented a probabilistic derivation of the complete monotonicity of the three-parameter +Mittag-Leffler function (also known as the Prabhakar function) by expressing it as the Laplace +transform of a distribution that we referred to as the three-parameter Pollard distribution. This +is a generalisation of a result due to Pollard for the one-parameter case. +References +[1] Julyan Arbel, Pierpaolo De Blasi, and Igor Pr¨unster. Stochastic Approximations to the +Pitman–Yor Process. Bayesian Analysis, 14(4):1201 – 1219, 2019. +[2] Ole E. Barndorff-Nielsen and Friedrich Hubalek. Probability measures, L´evy measures and +analyticity in time. Bernoulli, 14(3):764 – 790, 2008. +[3] David Blackwell and James B. MacQueen. Ferguson distributions via P´olya urn schemes. +The Annals of Statistics, 1(2):353–355, 1973. +[4] Lennart Bondesson. Generalized Gamma Convolutions and Related Classes of Distributions +and Densities. Lecture Notes in Statistics, 76. Springer-Verlag, New York, 1992. +[5] E. Capelas de Oliveira, F. Mainardi, and J. Vaz. Models based on Mittag-Leffler functions +for anomalous relaxation in dielectrics. +The European Physical Journal Special Topics, +193(1):161–171, Mar 2011. +[6] Luc Devroye. Random variate generation for exponentially and polynomially tilted stable +distributions. ACM Trans. Model. Comput. Simul., 19(4), nov 2009. +[7] William Feller. An Introduction to Probability Theory and its Applications, Vol. II. Wiley, +New York, 1971. +[8] Thomas S. Ferguson. A Bayesian analysis of some nonparametric problems. The Annals of +Statistics, 1(2):209–230, 1973. +[9] Christina Goldschmidt and B´en´edicte Haas. +A line-breaking construction of the stable +trees. Electronic Journal of Probability, 20:1 – 24, 2015. +[10] K. G´orska, Andrzej Horzela, Ambra Lattanzi, and Tibor Pog´any. On complete monotonicity +of three parameter Mittag-Leffler function. Applicable Analysis and Discrete Mathematics, +15:118–128, 04 2021. +[11] Man-Wai Ho, Lancelot F. James, and John W. Lau. Gibbs partitions, Riemann–Liouville +fractional operators, Mittag–Leffler functions, and fragmentations derived from stable sub- +ordinators. Journal of Applied Probability, 58(2):314–334, 2021. +[12] Lancelot F. James. Lamperti-type laws. Ann. Appl. Probab., 20(4):1303–1340, 2010. +[13] Svante Janson. Limit theorems for triangular urn schemes. Probability Theory and Related +Fields, 134(3):417–452, Mar 2006. +[14] Francesco Mainardi and Roberto Garrappa. On complete monotonicity of the Prabhakar +function and non-Debye relaxation in dielectrics. Journal of Computational Physics, 293:70– +80, 2015. Fractional PDEs. +18 + +[15] R. N. Pillai. On Mittag-Leffler functions and related distributions. Annals of the Institute +of Statistical Mathematics, 42(1):157–161, Mar 1990. +[16] J. Pitman. Combinatorial Stochastic Processes, volume 1875 of Lecture Notes in Mathemat- +ics. Springer-Verlag, Berlin, 2006. Lectures from the 32nd Summer School on Probability +Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard. +[17] Harry Pollard. The representation of e−xλ as a Laplace integral. Bulletin of the American +Mathematical Society, 52(10):908 – 910, 1946. +[18] Harry Pollard. The completely monotonic character of the Mittag-Leffler function Ea (−x). +Bulletin of the American Mathematical Society, 54(12):1115 – 1116, 1948. +[19] K. Sato. L´evy Processes and Infinitely Divisible Distributions. Cambridge University Press, +Cambridge, 1999. +[20] Ren´e L. Schilling, Renming Song, and Zoran Vondracek. Bernstein Functions: Theory and +Applications. De Gruyter, Berlin, Boston, 2012. +[21] F.W. Steutel and K. van Harn. Infinite Divisibility of Probability Distributions on the Real +Line. Marcel Dekker, New York, 2003. +[22] E.C. Titchmarsh. Introduction to the Theory of Fourier Integrals. Clarendon Press, 1948. +[23] David Vernon Widder. Laplace transform (PMS-6). Princeton university press, 2015. +19 + diff --git a/7NAzT4oBgHgl3EQfgPyo/content/tmp_files/load_file.txt b/7NAzT4oBgHgl3EQfgPyo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca8711c42c28ca1c2b4857c05b395bb27488bf80 --- /dev/null +++ b/7NAzT4oBgHgl3EQfgPyo/content/tmp_files/load_file.txt @@ -0,0 +1,434 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf,len=433 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='01466v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='PR] 4 Jan 2023 A Bayesian Perspective on Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function Nomvelo Karabo Sibisi sbsnom005@myuct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='za January 5, 2023 Abstract Pollard used contour integration to show that the Mittag-Leffler function is the Laplace transform of a positive function, thereby proving that it is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' He also cited personal communication by Feller of a discovery of the result by “methods of probability theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Feller used the two-dimensional Laplace transform of a bivariate distribution to derive the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We prove the result by a Bayesian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We proceed to prove the complete monotonicity of the multi- parameter Mittag-Leffler function, thereby generalising the Pollard result by meth- ods of Bayesian probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Keywords— Bayesian reasoning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' complete monotonicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' stable & gamma distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Mittag-Leffler function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Prabhakar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 1 Introduction The problem of interest in this paper is the study of the complete monotonicity of the Mittag- Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Complete monotonicity is an analytic property of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Accordingly, Pollard [18] used analytic methods to prove the property in the instance of the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Pollard also cited personal communication by Feller of a discovery of the result by “methods of probability theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' However, Pollard’s comment notwithstanding, the published proof by Feller [7] (XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='8) is also analytic rather than probabilistic (we discuss both approaches later in this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' This prompted us to ask the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' What might constitute a “method of probability theory” in proving an analytic property of a function, at least in the context of proving that the Mittag-Leffler function is completely monotone?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' What additional or complementary insight, if any, might the method of probability theory offer relative to an analytic method?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The approach of this paper is to assign appropriate probability distributions and use the sum and product rules of probability theory to explore analytic attributes of associated functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' This is an instance of Bayesian reasoning for analytic purposes, without the Bayesian inference step associated with data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We do not have cause to invoke Bayes’ rule to generate a posterior distribution from an assigned prior distribution and a prescribed likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Beyond reproducing known analytic results due to Pollard and Feller, we discuss the generalisation that flows from adopting such Bayesian reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We start with definitions of complete monotonicity and the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='1 Definitions An infinitely differentiable function ϕ(x) on x > 0 is completely monotone if its derivatives ϕ(n)(x) satisfy (−1)nϕ(n)(x) ≥ 0, n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Bernstein’s theorem states that ϕ(x) is completely monotone iff it may be expressed as ϕ(x) = � ∞ 0 e−xt dF(t) = � ∞ 0 e−xtf(t)dt (1) for a non-decreasing distribution function F(t) with density f(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' F(t) = � t 0 f(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The first integral in (1) is formally called the Laplace-Stieltjes transform of F and the latter the (ordinary) Laplace transform of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For bounded F(t), ϕ(x) is defined on x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Integrating (1) by parts in this case gives ϕ(x) in terms of the ordinary Laplace transform of F: ϕ(x) = x � ∞ 0 e−xtF(t) dt = � ∞ 0 e−tF(t/x) dt (2) The Mittag-Leffler function Eα(x) is defined by the infinite series Eα(x) = ∞ � k=0 xk Γ(αk + 1) α ≥ 0 (3) For later reference, the Laplace transform of Eα(−λxα) (λ > 0) is � ∞ 0 e−sxEα(−λxα) dx = sα−1 λ + sα Re(s) ≥ 0 (4) We may turn to the problem of proving the complete monotonicity of Eα(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We discuss the approaches due to Pollard and Feller in turn before turning to the Bayesian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='2 Pollard’s Method In a 1948 paper, Pollard [18] led with the opening remark: “W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Feller communicated to me his discovery – by the methods of probability theory – that if 0 ≤ α ≤ 1 the function Eα(−x) is completely monotonic for x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' This means that it can be written in the form Eα(−x) = � ∞ 0 e−xtdPα(t) where Pα(t) is nondecreasing and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In this note we shall prove this fact directly and determine the function Pα(t) explicitly.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [we use Pα where Pollard used Fα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' which we reserve for another purpose] 2 Having dispensed with E0(−x) = 1/(1 + x) and E1(−x) = e−x since “there is nothing to be proved in these cases”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Pollard used a contour integral representation of Eα(−x): Eα(−x) = 1 2πi � C sα−1es x + sα ds = 1 2πiα � C′ ez 1 α x + z dz (5) to prove that pα(t) ≡ P ′ α(t) = 1 α fα(t−1/α) t−1/α−1 0 < α < 1 (6) where fα(t) is defined by e−sα = � ∞ 0 e−stfα(t) dt 0 < α < 1 (7) Pollard [17] had earlier proved that fα(t) > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' so that pα(t) ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' thereby completing his proof that Eα(−x) is completely monotone for 0 ≤ α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Pollard stopped at the point of deriving (6), the density pα(t) ≡ P ′ α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' As per initial task, we proceed to discuss Pα(t) explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We first recognise fα(t) as the density of the stable distribution Fα on [0, ∞) Fα(t) = � t 0 fα(u) du 0 < α < 1 (8) with normalisation Fα(∞) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In turn, the Pollard distribution Pα(t) is Pα(t) = � t 0 pα(u) du = 1 α � t 0 fα(u−1/α) u−1/α−1 du (9) Janson [13] derived Pα(t) as a limiting distribution of a P´olya urn scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Pα(t) is known as the Mittag-Leffler distribution in the probabilistic literature (one of two distributions bearing the same name as discussed later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Setting y = u−1/α in (9) gives a simple relation between Pα and Fα: Pα(t) = � ∞ t−1/α fα(y) dy = 1 − � t−1/α 0 fα(y) dy ≡ 1 − Fα(t−1/α) (10) This ‘duality’ between the Mittag-Leffler and stable distributions is key to the discussion that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The Pollard result may accordingly be written in several equivalent forms: Eα(−x) = � ∞ 0 e−xtdPα(t) = � ∞ 0 e−tPα(t/x) dt or Eα(−xα) = � ∞ 0 e−tPα(x−αt) dt = � ∞ 0 e−t(1 − Fα(xt−1/α)) dt (11) Another representation arising from change of variable in Pollard’s original result is αEα(−xα) = � ∞ 0 e−xαu fα(u−1/α) u−1/α−1 du = x � ∞ 0 e−t fα(xt−1/α) t−1/α−1 dt (12) Setting aside Pollard’s contour integral proof,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' it is hard to evaluate directly any of the equivalent integral representations above to demonstrate that they do indeed generate Eα(−x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Eα(−xα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' A method that may be convenient to prove one representation effectively proves all other rep- resentations because they are interchangeable ways of stating the Pollard result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In particular, Feller followed an indirect route to prove the representation (11), discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='3 Feller’s Method In an illustration of the use of the two-dimensional Laplace transform, Feller [7](p453) considered 1 − Fα(xt−1/α) as a bivariate distribution over x > 0, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The Laplace transform over x, followed by that over t gives � ∞ 0 e−sx(1 − Fα(xt−1/α)) dx = 1 s − e−tsα s (13) 1 s � ∞ 0 e−λt � 1 − e−tsα� dt = 1 λ sα−1 λ + sα (14) By reference to (4), the right hand side of (14) is the Laplace transform of Eα(−λxα)/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Since the two-dimensional Laplace transform equivalently can be evaluated first over t then over x, Feller concluded that Eα(−λxα) = λ � ∞ 0 e−λt(1 − Fα(xt−1/α)) dt (15) which, for λ = 1, is the Pollard result in the form (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Feller’s proof is based on the interchange of the order of integration (Fubini’s theorem) and the uniqueness of Laplace transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We represent it by the commutative diagram below, where Ls|t denotes the one-dimensional Laplace transform of a bivariate source function at fixed t, to give a bivariate function of (s, t) where s is the Laplace variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 1 − Fα(xt−1/α) 1 s − e−tsα s 1 λEα(−λxα) 1 λ sα−1 λ + sα Ls|t easy Lλ|s easy Lλ|x hard L −1 x|λ easy (16) The desired proof is the “hard” direct path, which is equivalent to the “easy” indirect path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We will return to commutative diagram representation in a different context later in the paper when we discuss infinite divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Feller’s concise proof uses “methods of probability theory”, as cited by Pollard, only to the extent of choosing the bivariate distribution as input to the two-dimensional Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Short of any further insight, the methods by both Pollard and Feller might be described as analytic rather than probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We may now turn to an approach that may indeed be described as a “method of probability theory” in the context of the Pollard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='4 Purpose and Scope of Paper As stated earlier, the approach is this paper is that of Bayesian reasoning, involving strict use of the sum and product rules of probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The assignment of appropriate distribution in our context is guided by the task of proving that Eα(−x) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We first cast Feller’s argument in such terms before proceeding to a more general probabilistic discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The Mittag-Leffler function is of growing interest in probability theory and physics, with a diversity of applications, notably fractional calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' A comprehensive study of the properties 4 and applications of the Mittag-Leffler function and its numerous generalisations is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We consciously restrict the scope to the theme of complete monotonicity and Mittag-Leffler functions, underpinned by Bayesian reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Other studies that explicitly discuss complete monotonicity and Mittag-Leffler functions build upon complex analytic approaches similar to Pollard’s rather than the probabilistic underpin- ning discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For example, de Oliviera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [5] and Mainardi and Garrappa [14] studied the complete monotonicity of xβ−1Eγ α,β(−xα), whereas G´orska [10] explored the complete mono- tonicity of Eγ α,β(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Eγ α,β(x) is the three-parameter variant of the Mittag-Leffler function, also known as the Prabhakar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' These papers comment on the fundamental importance of the complete monotonicity of Mittag-Leffler functions used in the modelling of physical phenomena, such as anomalous dielectric relaxation and viscoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Finally, we are keenly aware that there are other views on the interpretation of “methods of probability theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We comment on this before discussing the Bayesian approach in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='5 Probabilistic Perspectives The phrase ‘methods of probability theory’ used by Pollard may suggest an experiment with random outcomes as a fundamental metaphor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' As noted earlier, Pα is derived as a limiting distribution of a P´olya urn scheme in the probabilistic literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Diversity of approach is commonplace in probability theory and mathematics more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For example, in a context of nonparametric Bayesian analysis, Ferguson [8] constructed the Dirichlet process based on the gamma distribution as the fundamental probabilistic concept, without invoking a random experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Blackwell and MacQueen [3] observed that the Ferguson approach “involves a rather deep study of the gamma process” as they proceeded to give an alternate construction based on the metaphor of a generalised P´olya urn scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Adopting the one approach is not to deny or diminish the other, but to bring attention to the diversity of thinking in probability theory, even when the end result is the same mathematical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We look upon this as healthy complementarity rather than undesirable contestation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We discuss complete monotonicity by methods of probability theory in the sense of Bayesian reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For the purpose at hand, we have no need to invoke an underlying random ex- periment or indeed an explicit random variable, while not denying the latter as an alternative probabilistic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Hence, for example, we shall continue to express the Laplace transform of a distribution as an explicit integral rather than as an expectation E � e−sX� for a random variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 2 A Bayesian Method First, we note that the scale change s → t1/αs (t > 0) in (7) gives e−tsα = � ∞ 0 e−sxfα(x t−1/α)t−1/α dx ≡ � ∞ 0 e−sxfα(x|t) dx (17) 5 where fα(x|t) ≡ fα(x t−1/α)t−1/α is the stable density conditioned on the scale parameter t, with fα(x) ≡ fα(x|1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Correspondingly, the stable distribution conditioned on t is Fα(x|t) = � x 0 fα(u|t) du = � xt−1/α 0 fα(u) du ≡ Fα(xt−1/α) (18) with Laplace transform e−tsα/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We then assign a distribution G(t) to the scale parameter t of Fα(x|t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Then, by the sum and product rules of probability theory, the unconditional or marginal distribution Mα(x) over x is Mα(x) = � ∞ 0 Fα(x|t)dG(t) (19) with Laplace transform � ∞ 0 e−sxMα(x) dx = 1 s � ∞ 0 e−tsα dG(t) (20) Mα is also referred to as a mixture distribution, arising from randomising or mixing the parame- ter t in Fα(x|t) with G(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' This has the same import as saying that we assign a prior distribution G(t) on t and we shall continue to use the latter language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' G may depend on one or more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' A notable example is the gamma distribution G(µ, λ) with shape and scale parameters µ > 0, λ > 0 respectively: dG(t|µ, λ) = λµ Γ(µ) tµ−1e−λt dt (21) λ is not fundamental and may be set to λ = 1 by change of scale t → λt, while µ controls the shape of G(t|µ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The marginal (19) becomes Mα(x|µ, λ), with Laplace transform � ∞ 0 e−sxMα(x|µ, λ) dx = 1 s � λ λ + sα �µ = 1 s � 1 − sα λ + sα �µ (22) We may now state Feller’s approach from a Bayesian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='1 A Bayesian View of Feller’s Approach The case µ = 1 in (21) gives the exponential distribution dG(t|λ) = λe−λtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Then Mα(x|λ) ≡ Mα(x|µ = 1, λ) is Mα(x|λ) = � ∞ 0 Fα(x|t)dG(t|λ) = λ � ∞ 0 Fα(x|t)e−λt dt (23) The Laplace transform of Mα(x|λ), read from (22) with µ = 1, is � ∞ 0 e−sxMα(x|λ) dx = 1 s − sα−1 λ + sα (24) =⇒ Mα(x|λ) = 1 − Eα(−λxα) (25) =⇒ Eα(−λxα) = 1 − Mα(x|λ) = λ � ∞ 0 (1 − Fα(x|t))e−λt dt (26) This reproduces Feller’s result (15) from a Bayesian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The difference is purely a matter of conceptual outlook: 6 Feller: Study the two-dimensional Laplace transform of the bivariate distribution 1−Fα(xt−1/α), where Fα is the stable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Deduce that Eα(−λxα)/λ is the Laplace transform of 1 − Fα(xt−1/α) over t at fixed x, where λ is the Laplace variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Bayes: Assign an exponential prior distribution G(t|1, λ) to the scale factor t of Fα(x|t) ≡ Fα(xt−1/α), where G(t|µ, λ) is the gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Marginalise over t to generate the Feller result directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Feller himself might also have established the result by the latter reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Under subordination of processes, Feller [7](p451) discussed mixture distributions but he did not specifically discuss the Mittag-Leffler function in this context in his published work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The task fell on Pillai [15] to study Mα(x|µ) ≡ Mα(x|µ, λ = 1), including its infinite divisibility and the corresponding Mittag- Leffler stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' He also proved that Mα(x|1) = 1 − Eα(−xα) (as discussed above), which he referred to as the Mittag-Leffler distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' There are thus two distributions bearing the name “Mittag-Leffler distribution”: Mα(x) = 1 − Eα(−xα) and Pα(t) = 1 − Fα(t−1/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The natural question arising from the Bayesian approach is whether there might be other choices of µ in G(µ, λ) (or indeed other choices of G altogether) that yield the Pollard result and, if so, what insight they might offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' At face value, there would appear to be nothing further to be said since other choices of µ can be expected to lead to different results, beyond the study of the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The main contribution of this paper is that, in fact, there is a limit relationship that generates the Pollard result for any µ > 0, as discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We first note, given the definition of the conditional stable density fα(x|t) ≡ fα(x t−1/α)t−1/α =⇒ fα(1|t) ≡ fα(t−1/α)t−1/α that we may write Pα(t) of (9) and the representation (12) of the Pollard result as Pα(t) = � t 0 pα(u) du = 1 α � t 0 fα(1|u) u−1 du (27) αEα(−λxα) = x � ∞ 0 fα(x|t) t−1e−λt dt 0 < α < 1 (28) u = x−αt : Eα(−λxα) = � ∞ 0 e−λxαu dPα(u) (29) The intent is to generate this representation using the general G(µ, λ) prior distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' without reference to Pollard’s analytic method and without explicit restriction to the G(µ = 1, λ) case that is equivalent to Feller’s approach, as demonstrated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 3 Main Contribution We first state Theorem 1, which warrants dedicated discussion, even though it is actually a special case of the more general Theorem 3 stated later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We note first that the density of the marginal distribution Mα(x|µ, λ) of Section 2 is mα(x|µ, λ) = � ∞ 0 fα(x|t) dG(t|µ, λ) µ > 0, λ > 0 = λµ Γ(µ) � ∞ 0 fα(x|t) tµ−1e−λt dt = µλµ Γ(µ + 1) � ∞ 0 fα(x|t) tµ−1e−λt dt (30) 7 where the latter expression follows from the identity µΓ(µ) = Γ(µ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The limit lim n→∞ n µ x mα(x|µ n, λ) = lim n→∞ n µ x � ∞ 0 fα(x|t) dG(t|µ n, λ) (31) is finite and independent of µ for any µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' This limit yields the following integral representa- tion of the Mittag-Leffler function Eα(−λxα) αEα(−λxα) = x � ∞ 0 fα(x|t) t−1e−λt dt (32) u = x−αt : Eα(−λxα) = � ∞ 0 e−λxαu dPα(u) (33) where Pα(t) is the (one-parameter) Pollard distribution Pα(t) = 1 α � t 0 fα(1|u) u−1 du = 1 α � t 0 fα(u−1/α) u−1/α−1 du Hence Eα(−x) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The Laplace transform of x mα(x|µ, λ) is � ∞ 0 e−sxx mα(x|µ, λ) dx = � ∞ 0 e−sxx � ∞ 0 fα(x|t) dG(t|µ, λ) dx = − d ds � ∞ 0 � ∞ 0 e−sxfα(x|t) dx dG(t|µ, λ) = − d ds � ∞ 0 e−tsα dG(t|µ, λ) = αsα−1 � ∞ 0 t e−tsα dG(t|µ, λ) = αsα−1 λµ Γ(µ) � ∞ 0 tµ e−(λ+sα)t dt = αsα−1 λµ Γ(µ) Γ(µ + 1) (λ + sα)µ+1 = λµµα sα−1 (λ + sα)µ+1 =⇒ lim n→∞ n µ � ∞ 0 e−sxx mα(x|µ n, λ) dx = α sα−1 λ + sα which is the Laplace transform of αEα(−λxα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' With the aid of (30), it also readily follows that the limit (31) is lim n→∞ n µ x mα(x|µ n, λ) = x � ∞ 0 fα(x|t) t−1e−λt dt The integral representations (32) and (33) of Eα(−λxα) follow, hence the conclusion that Eα(−x) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Pursuing the Bayesian theme, we turn next to Laplace convolution to demonstrate the complete monotonicity of the two and three parameter Mittag-Leffler functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 8 4 A Convolution Representation Toward a more general discussion, we first present an alternative representation of xfα(x|t) using Laplace convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The convolution {ρ ⋆ f}(x) of ρ(x), f(x) is given by {ρ ⋆ f}(x) = � x 0 ρ(x − u)f(u) du (34) The convolution theorem states that the Laplace transform of {ρ⋆f} is a product of the Laplace transforms of ρ, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='1 One Parameter Case Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Let ρα(x) = x−α/Γ(1 − α), 0 < α < 1 with Laplace transform sα−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Let {ρα ⋆ fα(·|t)}(x) be the convolution of ρα(x) and fα(x|t) with Laplace transform sα−1e−tsα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Then x fα(x|t) = α t{ρα ⋆ fα(·|t)}(x) = α {ρα ⋆ fα}(xt−1/α) (35) where {ρα ⋆ fα}(x) is the convolution of ρα(x) and fα(x) ≡ fα(x|1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For compatibility with later discussion, we also use the name wα(x|t) defined by αwα(x|t) ≡ x fα(x|t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' By the convolution theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' {ρα ⋆ fα(·|t)}(x) has Laplace transform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='sα−1e−tsα = − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='αt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='dse−tsα = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='αt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='e−sxxfα(x|t) dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='=⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='α t {ρα ⋆ fα(·|t)}(x) = xfα(x|t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='The convolution {ρα ⋆ fα(·|t)}(x) takes the explicit form: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='{ρα ⋆ fα(·|t)}(x) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='� x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='ρα(x − u)fα(u|t) du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='� x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='ρα(x − u)fα(ut−1/α)t−1/α du ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='y = ut−1/α : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='� xt−1/α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='ρα(x − yt1/α)fα(y) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='� xt−1/α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='ρα(t1/α(xt−1/α − y))fα(y) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='= t−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='� xt−1/α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='ρα(xt−1/α − y)fα(y) dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='= t−1{ρα ⋆ fα}(xt−1/α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='so that αwα(x|t) ≡ x fα(x|t) = α t{ρα ⋆ fα(·|t)}(x) = α {ρα ⋆ fα}(xt−1/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Hence the following are equivalent representations of the Pollard distribution Pα(t): Pα(t) = � t 0 wα(1|t) u−1 du ≡ 1 α � t 0 fα(1|u) u−1 du = � t 0 {ρα ⋆ fα(·|u)}(1) du = � t 0 {ρα ⋆ fα}(u−1/α) u−1 du (36) 9 The motivation for the convolution representation is to facilitate generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Specifically, the Laplace transform αtsα−1e−tsα of xfα(x|t) is the derivative of −e−tsα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' However, a more general term like tsα−βe−tsα cannot arise from simple derivatives of e−tsα for non-integer β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' It might be interpreted as a fractional derivative, as can be represented instead by convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Accordingly, we proceed to consider more general convolutions than the convolution form (35) for xfα(x|t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='2 Two Parameter Case First, we introduce the two-parameter Mittag-Leffler function Eα,β(x) = ∞ � k=0 xk Γ(αk + β) (37) The Laplace transform of xβ−1Eα,β(−λxα) is � ∞ 0 e−sxxβ−1Eα,β(−λxα) dx = sα−β λ + sα (38) We may now proceed to prove that Eα,β(−x) is completely monotone by showing that it is the Laplace transform of a two-parameter variant Pα,β(t) of the Pollard distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We follow a corresponding two-parameter variant of the convolution argument presented above for the one-parameter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Let ρα,β(x) = xβ−α−1/Γ(β − α) β > α, with Laplace transform sα−β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Let {ρα,β ⋆ fα(·|t)}(x) be the convolution of ρα,β(x) and fα(x|t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Then wα,β(x|t) ≡ t {ρα,β ⋆ fα(·|t)}(x) = t(β−1)/α {ρα,β ⋆ fα}(xt−1/α) (39) (the name wα,β(x|t) is a shorthand adopted for convenience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' {ρα,β ⋆ fα(·|t)}(x) = � x 0 ρα,β(x − u)fα(u|t) du = � xt−1/α 0 ρα,β(t1/α(xt−1/α − u))fα(u) du = t(β−1)/α−1 � xt−1/α 0 ρα,β(xt−1/α − u)fα(u) du = t(β−1)/α−1{ρα,β ⋆ fα}(xt−1/α) Thus wα,β(x|t) ≡ t {ρα,β ⋆ fα(·|t)}(x) = t(β−1)/α{ρα,β ⋆ fα}(xt−1/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The two-parameter Mittag-Leffler function Eα,β(−λxα) has the integral represen- tation Eα,β(−λxα) = � ∞ 0 e−λxαt dPα,β(t) (40) 10 where Pα,β(t), which we refer to as the two-parameter Pollard distribution, is Pα,β(t) = � t 0 wα,β(1|u) u−1 du ≡ � t 0 {ρα,β ⋆ fα(·|u)}(1) du = � t 0 {ρα,β ⋆ fα}(u−1/α) u(β−1)/α−1 du (41) Hence Eα,β(−x) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The theorem is a particular case of the more general Theorem 3 below, hence the current proof is deferred to that of the latter theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='3 Three Parameter Case The three-parameter Mittag-Leffler function, also known as the Prabhakar function, is given by Eγ α,β(x) = 1 Γ(γ) ∞ � k=0 Γ(γ + k) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Γ(αk + β) xk (42) The Laplace transform of xβ−1Eγ α,β(−λxα) is � ∞ 0 e−sxxβ−1Eγ α,β(−λxα) dx = sαγ−β (λ + sα)γ (43) We may now proceed to prove that Eγ α,β(−x) is completely monotone by showing that it is the Laplace transform of a three-parameter variant P γ α,β(t) of the Pollard distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In principle, we need only have discussed the three-parameter case from the outset because the two and one- parameter instances are the special cases γ = 1 and γ = β = 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We chose instead to present in sequential order for clarity of exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We devote a separate section to the three-parameter case, which subsumes all prior discussion, by restating Theorem 1 in the three-parameter context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 5 Main Theorem We start with a proposition required for the general theorem that follows: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Let ργ α,β(x) = xβ−αγ−1/Γ(β − αγ) (0 < α < 1, γ > 0, β > αγ) and let {ργ α,β ⋆ fα(·|t)}(x) be the convolution of ργ α,β(x) and the stable density fα(x|t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Then wγ α,β(x|t) ≡ tγ {ργ α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ α,β ⋆ fα}(xt−1/α) (44) 11 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' {ργ α,β ⋆ fα(·|t)}(x) = � x 0 ργ α,β(x − u)fα(u|t) du = � xt−1/α 0 ργ α,β(t1/α(xt−1/α − u))fα(u) du = t(β−1)/α−γ � xt−1/α 0 ργ α,β(xt−1/α − u)fα(u) du = t(β−1)/α−γ{ργ α,β ⋆ fα}(xt−1/α) Thus wγ α,β(x|t) ≡ tγ {ργ α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ α,β ⋆ fα}(xt−1/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Let ργ α,β(x), wγ α,β(x|t) (0 < α < 1, γ > 0, β > αγ) be as defined in Proposition 3 and let G(µ, λ) be the gamma distribution with shape and scale parameters µ > 0, λ > 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Let the distribution Mγ α,β(x|µ, λ) have density mγ α,β(x|µ, λ) = � ∞ 0 wγ α,β(x|t) dG(t|µ, λ) = λµ Γ(µ) � ∞ 0 wγ α,β(x|t) tµ−1e−λt dt (45) ≡ λµ Γ(µ) � ∞ 0 {ργ α,β ⋆ fα(·|t)}(x) tγ+µ−1e−λt dt (46) = λµ Γ(µ) � ∞ 0 {ργ α,β ⋆ fα}(xt−1/α) t(β−1)/α+µ−1e−λt dt (47) where the latter two forms follow from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Then the following limit is finite and independent of µ for any µ > 0 lim n→∞ n µ mγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(x|µ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' λ) (48) This limit yields the following integral representation of the three-parameter Mittag-Leffler or Prabhakar function Eγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(−λxα) Eγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(−λxα) = � ∞ 0 wγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(x|t) t−1e−λt dt = � ∞ 0 e−λxαt dP γ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(t) (49) where P γ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' which we refer to as the three-parameter Pollard distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' is P γ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(t) = � t 0 wγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(1|u) u−1 du ≡ 1 Γ(γ) � t 0 {ργ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β ⋆ fα(·|u)}(1) uγ−1 du = 1 Γ(γ) � t 0 {ργ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β ⋆ fα}(u−1/α) u(β−1)/α−1 du (50) Hence Eγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(−x) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 12 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The Laplace transform �mγ α,β(s|µ, λ) of (45) is �mγ α,β(s|µ, λ) ≡ � ∞ 0 e−sx mγ α,β(x|µ, λ) dx = sαγ−β λµ Γ(µ) � ∞ 0 tγ+µ−1e−(λ+sα)t dt = λµ Γ(γ + µ) Γ(µ) sαγ−β (λ + sα)γ+µ (51) =⇒ lim n→∞ n µ � ∞ 0 e−sxmγ α,β(x|µ n, λ) dx = Γ(γ) sαγ−β (λ + sα)γ (52) By (43), the right hand side is the Laplace transform of Γ(γ) xβ−1Eγ α,β(−λxα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Given (46) and (47), it also readily follows that the limit (48) is � ∞ 0 tγ{ργ α,β ⋆ fα(·|t)}(x)t−1e−λtdt = � ∞ 0 {ργ α,β ⋆ fα}(xt−1/α) t(β−1)/α−1e−λtdt =⇒ Eγ α,β(−λxα) = x1−β Γ(γ) � ∞ 0 {ργ α,β ⋆ fα}(xt−1/α) t(β−1)/α−1e−λt dt u = x−αt : = 1 Γ(γ) � ∞ 0 e−λxαu {ργ α,β ⋆ fα}(u−1/α) u(β−1)/α−1 du = � ∞ 0 e−λxαu dP γ α,β(u) Hence Eγ α,β(−x) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Theorem 3 may be visually represented by the following commutative diagram, where mγ α,β(x|µ, λ) and its Laplace transform �mγ α,β(s|µ, λ) are given by (45) and (51) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The equivalence of the two routes from the top left node to the bottom left node induces the integral represen- tation of the Mittag-Leffler function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' mγ α,β(x|µ, λ) �mγ α,β(s|µ, λ) Γ(γ)xβ−1Eγ α,β(−λxα) Γ(γ) sαγ−β (λ + sα)γ L lim n→∞ n µ �mγ α,β(s|µ n, λ) lim n→∞ n µ mγ α,β(x|µ n, λ) L −1 (53) The representation (49) of Eγ α,β(x), with P γ α,β(t) given by (50), is equivalent to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='4) in G´orska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The difference is one of approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' This paper offers a fundamentally probabilistic argument, while G´orska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [10] follows a complex analytic route inspired by Pollard [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The balance of G´orska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [10] is devoted to finding an explicit formula for a function f γ α,β(x) featuring in the paper in terms of the Meijer G function and associated confluent Wright function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In turns out that f γ α,β(x) in G´orska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [10] is identical to {ργ α,β ⋆ fα}(x) in 13 this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We are content to leave it in the conceptually simple convolution form: {ργ α,β ⋆ fα}(x) = � x 0 ργ α,β(x − u)fα(u) du = 1 Γ(β − αγ) � x 0 (x − u)β−αγ−1fα(u) du (54) rather than express it in terms of special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In our context, we have actually worked with the conditional density wγ α,β(x|t) ≡ tγ {ργ α,β ⋆ fα(·|t)}(x) = t(β−1)/α{ργ α,β ⋆ fα}(xt−1/α) where we assigned a gamma prior distribution to the scale parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The density wγ α,β(x|t) reduces to (54) for the particular choice t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We have completed the task of proving that the three-parameter Mittag-Leffler function Eγ α,β(−x) is completely monotone by methods of probability theory, using Bayesian reasoning to derive an explicit form for P γ α,β(t), whose Laplace transform is Eγ α,β(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Beyond that, we draw conclu- sions on the complete monotonicity of related functions, notably xβ−1Eγ α,β(−xα) and Eγ α,β(−xα) in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' First, we discuss xβ−1Eγ α,β(−xα), the bottom left node of the commutative dia- gram (53), in the Bayesian context of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The discussion involves an alternative repre- sentation of the fundamental probabilistic object – the convolution density {ργ α,β ⋆ fα(·|t)}(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 6 An Alternative Representation For xβ−1Eγ α,β(−λxα) to be completely monotone, there must exist a distribution Rγ α,β(u|λ) defined by the Laplace transform xβ−1Eγ α,β(−λxα) = � ∞ 0 e−xu dRγ α,β(u|λ) (55) In turn, the Laplace transform of (55) is the Stieltjes transform (or iterated Laplace transform) of Rγ α,β(u|λ): sαγ−β (λ + sα)γ = � ∞ 0 1 s + u dRγ α,β(u|λ) (56) Then, as de Oliviera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [5], Mainardi and Garrappa [14] show, the Stieltjes inversion formula (Titchmarsh [22](11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='8, p318), Widder [23](VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='7, p342)) gives dRγ α,β(u|λ) = 1 π Im � (e−iπu)αγ−β (λ + (e−iπu)α)γ � du (57) The expression in braces on the RHS of (57) is (56) at s = e−iπu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In particular, for γ = β = 1, (57) reduces to dRα(u|λ) = 1 π λ uα−1 sin πα λ2 + 2λ uα cos πα + u2α du (58) which has been discussed in various contexts in the fractional calculus and probabilistic literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' James [12] in the latter context).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 14 We have mentioned (55) for completeness but it was not the core of our probabilistic discussion, whose focus was to determine P γ α,β(t), with Laplace transform Eγ α,β(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' That said, we can offer a ‘hybrid’ derivation of (55) that combines the core of the probabilistic argument in the form of the convolution density {ργ α,β ⋆ fα(·|t)}(x) with the complex analytic Stieltjes inversion argument presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Assume {ργ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β ⋆ fα(·|t)}(x) to be the Laplace transform of a distribution Sγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(u|t): {ργ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β ⋆ fα(·|t)}(x) = � ∞ 0 e−xu dSγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(u|t) (59) In turn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' the Laplace transform of (59) is the Stieltjes transform of Sγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(u|t): sαγ−βe−tsα = � ∞ 0 1 s + u dSγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(u|t) (60) By the Stieltjes inversion formula: dSγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(u|t) = 1 π Im � (ue−iπ)αγ−βe−t(ue−iπ)α� du (61) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' using the representation (59) in the proof of Theorem 3: Γ(γ) xβ−1Eγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(−λxα) = � ∞ 0 tγ{ργ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β ⋆ fα(·|t)}(x) t−1e−λt dt = � ∞ 0 dt tγ−1e−λt � ∞ 0 e−xu dSγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(u|t) = 1 π Im � ∞ 0 du e−xu(ue−iπ)αγ−β � ∞ 0 tγ−1e−(λ+(ue−iπ)α)t dt = Γ(γ) π Im � ∞ 0 e−xu (e−iπu)αγ−β (λ + (e−iπu)α)γ du = Γ(γ) � ∞ 0 e−xu dRγ α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='β(u|λ) (62) thereby reproducing (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The Stieltjes transform and its complex analytic inverse are not unfamiliar in probability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In his study of a family of distributions known as generalised gamma convolutions, Bondesson [4] used the concept under the guise of Pick functions (also known as Nevanlinna functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We turn next to the complete monotonicity of Eγ α,β(−λxα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 7 A Further Consequence There is a well-known property of completely monotone functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Schilling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [20]) that we state without proof in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We start with a definition: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' A Bernstein function is a nonnegative function η(x), x ≥ 0 with a completely monotone derivative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' η(x) ≥ 0 and (−1)k−1η(k)(x) ≥ 0, k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For example, η(x|λ) = λxα (0 ≤ α ≤ 1, λ > 0) is a Bernstein function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 15 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' If ϕ(x) is completely monotone and η is a Bernstein function, ϕ(η) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Given a Bernstein function η, the Mittag-Leffler function Eγ α,β(−η) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For example, Eγ α,β(−λxα) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We have already shown that Eγ α,β(−x) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Hence, by Proposition 4, Eγ α,β(−η) is completely monotone for a Bernstein function η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Specifically, η(x|λ) = λxα (0 ≤ α ≤ 1, λ > 0) is a Bernstein function, hence Eγ α,β(−λxα) is completely monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The complete monotonicity of Eγ α,β(−λxα) implies that there exists a distribution Qγ α,β(t|λ) whose Laplace transform is Eγ α,β(−λxα): Eγ α,β(−λxα) = � ∞ 0 e−xt dQγ α,β(t|λ) (63) Qγ α,β(t|λ) is to Eγ α,β(−λxα) what P γ α,β(t) is to Eγ α,β(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' However, determining Qγ α,β(t|λ) appears to be a challenging problem, whether the approach is analytic or probabilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Clearly, (63) and (57) are identical for β = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Qγ α,1(t|λ) ≡ Rγ α,1(t|λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' But, to our awareness, determining Qγ α,β(t|λ) for β ̸= 1 is an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We shall not pursue it further here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Our primary purpose in this section was to bring attention to Theorem 4 and hence the existence of a distribution Qγ α,β(t|λ) defined by (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 8 A Different Generalisation As mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='5, the Pollard distribution Pα is known as the Mittag-Leffler distri- bution in probabilistic literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For completeness, we briefly discuss a different generalisation of Pα that features extensively in such literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' It is known as the generalised Mittag-Leffler distribution Pα,θ (Pitman [16], p70 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='27)), also denoted by ML(α, θ) (Goldschmidt and Haas [9], Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Despite its name, Pα,θ(t) is different from the two-parameter Pollard distribution Pα,β(t) dis- cussed above, whose Laplace transform is the Mittag-Leffler function Eα,β(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Janson [13] showed that Pα,θ may be constructed as a limiting distribution of a P´olya urn scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' It is also intimately linked to a concept known as ‘polynomial tilting’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' For some parameter θ, fα,θ(x) ∝ x−θfα(x) is said to be a polynomially tilted variant of fα(x) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Arbel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [1], De- vroye [6], James [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Here, we consider the polynomially tilted density fα,θ(x|t) ∝ x−θfα(x|t) conditioned on a scale factor t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Normalisation gives fα,θ(x|t) = Γ(θ + 1) Γ(θ/α + 1)tθ/α x−θfα(x|t) (64) so that fα,θ(x|t) is defined for θ/α + 1 > 0, or θ > −α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We then consider a two-parameter 16 function hα,θ(x|λ) defined by: α hα,θ(x|λ) = x � ∞ 0 fα,θ(x|t) t−1e−λt dt (65) = Γ(θ + 1) Γ(θ/α + 1) x1−θ � ∞ 0 fα(x|t) tθ/α−1 e−λt dt u = x−αt : hα,θ(x|λ) = � ∞ 0 e−λxαu dPα,θ(u) (66) where Pα,θ(t) = Γ(θ + 1) Γ(θ/α + 1) 1 α � t 0 fα(u−1/α) u(θ−1)/α−1 du (67) or dPα,θ(t) = Γ(θ + 1) Γ(θ/α + 1) tθ/α dPα(t) (68) It is clear from (66) that hα,θ(x|λ) may be written as hα,θ(λxα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' It follows that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' hα,θ(x) is completely monotone 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' θ = 0: Pα,0(t) = Pα(t) =⇒ hα,0(x) = Eα(−x), as directly apparent from comparing (32) and (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' hα,θ(η) is completely monotone where η is a Bernstein function as discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' In particular, hα,θ(λxα) is completely monotone and thus expressible as the Laplace trans- form of a corresponding distribution Qα,θ(t|λ) (distinct from Qα,β(t|λ) discussed in Sec- tion 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' We are not aware of a representation of hα,θ other than that generated by Pα,θ in (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' By comparison, the two-parameter Mittag-Leffler function Eα,β has a well-established infinite se- ries representation (37), in addition to the representation (40) generated by the two-parameter Pollard distribution Pα,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 9 Discussion The integral representation (49) of Eγ α,β(−λxα) in Theorem 3, arising from the limit (48), con- tains the L´evy measure t−1e−λtdt of the infinitely divisible gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' There is indeed an intimate relationship between completely monotone functions and the theory of infinitely di- visible distributions on the nonnegative half-line R+ = [0, ∞) (Feller [7] (XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='4, XIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='7), Steutel and van Harn [21] (III)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Sato [19] considers infinitely divisible distributions on Rd, but the de- liberate restriction to R+ makes for simpler discussion and relates directly to the core concept of complete monotonicity that is of interest here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' There is also an intimate link to the generalised gamma convolutions studied by Bondesson [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The limit (48) of Theorem 3 is an instance of a limit rule to generate the L´evy measure of an infinitely divisible distribution given in Steutel and van Harn [21] (III(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='7)) and Sato [19] (Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content='9 restricted to R+ rather than Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Barndorff-Nielsen and Hubalek [2] also cite Sato’s Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Further exploration using the probabilistic machinery of this paper possibly includes the ex- plicit determination of the three-parameter distribution Qγ α,β(t|λ), whose Laplace transform is Eγ α,β(−λxα), as per (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 17 10 Conclusion We have presented a probabilistic derivation of the complete monotonicity of the three-parameter Mittag-Leffler function (also known as the Prabhakar function) by expressing it as the Laplace transform of a distribution that we referred to as the three-parameter Pollard distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' This is a generalisation of a result due to Pollard for the one-parameter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' References [1] Julyan Arbel, Pierpaolo De Blasi, and Igor Pr¨unster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Stochastic Approximations to the Pitman–Yor Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Bayesian Analysis, 14(4):1201 – 1219, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [2] Ole E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Barndorff-Nielsen and Friedrich Hubalek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Probability measures, L´evy measures and analyticity in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Bernoulli, 14(3):764 – 790, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [3] David Blackwell and James B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' MacQueen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Ferguson distributions via P´olya urn schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The Annals of Statistics, 1(2):353–355, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [4] Lennart Bondesson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Generalized Gamma Convolutions and Related Classes of Distributions and Densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Lecture Notes in Statistics, 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Springer-Verlag, New York, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Capelas de Oliveira, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Mainardi, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Vaz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' The European Physical Journal Special Topics, 193(1):161–171, Mar 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' [6] Luc Devroye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Random variate generation for exponentially and polynomially tilted stable distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQfgPyo/content/2301.01466v1.pdf'} +page_content=' 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PETROV-GALERKIN REDUCED ORDER +METHOD FOR ADVECTION-DOMINATED PARTIAL DIFFERENTIAL +EQUATIONS UNDER OPTIMAL CONTROL +FABIO ZOCCOLAN1, MARIA STRAZZULLO2, AND GIANLUIGI ROZZA3 +Abstract. In this paper we will consider distributed Linear-Quadratic Optimal Control Problems +dealing with Advection-Diffusion PDEs for high values of the P´eclet number. In this situation, +computational instabilities occur, both for steady and unsteady cases. +A Streamline Upwind +Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects. +We will apply a finite element method discretization in a optimize-then-discretize approach. For +the parabolic case, a space-time framework will be considered and stabilization will also occur in +the bilinear forms involving time derivatives. Then we will build Reduced Order Models on this +discretization procedure and two possible settings can be analyzed: whether or not stabilization is +needed in the online phase, too. In order to build the reduced bases for state, control, and adjoint +variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach. +The discussion is supported by computational experiments, where relative errors between the FEM +and ROM solutions are studied together with the respective computational times. +1. Introduction +The main goal of Optimal Control theory is to modify a physical or engineering system through an +input, called control, to obtain a desired output. From a theoretical point of view, one can describe +the state problem through partial differential equations (PDEs), following the approach of J.L. Lions +[30, 31]. Applying an optimal control means to solve a constrained optimization problem, where +a cost functional has to be minimized. This process translates into an optimality system, which +will be discretized for numerical simulations, that, in this framework, are more and more needed. +Thus, effective and fast numerical techniques are required to exploit optimal control in scientific and +industrial applications. +In this work, we will consider Advection-Diffusion equations [42] for large P´eclet numbers. These +equations are widespread in many engineering contexts since they can model transfer of particles, +of energy, of heat and so on. In the case of high values of the P´eclet number, numerical instabilities +occur during discretization: this can happen for related optimal control problems, too. Thus, it +becomes necessary to introduce some stabilization techniques to overcome this undesired behaviour. +We exploit a Streamline Upwind Petrov–Galerkin (SUPG) technique over a finite element method +(FEM) [11, 26, 38] in a optimize-than-discretize approach, as done in [14], to provide strongly- +consistency to the discretization. When we deal with unsteady problems, a space-time discretization +[21, 46, 50, 51, 52, 57] will be used together with the SUPG stabilization for bilinear forms related +to the derivative over time. +The discretization procedure can easily request a huge amount of +computational resources, especially for parametric time-dependent problems. The parameters can +represent physical or geometrical features of the system at hand. In this scenario, we decide to exploit +the parameter dependence of the equations to build Reduced Order Models (ROMs) [22, 40, 39, 43] +by means of Proper Orthogonal Decomposition (POD) algorithm in a partitioned approach. Namely, +1 Section de Math´ematiques, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland, +email: fabio.zoccolan@epfl.ch +2 DISMA, Politecnico di Torino, Corso Duca degli Abruzzi 24, Turin, Italy. +email: maria.strazzullo@polito.it +3 mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy. +email: gianluigi.rozza@sissa.it +1 +arXiv:2301.01973v1 [math.NA] 5 Jan 2023 + +2 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +the discretization process is divided in two phases: an offline stage where a low-dimensional space is +built through FEM solutions computed in properly chosen parameters, and an online stage, where +the system is solved for a new parametric instance in the new low-dimensional framework. Thus, +we consider two possible strategies: the former is to stabilize the system only in the offline phase; +the latter uses SUPG in the online one, too. This setting was considered for problems without +optimal control in [37, 55]. To the best of our knowledge, it is the first time that SUPG stabilization +for time-dependent Advection-Dominated problems under distributed control is analyzed in a ROM +setting. +This work is organized as follows. The first section will illustrate some theoretical aspects about +Optimal Control Theory for PDEs. Section 3 shows the FEM discretization that will be used for +numerical experiments, an introduction to Advection-Dominated problems, and SUPG technique +in an optimize-than-discretize approach. Instead, in Section 4, we will focus on the ROM setting +and Section 5 refers to the related numerical simulations. Firstly, we will introduce two specific +examples of Advection-Diffusion problems: the Graetz-Poiseuille and the Propagating Front in a +Square Problems. The former was studied in various forms without optimal control in [18, 37, 44, 55] +and with optimal control but without stabilization in [34, 50]. The latter is studied without optimal +control in a similar version in [37, 55]. Here, both the problems will be analyzed under a distributed +optimal control for high values of the P´eclet number, both in the steady and unsteady cases. Relative +errors between FEM and ROM solutions will be shown, as well as an analysis on the computational +times. +2. Problem Formulation +In this Section we will illustrate the fundamentals of Linear-Quadratic Optimal Control Problem +(OCP) for steady and unsteady PDEs. +The aim of Optimal Control is to achieve a prescribed +optimality condition by minimizing a suitable cost functional under the constraint of satisfying the +PDE Problem. The proposed framework follows the J.L. Lions theory [30, 31]. +2.1. Parametric Optimal Control Problems governed by PDEs. The main features of an +OCP are: +(1) a controlled system, i.e. an input-output process given by a system of PDEs; +(2) the output of the system, or an observation of it, when the output cannot be measured +directly. In our case, we will consider the solution of the system as the output; +(3) a control, which constitutes the input of the system. It influences the output which can be +expressed as a function of it. In this work we will only consider distributed control; +(4) an objective condition to be fulfilled, which can be represented by a real functional. +Therefore, from a mathematical perspective, we can state that an OCP is characterized by: +• e, the state equation function, which expresses the relationship between the output and the +control within the system in terms of a PDE problem or PDEs in a weak formulation. A +pair (y, u) ∈ X := Y × U is said to be physical or feasible if it is a solution of the state +equation e; y is called the state variable, the output, and u is the control variable, the input. +Xad is the set of all the feasible pairs (y, u); +• z(y) = Oy, a direct observation of the output. Here, a linear operator O is applied to the +state to describe the observation: we will denote the space of observation as Z. We will only +deal with state variables that can be measured on a portion of the domain; +• J, the objective functional, which describes the objective to achieve. +• suitable spaces Y and U, as the state space and control space respectively. +Domains of +definition for control and/or state can be taken smaller due to possible restrictions; hence +we have to introduce Yad ⊆ Y and Uad ⊆ U as the admissible state space and admissible +control space respectively. However, we will always consider unconstrained problems, i.e. +Xad = X. The theory of well-posedness can make use of the Lagrangian approach as in +[12, 34] or it can be consider as a particular case of the general Adjoint approach when we +can deal with Xad ⊂ Yad × Uad [24, 30, 38]. + +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +3 +Let us consider Ω ⊂ Rn, an open and bounded regular domain, and the time interval (0, T) ⊂ R+: +for us it will always be the case of n = 2. Let us denote with ΓD and ΓN the portions of the boundary +of ∂Ω where Dirichlet and Neumann boundary conditions are specified, respectively. We define the +observation domain Ωobs ⊆ Ω as the portion of the domain where we want that the state variable +assumes a desired value. P ⊆ Rp, for natural number p, is the parameter space and µ ∈ P is a +p-vector which can represent physical or geometrical parameter of interest. In this work we deal +with Parametric Optimal Control Problems (OCP(µ)s), i.e. systems where there is a dependency on +the parameter µ. +Problem 2.1.1 (Parametric Optimal Control Problem). Given Y, U real Banach spaces, consider +the state equation e : Y × U → Q, with Q a Banach space, which fulfills a set of boundary and/or +initial conditions, and the objective functional J : Y ×U → R. Given µ ∈ P, then find +� +y(µ), u(µ) +� +∈ +X such that the cost functional J(y(µ), u(µ); µ) is minimized subject to e(y(µ), u(µ); µ) = 0. +2.2. Lagrangian Approach. We refer to the Lagrangian approach to state the well-posedness of +OCP(µ)s in full admissibility setting, i.e. when Xad = Y × U. We want to solve: +min +(y(µ),u(µ))∈Y ×U J(y(µ), u(µ); µ) s.t. e(y(µ), u(µ); µ) = 0, +thus we define the Lagrangian operator L : Y × U × Q∗ → R as: +(1) +L(y(µ), u(µ), p(µ); µ) = J(y(µ), u(µ); µ) + ⟨p(µ), e(y(µ), u(µ); µ)⟩Q∗Q, +where p(µ) is a Lagrange multiplier belonging to Q∗, the dual space of Q. For the sake of notation, +we write y := y(µ), u := u(µ) and p := p(µ): we will explicit the parameter dependence only when +necessary. The discussion inherent to the Lagrangian approach is based on [12], the same reference +presents a comparison between this approach and the adjoint one. For the sake of simplicity, we +make some regularity assumptions [12]: +Assumption 2.2.1. The objective functional J and the state equation e are Fr´echet differentiable, +more precisely the differential operator related to J is continuous, i.e. J′(µ) ∈ C(Y ×U, B(Y ×U, R)), +where B(V , ˜V ) is the space of linear bounded operators between Banach spaces V and ˜V . +The following theorem and proposition claim that under Assumption 2.2.1 minimizers of the +function J, subject to equality constraints e, can be critical points of (1) [53]. +Theorem 2.2.2 (Lagrange Multipliers). Let X := Y × U and V ⊆ X be an open subset such that +J and e are Frech´et differentiable on V. Assume x = (y, u) ∈ V to be a minimizer of J subject to +the constraint e(x; µ) = 0, and e′(x; µ) ∈ B(X, Q) to be surjective. Then, there exists a Lagrange +multiplier p ∈ Q∗ such that (x, p) is an unconstrained stationary point of the Lagrangian L in (1). +Therefore, in order to find a stationary point (y, u, p) of L, one has to solve the following optimality +system [12]: +(2) +� +� +� +� +� +Ly(y, u, p; µ)(¯y) = Jy(y, u; µ)(¯y) + ⟨p, ey(y, u; µ)(¯y)⟩Q∗Q = 0, +∀¯y ∈ Y, +Lu(y, u, p; µ)(¯u) = Ju(y, u; µ)(¯u) + ⟨p, eu(y, u; µ)(¯u)⟩Q∗Q = 0, +∀¯u ∈ U, +Lp(y, u, p; µ)(¯p) = ⟨¯p, e(y, u; µ)⟩Q∗Q = 0, +∀¯p ∈ Q∗. +In the Lagrangian formulation Q∗ is said the adjoint space. The above result easily implies the +following useful proposition [38], where we derive another system of three equations that we will use +in the numerical simulations. +Proposition 2.2.3 (Optimality System). Suppose Xad = Y × U and Assumption 2.2.1 holds, then +for some p ∈ Q∗ a minimizer x = (y, u) of 2.1.1 where e′(y, u; µ) is surjective must satisfy +(3) +� +� +� +� +� +Ly(y, u, p; µ) = Jy(y, u; µ) + ey(y, u; µ)∗p = 0, +in Y ∗, +Lu(y, u, p; µ) = Ju(y, u; µ) + eu(y, u; µ)∗p = 0, +in U ∗, +Lp(y, u, p; µ) = e(y, u; µ) = 0, +in Q. + +4 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +In (3), the first equation is called the adjoint equation, the second one is the gradient equation +and, as we have already seen, the state equation is the third one. We remark that we will always +consider Linear-Quadratic problems. +Definition 2.2.4 (Linear-Quadratic Problem). Consider a Banach space Z and α > 0. Let the +Observation map O : Y → Z be a linear and bounded operator. Consider an element zd(µ) ∈ Z, +which is the so-called desired solution profile (the desired observed output). Let J be a quadratic +objective functional of the form +(4) +J(y, u; µ) = 1 +2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α +2 n(u(µ), u(µ)), +where m : Z × Z → R and n : U × U → R are symmetric and continuous bilinear forms. Let e be +affine, i.e. there exist A(µ) ∈ B(Y, Q), B(µ) ∈ B(U, Q) and f(µ) ∈ Q such that +(5) +e(y, u; µ) = A(µ)y + B(µ)u − f(µ), +∀ +� +y(µ), u(µ) +� +∈ Y × U. +Then an OCP(µ)s with the above properties is said a Linear-Quadratic Optimal Control Problem. +For Linear-Quadratic OCP(µ)s Proposition 2.2.3 implies that a solution (y, u) to Problem 2.1.1 +must satisfy, for some p ∈ Q∗ [12], +(6) +� +� +� +� +� +m(Oy, O¯y; µ) + ⟨A∗(µ)p, ¯y⟩Y ∗Y = m (O¯y, zd; µ) , +∀¯y ∈ Y, +αn(u, ¯u; µ) + ⟨B∗(µ)p, ¯u⟩U ∗U = 0, +∀¯u ∈ U, +⟨¯p, A(µ)y + B(µ)u⟩Q∗Q = ⟨¯p, f(µ)⟩Q∗Q, +∀¯p ∈ Q∗. +In this context, if (y, u, p) is a saddle point of L [56], then (y, u) minimizes J over all zeroes of +e [12]. Moreover, under some precise hypotheses existence and uniqueness of a saddle point can be +provided using Brezzi Theorem [9, 10, 12]. Therefore, a possible strategy to prove well-posedness +of an Linear-Quadratic OCP(µ)s can be to demonstrate that a stationary point of (6) is a saddle +point. At this purpose, System (6) can also be recast in a saddle-point structure [7, 12, 36]. In order +to derive this structure, assume x ∈ X := Y × U. We define M(µ) ∈ B (Z, Z∗) , N(µ) ∈ B (U, U ∗) +as the unique operators that satisfy the following relations: +⟨M(µ)z, ¯z⟩Z∗Z = m(z, ¯z; µ), +⟨N(µ)u, ¯u⟩U ∗U = n(u, ¯u; µ), +∀z, ¯z ∈ Z, ∀u, ¯u ∈ U. +This directly implies that m(Oy, O¯y; µ) = ⟨O∗M(µ)Oy, ¯y⟩Y ∗Y . Using Proposition (2.2.3) and a +matrix notation as follows [12]: +(7) +E(µ) = +� A(µ) +B(µ) � +, +D(µ) = +� O∗M(µ)O +0 +0 +αN(µ) +� +, +E∗(µ) = +� A∗(µ) +B∗(µ) +� +, +defining also ¯g(µ) = O∗M(µ)zd, the optimality system (6) for Linear-Quadratic OCP(µ)s can be +written in a more compact form as +(8) +� D(µ) +E∗(µ) +E(µ) +0 +� � x +p +� += +� ¯g(µ) +f(µ) +� +in X∗, +in Q. +For Linear-Quadratic Problems, a saddle point of L is a stationary point [56], so it satisfies (6). +For Linear-Quadratic problems the solution to system (8), and hence to (6), is a saddle point of L +when D(µ) is self-adjoint [12]. In this case Brezzi Theorem gives us well-posedness [9, 10, 12]. +Lemma 2.2.5. [12] If Y is reflexive so that D(µ) = D∗(µ), then (x, p) = (y, u, p) is a saddle point +of L if and only if it solves the system (8). +Assumption 2.2.6. We assume that Y, U are reflexive, A(µ) is weakly coercive, the operator B(µ) +is not null, αN(µ) is coercive with constant α > 0 and m(z, z; µ) ≥ 0, ∀z ∈ Z. +Considering Linear-Quadratic OCP(µ)s and Assumption 2.2.6, it follows that E(µ) is inf-sup +stable and D(µ) is coercive over the kernel of E(µ). Consequently, the well-posedness of the system +(8) is assured by Theorem 2.2.7. +Theorem 2.2.7 (Brezzi). [9, 10, 12] Let X be a reflexive Banach space. Then the equivalence of +the following statements holds: + +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +5 +(1) D(µ) ∈ B (X, X∗) , E(µ) ∈ B(X, Q) with the following properties: +• D(µ) is weakly coercive over the kernel of E(µ), +• E(µ) is inf-sup stable. +(2) The system (8) has a unique solution (x, p) ∈ X × Q∗ for all ¯g(µ) ∈ X∗, f(µ) ∈ Q, which +satisfies for some constant C > 0 ∥x∥X + ∥p∥Q∗ ≤ C (∥¯g(µ)∥X∗ + ∥f(µ)||Q) . +(3) The operator S(µ) := +� +D(µ) +E∗(µ) +E(µ) +0 +� +is an isomorphism in X∗ × Q. +Remark 2.2.8 (Notation). From now on, we will always involve Hilbert spaces. For the sake of +notation, there we will denote the various bilinear forms defined by A(µ), B(µ) and their adjoints +ones in the following unique way: +⟨A(µ)y, p⟩QQ∗ := a(y, p; µ) +⟨B(µ)u, p⟩QQ∗ := b(u, p; µ). +2.3. Unsteady Problems. We briefly recall results on well-posedness for time-dependent Linear- +Quadratic OCP(µ)s based on [50, 51]. We consider saddle-point formulation in order to prove well- +posedness by using tools of the previous Sections in the case of null initial conditions. Differently +from the steady case, here we will make some more technical assumptions, which will be fulfilled by +both “Graetz-Poiseuille” and “Propagating Front in a Square” problems. +Consider two separable Hilbert spaces Y and H satisfying Y �→ H �→ Y ∗ and, moreover, other +two Hilbert spaces U and Z ⊇ Y , where Y and U are the usual state and control spaces, and Z is +the space of observation. We endow them with the standard norms inherited from their respectively +scalar products: (·, ·)Y , (·, ·)Z, (·, ·)U and (·, ·)H. We define the following Hilbert spaces: +Y = L2(0, T; Y ), +Y∗ = L2 (0, T; Y ∗) , +U = L2(0, T; U) +Z := L2(0, T; Z) ⊇ Y. +with respective norms, for instance in the case of Y and U given by +(9) +∥y∥2 +Y := +T +� +0 +∥y∥2 +Y dt, +and +∥u∥2 +U := +T +� +0 +∥u∥2 +Udt +and similarly for the others. Furthermore, let us define the Hilbert space Yt with its scalar product +(·, ·)Yt: +Yt := +� +y ∈ Y +s.t. +∂y +∂t ∈ Y∗ +� +, +(y, z)Yt := +T +� +0 +(y, z)Y dt + +T +� +0 +�∂y +∂t , ∂z +∂t +� +Y ∗dt. +Our aim is to solve the following unconstrained Linear-Quadratic Parametric Parabolic OCP(µ): +Problem 2.3.1 (Parametric Parabolic OCP(µ)). For a given µ ∈ P find the pair (y(µ), u(µ)) ∈ +Yt × U that satisfies +(10) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∂y(µ) +∂t ++ A(µ)y(µ) + B(µ)u(µ) − f(µ) = 0, +in Ω × (0, T), +∂y(µ) +∂n += 0, +on ΓN × (0, T), +y(µ) = l, +on ΓD × (0, T), +y(µ)(0) = y0, +in Ω, +and minimizes +min +(y(µ),u(µ))∈Yt×U J(y, u; µ) = 1 +2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α +2 n(u(µ), u(µ)), +where m : Yt × Yt → R and n : U × U → R are symmetric and continuous bilinear forms, zd(µ) ∈ Z +is the observed desired solution profile and α > 0 is the fixed penalization parameter. In our test +case we will always take y0 ≡ 0. + +6 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +Also in this case, we denote y := y(µ) and u := u(µ) omitting the parameter dependence. We +can state the weak formulation of (10) as +� +� +� +� +� +� +� +T +� +0 +�∂y +∂t , q +� +Y∗Y +dt + +T +� +0 +⟨A(µ)y, q⟩Y∗Ydt + +T +� +0 +⟨B(µ)u, q⟩Y∗Ydt − +T +� +0 +⟨f(µ), q⟩Y ∗Y dt = 0, +∀q ∈ Yt, +y(0) = y0, +in Ω, +where f(µ) ∈ Y∗ gathers all forcing, boundary and, eventually, lifting terms of the state equation. +Nevertheless, for the sake of notation, we will consider a : Yt × Yt → R and b : U × Yt → R the +bilinear forms defined as a(y, q; µ) = ⟨A(µ)y, q⟩Y∗Y and b(u, q; µ) = ⟨B(µ)u, q⟩Y∗Y, respectively. +For a proper definition of the adjoint variable, it is opportune to take q ∈ Yt rather than q ∈ Y [50]. +Let us define the state-control product space X = Yt × U. Then we define the operators E,D and ¯g +similarly as made in the steady case in order to make the formulation more compact [50]: +(11) +D(µ) : X × X → R, +D(x, ¯x, µ) =m(Oy, O¯y; µ) + αn(u, ¯u; µ); +E(µ) : X × Yt → R, +E(x, q, µ) = +T +� +0 +�∂y +∂t , q +� +Y∗Y +dt + +T +� +0 +a(y, q, µ)dt + +T +� +0 +b(u, q, µ)dt; +¯g(µ) ∈ X ∗, +T +� +0 +⟨¯g(µ), ¯x⟩dt =m (O¯y, zd(µ)) . +Denoting p := p(µ) and considering Q∗ = Yt [50], the Lagrangian and objective functionals are, +respectively: +(12) L(x, p; µ) = J(x; µ)+E(x, p; µ)− +T +� +0 +⟨f(µ), p⟩Y ∗Y dt, +J(x, µ) = 1 +2D(x, x; µ)− +T +� +0 +⟨¯g(µ), x⟩dt. +As made in the steady case, the minimization of Problem 2.3.1 means to seek the solution of the +following system: given µ ∈ D, find (y, u, p) = (x, p) ∈ X × Yt which solve +(13) +� +� +� +� +� +Ly(y, u, p; µ)[¯y] = 0, +∀¯y ∈ Yt, +Lu(y, u, p; µ)[¯u] = 0, +∀¯u ∈ U, +Lp(y, u, p; µ)[¯p] = 0, +∀¯p ∈ Yt, +and satisfy boundary and initial conditions in Problem 2.3.1 with p(T) = 0 [30]. The saddle-point +structure of steady Linear-Quadratic OCP(µ)s (8) can be derived in the parabolic case, too (here +expressed in the weak formulation) [50]: +(14) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +D(x, w; µ) + E(w, p; µ) = +T +� +0 +⟨¯g(µ), w⟩dt, +∀w ∈ X, +E(x, q; µ) = +T +� +0 +⟨f(µ), q⟩Y ∗Y dt, +∀q ∈ Yt. +The equivalence between the optimality system and saddle-point formulation for Linear-Quadratic +Parabolic OCP(µ)s is straighforward. For well-posedness the following assumption is needed [50]. +Assumption 2.3.2. The bilinear forms n(·, · ; µ), m(·, · ; µ), b(·, · ; µ), and a(·, · ; µ) satisfy the +following features: +(1) m(·, · ; µ) is positive definite, continuous, and symmetric. +(2) n(·, · ; µ) is coercive, continuous, and symmetric; +(3) there exists Ca > 0 s.t. a(w, w; µ) ≥ Ca(µ)∥w∥2 +Y , +∀w ∈ Yt; +(4) there exists ca > 0 s.t. |a(w, p; µ)| ≤ ca(µ)∥w∥Y ∥p∥Y , +∀w, p ∈ Yt; + +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +7 +(5) there exists cb > 0 s.t. |b(v, p; µ)| ≤ cb(µ)∥v∥U∥p∥Y , +∀v ∈ U and ∀p ∈ Yt; +Finally, one can prove the well-posedness of Problem 2.3.1 (for more details, we refer to [50]). +Theorem 2.3.3 (Well-posedness of Parabolic OCP(µ)s). [50] Under Assumption 2.3.2 the saddle- +point formulation (14) satisfies the hypothesis (1) of Theorem 2.2.7, hence the solution is unique. +Assumption 2.3.4. For both steady and unsteady problems, we will consider the Identity operator +restricted to our observation domain Ωobs as the Observation function O. Therefore, Z = Y is +assumed and our desired state will be denoted by yd. +3. Truth Discretization +In this Section we firstly pursue a numerical method for the solution of an OCP: a discretization +of the optimality sistem (6) will be given following an one shot or all-at-once approach [23, 46, 47]. +Secondly, we will consider SUPG stabilization for Advection-Dominated equations in case of high +P´eclet number. An optimize-then-discretize approach is followed, i.e. at first we derive optimality +conditions as system (6) and then we discretize it. Therefore, we obtain a discretized system: +(15) +� +� +� +� +� +LyN (yN , uN , pN ) = JyN (yN , uN ) + eyN (yN , uN )∗pN = 0 +in +� +Y N �∗ +LuN (yN , uN , pN ) = JuN (yN , uN ) + euN (yN , uN )∗pN = 0 +in +� +U N �∗ +LpN (yN , uN , pN ) = e(yN , uN ) = 0 +in QN , +where LyN , LuN , LpN are the discretizations of partial derivatives of L and Y N , U N , QN are the +approximation of Y, U, Q, respectively. +Let us start our discussion from the steady case. From now on we will always assume to work with +Y, U, Q Hilbert spaces. We employ a FEM discretization, which will be named as the high-fidelity +or truth approximation. We consider Ωh as a quasi-uniform mesh on the domain Ω, for which the +parameter h indicates the mesh size, i.e. maximum diameter of an element of the grid. Th is a +regular triangularization on Ω and +Ωh := int +� � +K∈Th +K +� +, +where K is a triangle of Th. We define the FEM spaces Y N = Y ∩ XN ,r, U N = U ∩ XN ,r and +� +QN �∗ = Q∗ ∩ XN ,r, where +XN ,r = +� +vN ∈ C0(¯Ω) : vN +|K ∈ Pr(K), ∀K ∈ Th +� +and Pr(K) represents the space of polynomials of degree at most equal to r defined on a triangle K. +As we will remark later, we will always use the same triangulation Th and a P1-FEM approximation +for state, control and adjoint variables. The dimensions of Y N , U N , QN are all equal to N. The +overall dimension of the discrete problem is Ntot = 3 · N. For the sake of simplicity, we assume +Q∗ +h = Y N . Moreover, we indicate with XN = Y N × U N ⊂ X. From now on we will refer to the +same symbol yd to also indicate the FEM discretization version of the desired state. +The discretization of a Linear-Quadratic OCP of Problem 2.1.1 reads as +min +(yN ,uN )∈Y N ×U N J +� +yN , uN � += 1 +2m +� +yN − yd, yN − yd +� ++ α +2 n(uN , uN ) s.t. e +� +yN , uN � += 0. +Moreover, the operators m and n will be the L2 product on Ωobs and on Ω, respectively. +For the saddle-point system, we define the operators ¯gN : XN → R, f N : Y N → R, DN : XN → +� +XN �∗ , and EN : XN → +� +QN �∗ as just the usual restrictions +(16) +� +¯gN , ¯xN � +(XN )∗XN = +� +¯g, ¯xN � +X∗X , +� +DN xN , ¯xN � +(XN)∗XN = +� +DxN , ¯xN � +X∗X , +⟨f N , ¯pN ⟩(Y N )∗Y N = +� +f, ¯pN � +Q∗∗Q∗ , +� +EN xN , ¯pN � +(Y N )∗Y N = +� +ExN , ¯pN � +Q∗∗Q∗ , +for all xN ∈ XN , ¯pN ∈ Y N . + +8 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +We highlight the algebraic structure of the discretize optimality system. We define the basis of +the finite spaces XN and Y N as below: +(17) +� +ϕj ∈ XN �2N +j=1 , +� +ψk ∈ Y N �N +k=1 . +As a result, we can rewrite a pair +� +xN , pN � +∈ XN × Y N in the following way: +� +�xN = +2N +� +j=1 +xjϕj, +pN = +N +� +k=1 +pkψk +� +� . +Therefore, we can define D ∈ R2N ·2N , E ∈ RN ·2N , ¯g ∈ R2N and f ∈ RN as follows: +(18) +Dij = ⟨DN ϕi, ϕj⟩(XN )∗XN , +Elm = ⟨EN ϕl, ψm⟩(Y N )∗Y N , +¯gk = +� +¯gN , ϕk +� +(XN )∗XN , +fn = +� +f N , ψn +� +(Y N )∗Y N . +Finally, we can build the following saddle point system, with a block structure: +(19) +� +D +ET +E +0 +� � x +p +� += +� ¯g +f +� +, +where (x)i = xi, i = 1, · · · 2N and (p)k = pk, k = 1, · · · N. For this purpose, let us denote with y, +u and p the vectors of coefficients of yN , uN and pN , expressed in terms of the nodal basis (17) +by splitting components of XN in those of Y N and U N . We express with yd the vector with the +components of the discretized desired state, i.e. the Galerkin projection of yd on Y N . Moreover, +let us indicate the stiffness matrix derived from the bilinear form a(·, ·) with K, KT is the stiffness +matrix related to a∗ and the mass matrix is denoted with M. In addition, we call B, BT is the mass +matrix related to the forms b and b∗. We have that: +D = +� M +0 +0 +αM +� +, +E = +� K +B � +, +x = +� y +u +� +, +¯g = +� Myd +0 +� +. +and the optimality system shows this block structure: +(20) +� +� +M +0 +KT +0 +αM +BT +K +B +0 +� +� +� +� +y +u +p +� +� = +� +� +Myd +0 +f +� +� . +3.1. SUPG stabilization for Advection-Dominated OCP(µ)s. In this Section we illustrate +Advection-Dominated OCP(µ)s and the SUPG technique applied to an optimize-then-discretize +approach. From now, we recall the dependence on parameters of our operators. Let us start from +our definition of an Advection-Diffusion equation. +Definition 3.1.1 (Advection-Diffusion Equations). Let us consider the following problem: +(21) +L(µ)y := −ε(µ)∆y + b(µ) · ∇y = f(µ) in Ω ⊂ R2, +with suitable boundary conditions on ∂Ω. Let us suppose that: +• the diffusion coefficient ε : Ω → R belongs to L∞(Ω) and depends on the parameter µ. We +assume there exists a constant ¯ε > 0 such that ε(x) ≥ ¯ε, ∀x ∈ Ω; +• the advection field b : Ω → R2 belongs to (L∞(Ω))2 and depends on the parameter µ. We +suppose that 0 ≥ div b(x) ≥ −˜k, holds for all x ∈ Ω, with ˜k ∈ R+ +0 ; +• f(µ) : Ω → R is an L2(Ω)-function that can depend on the parameter µ. +In this case, (21) is an Advection-Diffusion problem and the operator L(µ)y := −ε(µ)∆y +b(µ)· +∇y is said the Advection-Diffusion operator. +From (21), we can easily derive the weak formulation of an Advection-Diffusion problem: +(22) +find y ∈ Y s.t. a (y, q; µ) = F (q; µ) +∀q ∈ Q∗, + +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +9 +where +(23) +a (y, q; µ) := +� +Ω +ε(µ)∇y∇q + b(µ) · ∇yq dx, +F(q; µ) := +� +Ω +f(µ)q dx, +y ∈ Y, q ∈ Q∗. +From a numerical point of view, when the advection term b(µ) · ∇u “dominates” the diffusive +one −ε(µ)∆u, i.e. when |b(µ)| ≫ ε(µ), the approximated solution can show instability phenomena +along the direction of the advection field [42]. In order to give an indicator of the instability, let us +consider the regular triangulation Th related to FEM discretization. For any element K ∈ Th, we +can then define the local P´eclet number as [42, 38]: +(24) +PeK(x) := |b(x)|hK +2ε(x) +, +∀x ∈ K, +where hK is the diameter of K. +Definition 3.1.2 (Advection-Dominated problem). Considering Definition 3.1.1 we are dealing +with an Advection-Dominated problem if PeK(x) > 1, ∀x ∈ K, ∀K ∈ Th. +To solve the issue of the instability, we will exploit the SUPG method [11, 25, 26, 42], which is a +strongly consistent stabilization technique; i.e. is consistent for weak PDEs and its order of accuracy +can be greater than one. Let us now consider the Advection-Diffusion operator (21): for the sake of +simplicity, we define it on H1 +0(Ω) and we do not indicate the parameter dependence. The operator +L can be split into its symmetric and skew-symmetric parts [42], defined as: +(25) +symmetric part: LSy = −ε∆y − 1 +2(div b)y, +∀y ∈ H1 +0(Ω), +skew-symmetric part: LSSy = b · ∇y + 1 +2(div b)y, +∀y ∈ H1 +0(Ω), +i.e. L = LS +LSS. Symmetric and skew-symmetric parts can be directly derived using the formulae: +(26) +LS = L + L∗ +2 +, +LSS = L − L∗ +2 +, +where L∗ is the adjoint operator related to L. +Now, let us analyze our OCP problem (6): we follow the optimize-then-discretize approach in +[14]. The discretized state equation is described as follows, where the control is distributed, i.e. it +acts on the whole domain Ω: +(27) +as +� +yN , qN � ++ bs +� +uN , qN � += Fs(qN ), +∀qN ∈ +� +QN �∗ , +with +(28) +as +� +yN , qN � +:= a +� +yN , qN � ++ +� +K∈Th +δK +� +LyN , hK +|b| LSSqN +� +K +, +(29) +bs +� +uN , qN � +:= − +� +Ω +uN qN − +� +K∈Th +δK +� +uN , hK +|b| LSSqN +� +K +, +and +(30) +Fs(qN ) := F +� +qN � ++ +� +K∈Th +δK +� +f, hK +|b| LSSqN +� +K +, +where +� +·, · +� +K indicates the usual L2(K)-product, f collects all the forcing and lifting terms, and δK +denotes a positive dimensionless stabilization parameter related to an element K ∈ Th. In principle, +since δK is local, it can be different for each K. Considering the adjoint equation, we can see that +it is also an Advection-Dominated equation, but with an advective term with opposite sign with +respect to the state one. As a matter of fact, from (26) we obtain that L∗ = LS − LSS. The SUPG +method leads to the discretized adjoint equation +(31) +a∗ +s +� +zN , pN � ++ +� +yN − yd, zN � +s = 0, +∀zN ∈ Y N , + +10 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +with +(32) +a∗ +s +� +zN , pN � +:= a∗ � +zN , pN � ++ +� +K∈Th +δa +K +� +(LS − LSS)pN , hK +|b| (−LSS) zN +� +K +, +� +yN − yd, zN � +s := +� +Ωobs +(yN − yd)zN dx + +� +K∈Th|Ωobs +δa +K +� +yN − yd, hK +|b| (−LSS) zN +� +K +, +where a∗ is the adjoint form of a and δa +K is the parameter related to the stabilized adjoint bilinear +forms. As in this work we consider δK = δa +K in numerical simulations, from now on we will always +denote both stabilization parameter with δK. Instead, the discretized gradient equation is not affected +by the SUPG and it remains untouched: +(33) +b∗� +vN , pN � ++ αn +� +uN , vN � += 0, +∀vN ∈ U N . +With this setting it follows a nonsymmetric system for the computation of the numerical solution, +but we gain the strongly-consistency of the method for the optimality system if y, u, p are regular +[14]. To summarize, the SUPG optimality system for a steady OCP is the following: +(34) +discretized adjoint equation: +a∗ +s +� +zN , pN � ++ +� +yN − yd, zN � +s = 0, +∀zN ∈ Y N , +discretized gradient equation: +b∗� +vN , pN � ++ αn +� +uN , vN � += 0, +∀vN ∈ U N , +discretized state equation: +as +� +yN , qN � ++ bs +� +uN , qN � += Fs(qN ), +∀qN ∈ +� +QN �∗ , +and, referring to (20), the discretized algebraic system reads as: +(35) +� +� +Ms +0 +KT +s +0 +αM +BT +Ks +Bs +0 +� +� +� +� +y +u +p +� +� = +� +� +Msyd +0 +fs +� +� , +where Ms is the stabilized mass matrix related to m, M is the not-stabilized mass matrix related +to n, Ks and KT +s are the stiffness matrices related to as and a∗ +s, respectively, Bs is the stabilized +mass matrix related to bs, BT is the block linked to b∗ and fs is the vector whose components are +the coefficients of the stabilized force term. Every block is derived as in (18). +We indicate with |∥ · ∥| the energy norm related to the bilinear form a belonging to Advection- +Diffusion equations (3.1.1), i.e. +(36) +|∥w∥|2 := ε∥∇w∥2 +L2(Ω) + 1 +2 +���(div b) +1 +2 w +��� +2 +L2(Ω) , +∀w ∈ H1 +0(Ω). +Therefore, we define the SUPG norm on H1 +0(Ω) as +(37) +∥w∥2 +SUP G := |∥w∥|2 + +� +K∈Th +δK +� +LSSw, hK +|b| LSSw +� +K +, +∀w ∈ H1 +0(Ω). +Considering that (38) holds true, it is immediate to see that the SUPG bilinear form (28) is coercive +with respect to the SUPG norm [42]. Finally, we can illustrate an estimate of the error for the +adjoint and the state variables of the solution of an OCP [14]. +Theorem 3.1.3 (Error for state and adjoint variables). Let m, r ≥ 1 and (y, u, p) be the solution +of (6) with y ∈ Hm+1(Ω), p ∈ Hr+1(Ω). Furthermore, let yN , uN , pN be the numerical solution of +(34). If δK satisfies +(38) +0 < δK ≤ hK +εη2 +inv +and +δK = +� +� +� +δ1 +hK +ε , +PeK(x) ≤ 1, +δ2, +PeK(x) > 1, +where δ1, δ2 > 0 are chosen constant, and ηinv is defined as the following inverse constant +|yN |1,K ≤ ηinvh−1 +K ∥yN ∥L2(K) +and +∥∆yN ∥L2(K) ≤ ηinvh−1 +K ∥∇yN ∥L2(K) +∀yN ∈ Y N , + +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +11 +with | · |1,K, ∥ · ∥K seminorm and L2-norm on K, respectively, then there exists C > 0 such that +(39) +��y − yN �� +SUP G ≤ C +� +hm � +ε1/2 + h1/2� +|y|Hm+1(Ω) + +��uN − u +�� +L2(Ω) +� +, +∀h, +��p − pN �� +SUP G ≤ C +� +hr � +ε1/2 + h1/2� +|p|Hr+1(Ω) + +��yN − y +�� +L2(Ω) +� +, +∀h. +3.2. SUPG for Time-Dependent Advection-Dominated OCP(µ)s. We briefly discuss the +SUPG technique employed with time-dependent problems. Referring to (13), the main challenge +comes from the fact that the time derivative should also enter into stabilization framework to ensure +consistency [27]. However, other approaches have been proposed: in [45], for instance, the time- +derivative is not stabilized. Nevertheless, our discussion follows works inherent to Graetz-Poiseuille +and Propagating Front in a Square problems without optimal control [37, 54], where stabilization +is used for time derivative, too. This adds nonsymmetric terms to the discretized state and adjoint +equations for time derivatives. +To the best of our knowledge SUPG for Parabolic OCPs in an +optimize-then-discretized approach is still a novelty element in literature from a theoretical point of +view. However, we refer to [17, 20, 27] for SUPG applied to general Parabolic equations. +We firstly discretize the equation in time, considering each discrete time as a steady-state Advection- +Diffusion equation, in a space-time approach, and then stabilized it with the SUPG. The time interval +(0, T) is divided in Nt sub-intervals of equal length ∆t := ti − ti−1, i ∈ {1, . . . , Nt}. On the other +hand, all terms involving time-derivative go through a time discretization equivalent to a classical +implicit Euler approach [3, 23, 46, 50, 51, 52]. The backward Euler method is used to discretize the +state equation forward in time, instead the adjoint equation is discretized backward in time using +the forward Euler method, which is equivalent to the backward Euler with respect to time T − t, for +t ∈ (0, T) [16, 50]. The global dimension of the discrete spaces is Ntot = 3 · N · Nt. We recall that +Y, U, Q are Hilbert Spaces and that Y N ≡ (QN )∗. +For the state equation, the stabilized term added to the form related to the time derivative of the +state ∂y +∂t and the bilinear form a is the following [27, 37, 54]: +s +� +yN (t), qN � += +� +K∈Th +δK +�∂yN (t) +∂t ++ (LS + LSS) yN (t), hK +|b| LSSqN +� +K +, +where yN (t) ∈ Y N for each t ∈ (0, T) and qN ∈ Y N . Instead, the stabilized term added to the form +related to the time derivative of the adjoint ∂p +∂t and the bilinear form a∗ is: +s∗ � +zN , pN (t) +� += +� +K∈Th +δK +� +−∂pN (t) +∂t ++ (LS − LSS) pN (t), −hK +|b| LSSzN +� +K +. +We can write the discretized state formulation using a backward Euler approach as follows: +(40) +for each i ∈ {1, 2, · · · , Nt}, find yN +i +∈ Y N s.t. ∀qN ∈ Y N , +1 +∆tms +� +yN +i (µ) − yN +i−1(µ), qN ; µ +� ++ as +� +yN +i (µ), qN ; µ +� ++ bs +� +uN +i , qN ; µ +� += Fs +� +qN ; µ +� +, +given the initial condition yN +0 which satisfies +(41) +� +yN +0 , qN � +L2(Ω) = +� +y0, qN � +L2(Ω) , +∀qN ∈ Y N . +The stabilized term ms above is defined as: +(42) +ms +� +yN , qN ; µ +� += +� +yN , qN � +L2(Ω) + +� +K∈Th +δK +� +yN , hK +|b| LSSqN +� +K +and it is related to the time discretization; instead, as and Fs are defined as in the steady case. +Similarly we can derive the same for the adjoint forms applying a forward Euler method: +(43) +for each i ∈ {Nt − 1, Nt − 2, ..., 1}, find pN +i +∈ Y N s.t. +1 +∆tm∗ +s +� +pN +i (µ) − pN +i+1(µ), zN ; µ +� ++ a∗ +s +� +zN , pN +i (µ); µ +� += − +� +yN +i − ydi, zN � +s +∀zN ∈ Y N . + +12 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +The stabilized term m∗ +s above is defined as: +(44) +m∗ +s +� +pN , zN ; µ +� += +� +pN , zN � +L2(Ω) − +� +K∈Th +δK +� +pN , hK +|b| LSSzN +� +K +. +Now we give a look at the discretization scheme. As in the steady case, yi ∈ Y N , ui ∈ U N and +pi ∈ Y N , for 1 ≤ i ≤ Nt, represent the column vectors including the coefficients of the FEM dis- +cretization for state, control and adjoint, respectively. Therefore, we define y = +� +yT +1 , . . . , yT +Nt +�T , +u = +� +uT +1 , . . . , uT +Nt +�T and p = +� +pT +1 , . . . , pT +Nt +�T . The vector f s = +� +f T +s1, . . . , f T +sNt +�T +indicates the com- +ponents of the stabilized forcing term, yd = +� +yT +d1, . . . , yT +dNt +�T +is the vector made of discrete time +components of our desired state solution; instead, y0 = +� +yT +0 , 0T , . . . , 0T �T indicates the vector of ini- +tial condition for the state, where 0 is the zero vector in RN . The block matrix system is described +as follows. +• State equation. +We recall that Ks and Bs are the matrices associated to the stabilized +bilinear forms as and bs. Using the backward Euler along time, one has to solve +(45) +Msyi + ∆tKsyi + ∆tBsui = Msyi−1 + fsi∆t +for i ∈ {1, 2, . . . , Nt} , +where Ms is the stabilized mass matrix relative to the FEM discretization of ms. Therefore +the related block matrix subsystem is +� +���� +Ms + ∆tKs +0 +−Ms +Ms + ∆tKs +0 +... +... +0 +0 +−Ms +Ms + ∆tKs +� +���� +� +�� +� +As +y+∆t +� +�� +Bs +0 +0 +... +0 +0 +Bs +� +�� +� +�� +� +Cs +u = Msy0 + ∆tf s, +where Ms is a block diagonal matrix in RN ·Nt ×RN ·Nt whose element on the main diagonal +are [Ms, . . . , Ms]. Then everything can be recast in a more compact form as +(46) +Asy+∆tCsu = Msy0 + ∆tf s. +• Gradient equation. We recall that BT indicates the mass matrix related to the b∗ form and +hence at every time step we have to solve the equation +(47) +α∆tMui+∆tBT pi = 0, +∀i ∈ {1, 2, . . . , Nt} , +which translates into the following block system: +∆t · α +� +���� +M +M +... +... +M +� +���� +� +�� +� +M +� +���� +u1 +u2 +... +uNt +� +���� +∆t +� +���� +BT +0 +· · · +BT +... +BT +� +���� +� +�� +� +CT +� +���� +p1 +p2 +... +pNt +� +���� = +� +���� +0 +0 +... +0 +� +���� . +In a vector notation we have +(48) +α∆tMu+∆tCT p = 0. +• Adjoint equation: we have to solve the first equation of the optimality system (6) at each +time step as follows, considering M T +s the matrix formulation of m∗ +s: +M T +s pi = M T +s pi+1 + ∆t +� +−M T +s yi − KT +s pi + M T +s ydi +� +for i ∈ {Nt − 1, Nt − 2, . . . , 1} . + +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +13 +As did in previous steps, we derive the following block system: +� +���� +M T +s + ∆tKT +s +−M T +s +... +... +M T +s + ∆tKT +s +−M T +s +M T +s + ∆tKT +s +� +���� +� +�� +� +AT +s +p + +� +����� +∆tM T +s y1 +... +... +∆tM T +s yNt +� +����� += +� +����� +∆tM T +s yd1 +... +... +∆tM T +s ydNt +� +����� +. +Then, defining MT +s as the diagonal matrix in RN ·Nt × RN ·Nt which diagonal entries are +[M T +s , . . . , M T +s ], the adjoint system to be solved is: +∆tMT +s y + AT +s p = ∆tMT +s yd. +In the end, we solve system (49) via an one shot approach: +(49) +� +� +∆tMT +s +0 +AT +s +0 +α∆tM +∆tCT +As +∆tCs +0 +� +� +� +� +y +u +p +� +� = +� +� +∆tMT +s yd +0 +Msy0 + ∆tf s +� +� . +4. ROMs for advection-dominated OCP(µ)s +FEM simulations can be expensive in terms of computational time and memory storage: this issue +is obviously more evident in case of high-dimensional discrete spaces. Moreover, when we talk about +parametrized PDEs, one can require to repeat the simulations for several values of the parameter µ. +To overcome these difficulties, we will use ROMs approach. The basic idea of ROMs is to create a +low-dimensional space, called the reduced space, exploiting the parameter dependence of the problem +at hand, such that it is a good approximation of the discrete initial space [8, 22, 41, 40, 39]. Let us +consider a generic Parametrized OCPs described by the optimality conditions (6). We can define +the set of the parametric solutions of the optimality system with respect to the functional space +W = Y × U × Q∗ for steady OCP(µ)s and W = Yt × U × Yt for the unsteady ones as +(50) +M := {(y(µ), u(µ), p(µ)) solution of (6) | µ ∈ P}. +The extension to space-time formulation for time-dependent problem is straightforward [6, 50] and +requires small modifications, thus, we will exclusively refer to the steady framework. +Assumption 4.0.1 (Smoothness of the solution manifold). The continuous solution manifold M is +smooth with respect to the parameter µ ∈ P. +Let WN ⊂ W be our FEM approximation of the continuous space W, we call WN := Y N × +U N × +� +QN �∗ the high-fidelity space. Then, for stabilized problems we define the discrete parametric +solution manifold as +(51) +MN := +�� +yN (µ), uN (µ), pN (µ) +� +FEM solution of the (35) | µ ∈ P +� +. +Starting from MN , ROM techniques create a reduced space of low dimension N denoted with +WN, via a linear combination of snapshots, i.e. high-fidelity evaluations of the optimal solution +� +yN (µ), uN (µ), pN (µ) +� +computed in properly chosen parameters values µ. Obviously we have that +WN ⊂ WN and we denote WN = Y N × U N × +� +QN�∗. Here, Y N, U N and (QN)∗ are the reduced +spaces for the state, the control and the adjoint variables, respectively. The snapshots are collected by +a POD algorithm using a partitioned approach. This strategy is followed due to good results shown +in literature [28, 35, 49]. After having built these reduced function spaces, a standard Galerkin +projection is performed onto these ones [5, 38, 42]. + +14 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +4.1. Offline-Online Procedure for ROMs. ROM procedure is divided in two stages: +• offline phase: here the snapshots are collected by solving the high-fidelity system (35). +Secondly, the low-dimensional bases are created and hence all reduced spaces Y N, U N and +(QN)∗ are built and stored, too. Moreover, all the µ-independent quantities are assembled +and stored. It is potentially an expensive phase, which depends on N. +• online phase: here a parameter µ is chosen and all the previous store µ-independent quan- +tities are combined with the just-computed µ-dependent ones to build the reduced block +matrix system based on a Galerkin projection. +To be convenient, this phase should be +N-independent. Whereas in the offline phase stabilization is present due to stabilized snap- +shots, for the online phase this cannot be necessary. Therefore, we have two possibilities: if +stabilization is performed also here, we talk about Online-Offline stabilization, otherwise we +denote the setting as Only-Offline stabilization. +As already said, the online phase should be performed in a number of operations independent +of N. A sufficient condition is to admit the separation of the variables depending on µ and the +solution (y, u, p) in the affine decomposition [22]. +Assumption 4.1.1. We require that all the forms in (35) are affine in µ ∈ P. +In Section 4.2 we describe the POD algorithm used in the offline phase. Now, we illustrate the +explicit expression of the reduced solutions. Let us make clear the structure of the three reduced +spaces in terms of their bases. Therefore, we define +(52) +Y N = span {ηy +n, n = 1, . . . , N}, +U N = span {ηu +n, n = 1, . . . , N} , +(QN)∗ = span {ηp +n, n = 1, . . . , N} , +the reduced state, the reduced control and the reduced adjoint space, respectively. After having +built them, we consider an enriched space for state and adjoint variables. Therefore, let us denote +with {τn}2N +n=1 = {ηy +n}N +n=1 ∪ {ηp +n}N +n=1 the basis functions for the space ZN, with ZN ≡ Y N ≡ (QN)∗, +then we have ZN = span {τn, n = 1, . . . , 2N} [15, 19, 28, 29, 36, 35]. +4.2. Proper Orthogonal Decomposition. In this Section we briefly describe the Proper Orthog- +onal Decomposition (POD) Galerkin algorithm [6, 22, 49, 50] for the construction of a discrete +solution manifold and the relative reduced spaces. Since in the unsteady case we use a space-time +structure, this procedure can be described making no distinction between time-dependency and +steadiness. Firstly, we make a sampling of P by choosing Ntrain of its elements. Therefore, let us +define the set of the train samples as PNtrain: we have that obviously PNtrain ⊂ P and the cardinality +is |PNtrain| = Ntrain. The set PNtrain is denoted as the training set. We should pursue that Ntrain +is large enough so as to ensure that PNtrain is a good “approximation” of the parameter space P. +PNtrain is built through a Monte-Carlo sampling method with respect to a uniform density with +support equal to P. +Starting from the sampling, the POD algorithm manipulates Ntrain snapshots for the state, the +adjoint and the control variables: +(53) +�� +yN (µj), uN (µj), pN (µj) +��Ntrain +j=1 +with µj ∈ PNtrain. +After this step, a compressing stage is performed: from (53) we build N basis functions by only +considering the most important parametric information and throwing away the redundant ones, with +N ≤ Ntrain. A partitioned approach is used, which means that, after the deterministic sampling, + +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +15 +we perform the POD algorithm separately for all the three variables. Namely, we find three N- +dimensional reduced spaces Y N, U N and (QN)∗ that minimizes the following three quantities: +� +� +� +� +1 +Ntrain +� +µj∈PNtrain +min +¯y∈Y N +��yN � +µj +� +− ¯y +��2 +Y , +� +� +� +� +1 +Ntrain +� +µj∈PNtrain +min +¯u∈U N +��uN � +µj +� +− ¯u +��2 +U, +� +� +� +� +1 +Ntrain +� +µj∈PNtrain +min +¯p∈(QN)∗ +��pN � +µj +� +− ¯p +��2 +Q∗, +where obviously Y N ⊂ Y N , U N ⊂ U N and (QN)∗ ⊂ (QN )∗. +Let us discuss the data compression procedure of the POD for the state variable y(µ) [6, 22, 49, +50]. As we are following a partitioned approach, the control and the adjoint variables follow the +below discussion with usual modifications, as well. Firstly we collect a set of ordered parameters +µ1, . . . , µNtrain ∈ PNtrain, which the ordered snapshots yN (µ1) , . . . , yN � +µNtrain +� +are linked to. Let +us define Cy ∈ RNtrain×Ntrain as the correlation matrix of the snapshots for the state variable as +follows: +(54) +Cy +ij := +1 +Ntrain +� +yN (µi) , yN � +µj +�� +Y , +1 ≤ i, j ≤ Ntrain. +The next step is to find the pair eigenvalue-eigenvector (λy +n, ey +n), where ey +n has norm equal to one, of +the following problem: +Cyey +n = λy +ney +n, +1 ≤ n ≤ Ntrain. +For the sake of simplicity, we organise the eigenvalues λy +1, . . . , λy +Ntrain in decreasing order. Consider +the first N ones, specifically λy +1, . . . , λy +N together with the related eigenvectors ey +1, . . . , ey +N. We refer +to the k-th component of the state eigenvector ey +n ∈ RNtrain with the notation (ey +n)k. After having +finished the computation of the pair eigenvalue-eigenvector, the basis functions ηy +n for the state +equation are built through the following formula: +(55) +ηy +n = +1 +√ +λy +n +Ntrain +� +k=1 +(ey +n)k yN (µk) , +1 ≤ n ≤ N. +Therefore, our reduced spaces are built as (52) and, then, aggregated space technique is applied. +As both N and Ntrain can be chosen by us, we should find sharp criteria in order to decide them. +A possibility can be to set them in based on a study of the eigenvalues, using the estimate [22, 39, 58]: +(56) +� +� +� +� +1 +Ntrain +Ntrain +� +k=1 +∥yN (µk) − ΠN (yN (µk))∥2 +Y = +� +� +� +� +Ntrain +� +k=N+1 +λy +k, +where ΠN : Y → Y N is a Galerkin projector of functions from Y onto Y N. (56) holds for the +control and the adjoint in a partitioned approach, too. The second member of equation (56) can be +a measure of how well the FEM space is approximated by N reduced basis over the chosen training +set of cardinality Ntrain. We summarise the whole POD procedure in the below Algorithm 1. +Remark 4.2.1 (Time-dependent problems). When we are dealing with time-dependent OCPs, the +time instances are not separated in the POD procedure. Therefore, the space-time problem is studied +as a steady one and each snapshot carries all the time instances. + +16 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +Algorithm 1 POD algorithm for OCP problems in a partitioned approach +Input: parameter domain P, FEM spaces Y N , U N and (QN )∗ and Ntrain. +Output: reduced spaces Y N, U N and (QN)∗. +Starting from the high-fidelity spaces Y N , U N and (QN )∗: +1: Sample Ptrain ⊂ P; +2: for all µ ∈ Ptrain do +3: +Solve the high-fidelity OCP system (34) (in this case a stabilized one); +4: end for +5: Assemble the matrix Cy +ij := +1 +Ntrain +� +yN (µi) , yN � +µj +�� +Y , 1 ≤ i, j ≤ Ntrain. Do the same for u +and p variables; +6: Compute its eigenvalues λy +1, . . . , λy +Ntrain and the corresponding orthonormalised eigenvectors +ey +1, . . . , ey +Ntrain. Do the same for u and p variables; +7: After having chosen N according to a certain criterion, define Y N = span {ηy +n, n = 1, . . . , N}, +where ηy +n = +1 +√ +λy +n +�Ntrain +k=1 +(ey +n)k yN (µk). Do the same for u and p variables. +8: Define the aggregated space ZN = span +� +{ηy +n}N +n=1 ∪ {ηp +n}N +n=1 +� +and impose ZN ≡ Y N ≡ (QN)∗. +5. Numerical Results +In this Section we propose simulations regarding the Graetz-Poiseuille and the Propagating Front +in a Square problems. Regarding the steady case, the numerical experiments are coded through +the RBniCS library [2]; instead, the unsteady ones are implemented employing both RBniCS and +multiphenics [1] libraries. They are python-based libraries, built on FEniCS [32]. +When we will perform the Online-Offline stabilization procedure, we will always use the same +stabilization parameter δK of the high-fidelity approximation also at the reduced level, both in +steady and unsteady cases. +We will illustrate an analysis over relative errors between the FEM and the reduced solutions for +all three variables, defined as +(57) +ey,N(µ) := +��yN (µ) − yN(µ) +�� +Y +∥yN (µ)∥Y +, eu,N(µ) := +��uN (µ) − uN(µ) +�� +U +∥uN (µ)∥U +, ep,N(µ) := +��pN (µ) − pN(µ) +�� +Q∗ +∥pN (µ)∥Q∗ +, +for the state, the control and the adjoint, respectively. As we are dealing with parametrized OCPs, +we will evaluate a simple average of (57) for µ uniformly distributed in a testing set Ptest ⊆ P of size +Ntest for every dimension N = 1, . . . , Nmax of the reduced space obtained by our POD procedure. +More precisely, we will plot the base-10 logarithm of the average of (57). For parabolic problems we +will consider the sum of the errors with respect to each discretized instant of time t. +Regarding the efficiency of ROMs, we use the speedup-index to compare the computational cost +of the FEM solution with that of the reduced one. This quantity is defined as: +(58) +speedup-index = computational time of the high-fidelity solution +computational time of the reduced solution +, +which will be computed for each µ in the testing set with respect to the dimension N of the reduced +spaces. As made with the relative error, we will consider the sample average of this quantity with +respect to N; however, for the sake of completeness, we will add its minumum and maximum value +computed through the testing set. For each test case, we will use the same Ptest to compute relative +errors and the speedup-index. The steady results are obtained with 16GB of RAM and Intel Core +i7-7500U Dual Core, 2.7GHz for the CPU; instead, the FEM and ROM parabolic simulations are +run with 16GB of RAM and Intel Core i7 − 7700 Quad Core, 3.60GHz for the CPU. +5.1. Numerical Experiments for the Graetz-Poiseuille Problem. The Graetz-Poiseuille prob- +lem concerns the heat conduction in a straight duct, whose walls can be characterized by heat + +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +17 +exchange or maintained at a certain fixed temperature. This example is very well-known in the nu- +merical Advection-Dominated literature [18, 37, 44, 55]. We start by presenting the stationary case. +We apply a distributed control in the whole domain and the parameter µ = µ1 > 0 is a physical +component and characterizes the diffusion term. The spatial coordinates of the system are denoted +Ωobs +Ωobs +Ω +Γ1 +Γ2 +Γ3 +Γ4 +Γ5 +Γ6 +(0,0) +(1,0) +(2,0) +(2,0.2) +(2,0.8) +(2,1) +(1,1) +(0,1) +Figure 1. Geometry of the Graetz-Poiseuille Problem. +with (x0, x1). The boundary of Ω is Γ. We consider Dirichlet boundary conditions (BC) on sides +Γ1 := [0, 1] × {0}, Γ5 := [0, 1] × {1}, Γ6 := {0} × [0, 1] by imposing y = 0 and Γ2 := [1, 2] × {0} and +Γ4 := [1, 2] × {1} by imposing y = 1, referring to Figure 1. We deal with homogeneous Neumann +conditions on Γ3 := {2} × [0, 1]. The classic formulation of the problem is: +(59) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +− 1 +µ1 +∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u, +in Ω, +y(µ) = 0, +on Γ1 ∪ Γ5 ∪ Γ6, +y(µ) = 1, +on Γ2 ∪ Γ4, +∂y(µ) +∂ν += 0, +on Γ3. +Now we want to derive the optimality system. Ωobs := [1, 2]×[0.8, 1]∪[1, 2]×[0, 0.2] as illustrated +in Figure 1. In this case, the state belongs to the space: +˜Y := +� +v ∈ H1� +Ω +� +s.t. it satisfies the BC in (59) +� +. +For the sake of practice, it is better to introduce a lifting function Ry ∈ H1(Ω), such that it fulfills +the BC in (59). Therefore we define the variable ¯y := y − Ry, with ¯y ∈ Y , where +Y := +� +v ∈ H1 +0 +� +Ω +� +s.t. ∂¯y +∂ν = 0, on Γ3 and ¯y = 0 on Γ \ Γ3 +� +. +Nevertheless, without loss of generality, we will denote the new variable ¯y with y and we settle +U := L2(Ω) and Q := Y ∗. +Therefore, the adjoint variable p is null on Γ \ Γ3. The mathematical +formulation is described as follows (we omitted the dependence from µ). Fixed α > 0, find the pair +(y, u) ∈ Y × U that realizes +(60) +min +(y,u)∈Y ×U J(y, u) = 1 +2 +� +Ωobs +� +y − yd +�2 dx + α +2 +� +Ω +u2 dx +such that e (y, u, p; µ) = 0, +where e (y, u, p; µ) := a (y, p; µ)+b (u, p; µ)−⟨p, f(µ)⟩Y ∗Y . As explained in Sections 2 and 3, we follow +a Lagrangian approach and we use SUPG stabilization in a optimize-then-discretize framework. We +exploit P1-FEM approximation for the state, control and adjoint spaces. Here the stabilized forms +as and a∗ +s are, respectively: +as +� +yN , qN ; µ +� +:= a +� +yN , qN ; µ +� ++ +� +K∈Th +δK +� +K +� +4x1(1 − x1)∂x0yN � � +hK∂x0qN � +, +yN , qN ∈ Y N , +a∗ +s +� +zN , pN ; µ +� +:= a∗ � +zN , pN ; µ +� ++ +� +K∈Th +δK +� +K +� +4x1(1 − x1)∂x0pN � � +hK∂x0zN � +, +zN , pN ∈ Y N . + +18 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +We consider a parameter space P := +� +104, 106� +and a quite coarse mesh of size h = 0.029 for the +FEM spaces. The training set Ptrain has cardinality Ntrain = 100. We choose δK = 1.0 for all +K ∈ Th and the penalization term is α = 0.01. We pursue the convergence in the L2-norm of the +state to the desired solution profile yd(x) = 1.0, function defined on Ωobs of Figure 1. We perform +the POD algorithm for Nmax = 20 in a partitioned approach. We illustrate the reduced solution for +the state and adjoint variables in the best relative error scenario in Figure 2. Namely, we plot the +Only-Offline and Online-Offline Stabilized solutions for N = 1 and N = 6. The values of N can be +deduced by Figure 3. From the gradient equation (34), we expect the distributed control u to be +equal to the adjoint p up to the multiplicative constant α. +Figure 2. (Top) Only-Offline stabilized state (left) and adjoint (right) for N = 1; +(Bottom) Online-Offline stabilized state (left) and adjoint (right) for N = 6; for the +Graetz-Poiseuille Problem; P = +� +104, 106� +, µ1 = 105, h = 0.029, δK = 1.0, α = 0.01. +We consider the relative errors between the FEM and the reduced solution in Figure 3. +We +use a testing set Ptest of 100 elements in P. As previously cited, at N = 6 we reach the minima +for all the three errors for the Online-Offline stabilization; more precisely for the state we touch +ey,6 = 9.65 · 10−9, for the adjoint ep,6 = 1.98 · 10−8 and the control eu,6 = 6.00 · 10−9. In contrast +with this situation, Only-Offline stabilization never falls under 10−2. This implies that the best +choice is to pursue the Online-Offline stabilization procedure for this problem. However, after N = 6 +the errors begin slightly to increase. Our interpretation to this fact relies on P. Despite the fact +that this parameter space might be too large, however the coefficient which multiplies the diffusion +operator is still absolutely low in value for every µ1 ∈ P, nearly 10−4 and 10−5. Therefore, also +thanks to SUPG stabilization and the distributed control action, the majority of snapshots can be +very similar referring to the solution for µ1 = 105: this translates in very few bases to reach a good +relative error. As a matter of fact, the eigenvalues λy +7, λu +7 are ≈ 10−15 and λp +7 ≈ 10−16; by their +decreasing order, all the subsequent eigenvalues are very close to zero machine. Thus, recalling (55) +it follows that all basis components with N ≥ 7 are affected by some rounding errors due to the +orthonormalization procedure of the POD (for details see [39]). +Finally, we take a look at the speedup-index in Table 1. All the average values are better for +the Only-Offline stabilized ROM procedure due to the fact that the stabilized forms are not taken +into account in the online phase. However, the Online-Offline stabilized reduced solution shows very +good behaviour, for instance, we have an average equal to 284.3 for N = 6. Generally, in this case +speedup-index takes average value around 2 · 102 order of magnitude for the first 20 basis elements. + +-3.9e-02 +0.2 +0.4 +0.6 +0.8 +1 1.1e+00-2.1e-03 +0 +0.0020.004.0.0060.0080.010.012 +1.5e-020.0e+00 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.91.0e+002.2e-03 +0.002 0.0040.0060.0080.01 0.012 +1.6e-02SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +19 +Figure 3. Relative errors between FEM and reduced solution for state (left), control +(center) and adjoint (right), for Online-Offline and Only-Offline stabilization, α = 0.01, +Ntest = 100, h = 0.029, P = +� +104, 106� +. Graetz-Poiseuille Problem. +Only-Offline Stabilization +Offline-Online Stabilization +N +min +average +max +min +average +max +1 +162.1 +296.6 +338.1 +170.8 +261.7 +285.9 +2 +172.2 +342.1 +391.3 +178.4 +298.5 +327.1 +3 +168.5 +336.2 +383.7 +192.0 +298.9 +325.3 +4 +165.1 +336.1 +385.6 +256, 7 +298.0 +322.6 +5 +164.8 +331.6 +376.6 +220.1 +287.6 +307.7 +6 +198.3 +321.0 +366.4 +192.3 +284.3 +305.7 +7 +186.1 +318.4 +348.6 +228.6 +282.6 +306.9 +Table 1. Speedup-index of the Graetz-Poiseuille Problem for Online-Offline and Only- +Offline stabilizations with Ptest sampled from P = +� +104, 106], Ntest = 100, α = 0.01, +h = 0.029, δK = 1.0, α = 0.01. +Let us give a look to the unsteady version of Problem (59) with null initial condition. +The +unsteady Graetz-Poiseuille problem without control has been presented in [37, 55], instead the OCP +Graetz Problem under boundary control without Advection-dominancy is studied in [50, 48]. +Recalling Figure 1, for a fixed T > 0 we state the parabolic Graetz-Poiseuille Problem as follows: +(61) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∂y(µ) +∂t +− 1 +µ1 +∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u, +in Ω × (0, T), +y(µ) = 0, +on Γ1 ∪ Γ5 ∪ Γ6 × (0, T), +y(µ) = 1, +on Γ2 ∪ Γ4 × (0, T), +∂y(µ) +∂ν += 0, +on Γ3 × (0, T), +y(µ)(0) = 0, +in Ω. +We do simulations in a space-time framework as discussed in Section 3.2 for a prearranged number of +time-steps Nt using a P1-FEM approximation for the high-fidelity solutions. The relative stabilized +forms in (49) for derivatives along time for state and adjoint are, respectively: +(62) +ms +� +yN , qN ; µ +� += +� +yN , qN � +L2(Ω) + +� +K∈Th +δKhK +� +yN , ∂x0qN � +K , +yN , qN ∈ Y N , +m∗ +s +� +pN , zN ; µ +� += +� +pN , zN � +L2(Ω) − +� +K∈Th +δKhK +� +pN , ∂x0zN � +K , +pN , zN ∈ Y N . +We consider a final time of T = 3.0 and a time step of ∆t = 0.1, hence we have Nt = 30. We choose +a quite coarse mesh of h = 0.038 and the overall high-fidelity dimension is Ntot = 314820. This +means that a single FEM space for a fixed t has a dimension of N = 3498. We consider a initial +condition of y0(x) = 0 for all x ∈ Ω referring to Figure 1. We want the state solution to converge + +FEM vs ROM averaged relative error - y (state) +Online stab +101 +Online not stab. +Log-Error +10-1 +10- +Relative L +10-5 +10-7 +1 +2 +3 +4 +5 +6 +7 +8 +NFEM vs ROM averaged relative error - u (control) +101 +Online stab +Online not stab +10-1 +10-5 +10-7 +1 +2 +3 +4 +5 +6 +7 +8 +NFEM vs ROM averaged relative error - p (adjoint) +Online stab. +Online not stab. +101 +10 +10-3 +10-5 +10-7 +1 +2 +3 +4 +5 +6 +7 +8 +N20 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +in the L2-norm to a desired solution profile yd(x, t) = 1.0, function defined for all t ∈ [0, 3.0] and +for all x in Ωobs. Here the SUPG stabilization is implemented with parameters δK = 1.0 for all +K ∈ Th. P := +� +104, 106� +and we choose a training set Ptrain of cardinality Ntrain = 100. Then, we +performed the POD algorithm with Nmax = 20. The penalization parameter is α = 0.01. +Figure 4. (Top) SUPG FEM solution for the state and (Bottom) for the adjoint at +t = 0.1, t = 1.5, t = 3.0. +Unsteady Graetz-Poiseuille Problem, µ1 = 105, Nt = 30, +h = 0.038, δK = 1.0, α = 0.01. +As we will see in Figure 5 the performance of the Only-Offline stabilized reduced solutions are not +so good in terms of accuracy, unlike the Online-Offline stabilized ones. We consider a testing set of +100 elements in P. As succeeded in the steady case in Section 5.1, after nearly N = 6 Online-Offline +stabilized errors begin to fluctuate due to the nature of the eigenvalues of the correlation matrix +(54) that are closed to zero machine. For this reason we present the trend of error from 1 to 10. +However, errors stays close to 10−7 for the state and the adjoint and 10−6 to the control. For N = 6 +we have ey,6 = 4.20 · 10−7, eu,6 = 1.10 · 10−6 and ep,6 = 3.18 · 10−7, instead for N = 20 we have +ey,20 = 1.93 · 10−7, eu,20 = 3.25 · 10−7 and ep,20 = 1.21 · 10−7 for the Online-Offline stabilization +ROM. +Figure 5. Relative errors between the FEM and Only-Offline and Online-Offline stabi- +lized solutions for the state (left), control (center) and adjoint (right), Unsteady Graetz- +Poiseuille problem, Nt = 30, Ntest = 100, P = +� +104, 106� +, h = 0.038. +Finally, we can see the speedup-index for some value of N in Table 2. In both situation we can +compute a huge number of reduced solutions in the time of a high-fidelity one: for the Offline-Online +stabilization we have an average speedup-index of nearly 26000 for N = 6. On the whole, average +speedup-index has an order of magnitude of 2 · 104 for N ≤ 20. +5.2. Numerical Experiments for Propagating Front in a Square Problem. In this Section, +we consider a problem studied in the Advection-Dominated form in [37, 55] from a numerical point +of view and we will add a distributed control to it. Let Ω be the unit square in R2. We consider +the representation in Figure 6. Also in this case, (x0, x1) are the coordinates of the square domain. + +1.0e+00 +0.8 +0.6 +0.4 +0.2 +-7.2e-021.2e-02 +0.01 +0.008 +0.006 +0.004 +0.002 +0 +-1.5e-03FEM vs ROM averaged relative error - y (state) +Online stab. +Online not stab. +100 +Relative Log-Error +10-2 +10-6 +1 +2 +3 +4 +5 +6 +7 +8 +NFEM vs ROM averaged relative error - u (control) +Online stab +Online not stab. +101 +Relative Log-Error +10-1 +10-3 +10-5 +1 +2 +3 +4 +5 +6 +7 +8 +NFEM vs ROM averaged relative error - p (adjoint) +Online stab. +Online not stab. +100 +Relative Log-Error +10-2 +10-4 +10-6 +1 +2 +3 +4 +5 +6 +7 +8 +NSUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +21 +Only-Offline Stabilization +Offline-Online Stabilization +N +min +average +max +min +average +max +1 +21588.3 +26588.8 +30971.5 +18968.4 +23588.0 +27062.7 +2 +23821.3 +29723.4 +34817.2 +20757.2 +26018.9 +29929.1 +3 +23571.0 +29468.6 +34349.5 +20547.9 +25698.2 +29662.5 +4 +23062.2 +28880.6 +33702.7 +21385.2 +25380.9 +28883.3 +5 +25762.9 +28767.9 +33488.8 +23021.5 +25882.4 +29388.9 +6 +27003.2 +29707.7 +34544.7 +23236.5 +26054.7 +29677.5 +7 +26658.5 +29481.1 +34277.3 +23206.6 +25879.5 +29505.4 +Table 2. Speedup-index of the Unsteady Graetz Problem for Online-Offline and Only- +Offline stabilization with P = +� +104, 106], α = 0.01, Nt = 30, Ntest = 100, h = 0.038. +Γ1 +Γ2 +Γ3 +Γ4 +Γ5 +Ω +Ωobs +(0,0.25) +(0,1) +(1,0.75) +(1,1) +(0.25,1) +(1,0) +(0,0) +Figure 6. Geometry of the Square Problem +Referring to Figure 6, Γ1 := {0} × [0, 0.25], Γ2 := [0, 1] × {0}, Γ3 := {1} × [0, 1], Γ4 := [0, 1] × {1}, +Γ5 := {0} × [0.25, 1]; Ωobs := [0.25, 1] × [0.75, 1]. Given µ = (µ1, µ2), the problem is formulated as +(63) +� +� +� +� +� +� +� +− 1 +µ1 +∆y(µ) + (cos µ2, sin µ2) · ∇y(µ) = u, +in Ω, +y(µ) = 1, +on Γ1 ∪ Γ2, +y(µ) = 0, +on Γ3 ∪ Γ4 ∪ Γ5. +We assume that the Identity restricted to Ωobs as the Observation operator and Z := L2(Ωobs). In +our test cases, P := +� +104, 105� +× +� +0, 1.57 +� +. In this case, we have that the domain of definition of our +state y is +˜Y := +� +v ∈ H1� +Ω +� +s.t. BC in (63) +� +. +Exactly as done in the previous paragraph, we define a lifting function Ry ∈ H1� +Ω +� +such that +satisfies BC in (63). We define ¯y := y − Ry, even though we denote ¯y as y again for the sake of +notation. We consider Y := H1 +0(Ω), U = L2(Ω) and Q := Y ∗, hence the adjoint p is such that p = 0 +on ∂Ω. We define the objective functional J exactly as in (60); instead, a and b are +a (y, p; µ) := +� +Ω +1 +µ1 +∇y · ∇p + (cos µ2, sin µ2) · ∇yp dx, and b (u, p; µ) := − +� +Ω +up dx. +and ⟨p, f(µ)⟩Y ∗Y = −a (Ry, p; µ) . Then we follow usual discussions of Sections 2 and 3. +We exploit a P1-FEM approximation for the optimality system by using the usual SUPG stabi- +lization technique, arriving to system (34). Here, for the sake of completeness, we remark that the + +22 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +stabilized forms as and a∗ +s are, respectively: +as +� +yN , qN ; µ +� +:= a +� +yN , qN ; µ +� ++ +� +K∈Th +δK +� +K +hK (cos µ2, sin µ2) · ∇yN (cos µ2, sin µ2) · ∇qN , +a∗ +s +� +zN , pN ; µ +� +:= a∗ � +zN , pN ; µ +� ++ +� +K∈Th +δK +� +K +hK (cos µ2, sin µ2) · ∇pN (cos µ2, sin µ2) · ∇zN , +for all yN , qN , zN , pN ∈ Y N . +As previously done, we build a training set Ptrain and a testing +set Ptest with both cardinality ntrain = 100. The mesh size h is 0.025 and therefore the overall +dimension of the high-fidelity approximation is 12087, which implies that state, control and adjoint +spaces have dimension equal N = 4029. The SUPG stabilization is implemented with parameters +δK = 1.0 for all K ∈ Th. The penalization parameter is α = 0.01 and we pursue the state solution +to be convergent in the L2-norm to a desired solution profile yd(x) = 0.5, defined for all x in Ωobs of +Figure 6. In Figure 7 we observe state and adjoint FEM solutions for µ = (2 · 104, 1.2). Instead, in +Figure 8 we illustrate Only-Offline and Online-Offline reduced solution for the state and the adjoint +variable with µ = (2 · 104, 1.2) for N = 50. +Figure 7. Numerical solution without stabilization and SUPG FEM solution with µ = +(2 · 104, 1.2) for state (left) and adjoint (right) variables in the Propagating Front in a +Square Problem, α = 0.01, h = 0.025, δK = 1.0. +Figure 8. Only-Offline stabilized and Online-Offline stabilized reduced solutions with +µ = (2 · 104, 1.2) or state (left) and adjoint (right) variables in the Propagating Front in a +Square Problem, α = 0.01, N = 50, h = 0.025, δK = 1.0, P = +� +104, 105] × +� +0, 1.57]. +These computational evidences and the analysis of the relative errors show that Online-Offline +stabilization procedure is preferable in this setting. In Figure 9, the trend is the same of all three +variables, where errors continue to decrease along all N: we have ey,50 = 1.20·10−3, eu,50 = 7.67·10−4 +and ep,50 = 3.16 · 10−3. +Concerning the speedup-index, the performance are quite good as seen in Table 3. For the best +approximation, we have that we can compute an average of 44 Online-Offline reduced solutions when +we build the associated FEM one. Obviously, the Only-Offline stabilized one is slightly better. On +the whole, speedup-index takes average value around 101, 102 order of magnitude for N ≤ 50. + +1.2e+00 +1.1 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +-1.8e-029.7e-03 +0.008 +- 0.006 +- 0.004 +0.002 +0 +-0.002 +-0.004 +-0.006 +8.4e-031.2e+00 +1.1 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +-1.8e-029.7e-03 +0.008 +0.006 +0.004 +0.002 +0 +-0.002 +-0.004 +-0.006 +8.4e-03SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +23 +Figure 9. Relative errors between FEM and reduced solutions with P = +� +104, 105] × +� +0, 1.57] for the state (left), control (center) and adjoint (right) in the Propagating Front +in a Square Problem, Ntest = 100, α = 0.01, h = 0.025, δK = 1.0. +Only-Offline Stabilization +Offline-Online Stabilization +N +min +average +max +min +average +max +1 +123.1 +198.8 +243.1 +110.0 +162.7 +181.9 +10 +132.2 +200.4 +244.2 +110.8 +158.3 +176.9 +20 +84.6 +158.3 +191.7 +60.1 +124.3 +141.3 +30 +78.8 +114.7 +137.8 +65.0 +92.5 +104.7 +40 +54.2 +78.6 +96.1 +46.9 +64.2 +72.3 +50 +33.2 +53.0 +64.8 +28.5 +44.0 +49.9 +Table 3. Speedup-index of the Propagating Front in a Square Problem for Online-Offline +and Only-Offline stabilization with training set P = +� +104, 105]× +� +0, 1.57], α = 0.01, Ntest = +100, h = 0.025, δK = 1.0. +Now we study the unsteady case of the Propagating Front in a Square Problem for a fix T > 0: +(64) +� +� +� +� +� +� +� +� +� +� +� +� +� +∂y(µ) +∂t +− 1 +µ1 +∆y(µ) + (cos µ2, sin µ2) · ∇y(µ) = u, +in Ω × (0, T), +y(µ) = 1, +on Γ1 ∪ Γ2 × (0, T), +y(µ) = 0, +on Γ3 ∪ Γ4 ∪ Γ5 × (0, T), +y(µ)(0) = 0, +in Ω, +with initial value y0(x) = 0 for all x ∈ Ω referring to the domain in Figure 6. We build a parabolic +problem for a final time T = 3.0 and a time-step ∆t = 0.1, hence Nt = 30. We choose a quite +coarse mesh of size h = 0.036 and the overall dimension of the space-time system is Ntot = 174780, +which means that a single FEM space has dimension N = 1942. Again, our aim is to achieve in a +L2-mean a desired solution profile yd(x, t) = 0.5, defined for all t ∈ [0, 3] and x in Ωobs of Figure 6. +The penalization parameter is α = 0.01. We set δK = 1.0 for all K ∈ Th. Here we have that the +stabilized forms in (49) for derivatives along time for state and adjoint equations are, respectively: +ms +� +yN , qN ; µ +� += +� +yN , qN � +L2(Ω) + +� +K∈Th +δKhK +� +yN , (cos µ2, sin µ2) · ∇qN � +K , +yN , qN ∈ Y N , +m∗ +s +� +pN , zN ; µ +� += +� +pN , zN � +L2(Ω) − +� +K∈Th +δKhK +� +pN , (cos µ2, sin µ2) · ∇zN � +K , +pN , zN ∈ Y N . +We consider a parameter space equal to the steady case, i.e. P := +� +104, 105� +× +� +0, 1.57 +� +. Our training +set has cardinality Ntrain = 100. In Figure 10 we show a representative stabilized FEM solution for +µ = (2·104, 1.2) for some instants of time. We choose to perform a POD procedure with Nmax = 30. +In Figure 11 one can see the relative errors of the three variables. As previously said, Only- +Offline procedure has not good error behaviour. Instead, it is worth to note that in a Online-Offline + +FEM vs ROM averaged relative error - y (state) +100 +10-2 +Online stab +Online not stab. +10-3 +10 +20 +30 +40 +50 +NFEM vs ROM averaged relative error - u (contro +100 +Relative Log-Error +10-2 +Online stab. +10-3. +Online not stab. +10 +20 +30 +40 +50 +NFEM vs ROM averaged relative error - p (adjoint) +Online stab. + Online not stab. +101 +100 +10-1 +10-2 +10 +20 +30 +40 +50 +N24 +SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +Figure 10. (Top) SUPG FEM state solution and (Bottom) SUPG FEM adjoint solution +for µ = (2·104, 1.2) for time t = 0.1 (left), t = 1.5 (center), t = 3.0 (right), in the Parabolic +Propagating Front in a Square Problem, h = 0.036, α = 0.01, δK = 1.0. +stabilization context, errors between the FEM and the reduced solutions decrease as N grows. The +fact that we deal with a two-dimensional parameter space implies to require more N basis for a +good approximation of the reduced solution. We have ey,30 = 2.17 · 10−3, eu,30 = 1.59 · 10−3 and +ep,30 = 5.62 · 10−3. Therefore, also for this case test we can state that the SUPG stabilization is an +efficient procedure for the ROMs. +Figure 11. Relative errors between the FEM and the Only-Offline and Online-Offline +stabilized reduced solution for the state (left), control (center) and adjoint (right) solutions, +respectively with P = +� +104, 105� +× +� +0, 1.57 +� +, Nt = 30, Ntest = 100, δK = 1.0, h = 0.036. +Finally, we show the results about the speedup-index in Table 4. For N = 30, not only we have +the best accuracy for the reduced problem, but we are able to computed, averagely, 3981 reduced +solution in the interval of a FEM simulation. Speedup-index has an average order of magnitude of +103 overall. +6. Conclusions and Perspectives +In this work, we presented the numerical experiments concerning Advection-Dominated OCPs in +a ROM context with high P´eclet number, both in the steady and the unsteady cases, under SUPG +stabilization. Concerning ROMs, we can have two possibilities of stabilization: we can apply SUPG +only to the offline phase or we can use it in both online and offline phases. We analyzed relative + +1.2e+00 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +-1.8e-029.7e-03 +0.008 +0.006 +0.004 +0.002 +0 +-0.002 +-0.004 +-0.006 +-8.4e-03FEM vs ROM averaged relative error - y (state) +100 +10-1 +Relative L +10-2 +Online stab +Online not stab +5 +10 +15 +20 +25 +30 +NFEM vs ROM averaged relative error - u (control) +100 +Relative Log-Error +10-2 +Online stab. +Online not stab +5 +10 +15 +20 +25 +30 +NFEM vs ROM averaged relative error - p (adjoint) +100 +10-1 +10-2 +-Online stab. +Online not stab. +5 +10 +15 +20 +25 +30 +NSUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL +25 +Only-Offline Stabilization +Offline-Online Stabilization +N +min +average +max +min +average +max +5 +6136.6 +8659.4 +12654.5 +4588.3 +6605.8 +9914.7 +10 +6018.7 +8278.8 +11989.5 +4282.9 +6231.7 +9353.3 +15 +5611.3 +7721.4 +11359.5 +3725.3 +5779.0 +8794.3 +20 +4820.0 +7041.2 +10143.0 +3567.7 +5318.9 +8091.1 +25 +3814.1 +5970.5 +9212.2 +2751.7 +4366.6 +6678.3 +30 +3432.0 +5420.1 +8462.8 +2424.7 +3981.3 +6147.9 +Table 4. Speedup-index of the unsteady Propagating Front in a Square Problem for +Online-Offline and Only-Offline stabilization with training set P := +� +104, 105] × +� +0, 1.57], +h = 0.036, α = 0.01, Nt = 30, Ntest = 100, δK = 1.0. +errors between the reduced and the high fidelity solutions and of the speedup-index concerning the +Graetz-Poiseuille and Propagating Front in a Square Problems, always under a distributed control. +A P1-FEM approximation for the state, control and adjoint spaces is used in a optimize-then- +discretize framework. Concerning parabolic problems, a space-time approach is followed and we +applied in a suitable way the SUPG stabilization. For the ROM, we considered a partitioned approach +for all three variables using the POD algorithm. In all the steady and unsteady experiments, the +ROM technique performed excellently in a Online-Offline stabilization framework. Especially for +parabolic problems, the speedup-index features large values thanks to the space-time formulation. +Only-Offline stabilization technique performed very poorly in terms of errors, despite the little +favorable speedup values. Thus, Online-Offline stabilization is preferable. +We also performed experiments inherent a geometrical parametrization and boundary control for +the Graetz-Poiseuille Problem that are not shown in this work. Results were quite good for Online- +Offline stabilization: we had some little oscillations regarding relative errors due to the complexity +of the problem. As a perspective, it might be interesting to create a strongly-consistent stabilization +technique that flattens all the fluctuation for these two configurations, since, to the best of our +knowledge, this topic is still a novelty in literature. +Regarding the SUPG stabilization for parabolic OCPs in a optimize-then-discretize framework, +it would be also worth to derive some theoretical results that gives us the accuracy of the numerical +solution with respect to the time-step and the mesh-size. +In conclusion, as another goal it might be interesting to study the performance of new stabilization +techniques for the online phases, such as the Online Vanishing Viscosity and the Online Rectification +methods [4, 13, 33]. Moreover, the extension of this setting to the uncertainty certification context +will be the topic of future research. +Acknowledgements +We acknowledge the support by European Union Funding for Research and Innovation – Horizon +2020 Program – in the framework of European Research Council Executive Agency: Consolidator +Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods +with Applications in Computational Fluid Dynamics”. We also acknowledge the PRIN 2017 “Nu- +merical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of +complex systems governed by Partial Differential Equations” (NA-FROM-PDEs) and the INDAM- +GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali”. 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Weighted Reduced Order Methods for parametrized PDEs in uncertainty quantification problems. +Master’s thesis, University of Trieste and SISSA, 2016. + diff --git a/FNA0T4oBgHgl3EQfBP9o/content/tmp_files/load_file.txt b/FNA0T4oBgHgl3EQfBP9o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..879cefdffe2194ba5de0c157b4961956c0ee72bd --- /dev/null +++ b/FNA0T4oBgHgl3EQfBP9o/content/tmp_files/load_file.txt @@ -0,0 +1,1554 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf,len=1553 +page_content='A STREAMLINE UPWIND PETROV-GALERKIN REDUCED ORDER METHOD FOR ADVECTION-DOMINATED PARTIAL DIFFERENTIAL EQUATIONS UNDER OPTIMAL CONTROL FABIO ZOCCOLAN1, MARIA STRAZZULLO2, AND GIANLUIGI ROZZA3 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this paper we will consider distributed Linear-Quadratic Optimal Control Problems dealing with Advection-Diffusion PDEs for high values of the P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this situation, computational instabilities occur, both for steady and unsteady cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' A Streamline Upwind Petrov–Galerkin technique is used in the optimality system to overcome these unpleasant effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We will apply a finite element method discretization in a optimize-then-discretize approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the parabolic case, a space-time framework will be considered and stabilization will also occur in the bilinear forms involving time derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then we will build Reduced Order Models on this discretization procedure and two possible settings can be analyzed: whether or not stabilization is needed in the online phase, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In order to build the reduced bases for state, control, and adjoint variables we will consider a Proper Orthogonal Decomposition algorithm in a partitioned approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The discussion is supported by computational experiments, where relative errors between the FEM and ROM solutions are studied together with the respective computational times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Introduction The main goal of Optimal Control theory is to modify a physical or engineering system through an input, called control, to obtain a desired output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' From a theoretical point of view, one can describe the state problem through partial differential equations (PDEs), following the approach of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Lions [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Applying an optimal control means to solve a constrained optimization problem, where a cost functional has to be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This process translates into an optimality system, which will be discretized for numerical simulations, that, in this framework, are more and more needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Thus, effective and fast numerical techniques are required to exploit optimal control in scientific and industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this work, we will consider Advection-Diffusion equations [42] for large P´eclet numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' These equations are widespread in many engineering contexts since they can model transfer of particles, of energy, of heat and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In the case of high values of the P´eclet number, numerical instabilities occur during discretization: this can happen for related optimal control problems, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Thus, it becomes necessary to introduce some stabilization techniques to overcome this undesired behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We exploit a Streamline Upwind Petrov–Galerkin (SUPG) technique over a finite element method (FEM) [11, 26, 38] in a optimize-than-discretize approach, as done in [14], to provide strongly- consistency to the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' When we deal with unsteady problems, a space-time discretization [21, 46, 50, 51, 52, 57] will be used together with the SUPG stabilization for bilinear forms related to the derivative over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The discretization procedure can easily request a huge amount of computational resources, especially for parametric time-dependent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The parameters can represent physical or geometrical features of the system at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this scenario, we decide to exploit the parameter dependence of the equations to build Reduced Order Models (ROMs) [22, 40, 39, 43] by means of Proper Orthogonal Decomposition (POD) algorithm in a partitioned approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Namely, 1 Section de Math´ematiques, ´Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland, email: fabio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='zoccolan@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='ch 2 DISMA, Politecnico di Torino, Corso Duca degli Abruzzi 24, Turin, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' email: maria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='strazzullo@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='it 3 mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' email: gianluigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='rozza@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='it 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01973v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='NA] 5 Jan 2023 2 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL the discretization process is divided in two phases: an offline stage where a low-dimensional space is built through FEM solutions computed in properly chosen parameters, and an online stage, where the system is solved for a new parametric instance in the new low-dimensional framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Thus, we consider two possible strategies: the former is to stabilize the system only in the offline phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' the latter uses SUPG in the online one, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This setting was considered for problems without optimal control in [37, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' To the best of our knowledge, it is the first time that SUPG stabilization for time-dependent Advection-Dominated problems under distributed control is analyzed in a ROM setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The first section will illustrate some theoretical aspects about Optimal Control Theory for PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Section 3 shows the FEM discretization that will be used for numerical experiments, an introduction to Advection-Dominated problems, and SUPG technique in an optimize-than-discretize approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Instead, in Section 4, we will focus on the ROM setting and Section 5 refers to the related numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Firstly, we will introduce two specific examples of Advection-Diffusion problems: the Graetz-Poiseuille and the Propagating Front in a Square Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The former was studied in various forms without optimal control in [18, 37, 44, 55] and with optimal control but without stabilization in [34, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The latter is studied without optimal control in a similar version in [37, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Here, both the problems will be analyzed under a distributed optimal control for high values of the P´eclet number, both in the steady and unsteady cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Relative errors between FEM and ROM solutions will be shown, as well as an analysis on the computational times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Problem Formulation In this Section we will illustrate the fundamentals of Linear-Quadratic Optimal Control Problem (OCP) for steady and unsteady PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The aim of Optimal Control is to achieve a prescribed optimality condition by minimizing a suitable cost functional under the constraint of satisfying the PDE Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The proposed framework follows the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Lions theory [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Parametric Optimal Control Problems governed by PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The main features of an OCP are: (1) a controlled system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' an input-output process given by a system of PDEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (2) the output of the system, or an observation of it, when the output cannot be measured directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In our case, we will consider the solution of the system as the output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (3) a control, which constitutes the input of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' It influences the output which can be expressed as a function of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this work we will only consider distributed control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (4) an objective condition to be fulfilled, which can be represented by a real functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, from a mathematical perspective, we can state that an OCP is characterized by: e, the state equation function, which expresses the relationship between the output and the control within the system in terms of a PDE problem or PDEs in a weak formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' A pair (y, u) ∈ X := Y × U is said to be physical or feasible if it is a solution of the state equation e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' y is called the state variable, the output, and u is the control variable, the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Xad is the set of all the feasible pairs (y, u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' z(y) = Oy, a direct observation of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Here, a linear operator O is applied to the state to describe the observation: we will denote the space of observation as Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We will only deal with state variables that can be measured on a portion of the domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' J, the objective functional, which describes the objective to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' suitable spaces Y and U, as the state space and control space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Domains of definition for control and/or state can be taken smaller due to possible restrictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' hence we have to introduce Yad ⊆ Y and Uad ⊆ U as the admissible state space and admissible control space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' However, we will always consider unconstrained problems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Xad = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The theory of well-posedness can make use of the Lagrangian approach as in [12, 34] or it can be consider as a particular case of the general Adjoint approach when we can deal with Xad ⊂ Yad × Uad [24, 30, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 3 Let us consider Ω ⊂ Rn, an open and bounded regular domain, and the time interval (0, T) ⊂ R+: for us it will always be the case of n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us denote with ΓD and ΓN the portions of the boundary of ∂Ω where Dirichlet and Neumann boundary conditions are specified, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We define the observation domain Ωobs ⊆ Ω as the portion of the domain where we want that the state variable assumes a desired value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' P ⊆ Rp, for natural number p, is the parameter space and µ ∈ P is a p-vector which can represent physical or geometrical parameter of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this work we deal with Parametric Optimal Control Problems (OCP(µ)s), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' systems where there is a dependency on the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 (Parametric Optimal Control Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Given Y, U real Banach spaces, consider the state equation e : Y × U → Q, with Q a Banach space, which fulfills a set of boundary and/or initial conditions, and the objective functional J : Y ×U → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Given µ ∈ P, then find � y(µ), u(µ) � ∈ X such that the cost functional J(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) is minimized subject to e(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Lagrangian Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We refer to the Lagrangian approach to state the well-posedness of OCP(µ)s in full admissibility setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' when Xad = Y × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We want to solve: min (y(µ),u(µ))∈Y ×U J(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' e(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = 0, thus we define the Lagrangian operator L : Y × U × Q∗ → R as: (1) L(y(µ), u(µ), p(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = J(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) + ⟨p(µ), e(y(µ), u(µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)⟩Q∗Q, where p(µ) is a Lagrange multiplier belonging to Q∗, the dual space of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the sake of notation, we write y := y(µ), u := u(µ) and p := p(µ): we will explicit the parameter dependence only when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The discussion inherent to the Lagrangian approach is based on [12], the same reference presents a comparison between this approach and the adjoint one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the sake of simplicity, we make some regularity assumptions [12]: Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The objective functional J and the state equation e are Fr´echet differentiable, more precisely the differential operator related to J is continuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' J′(µ) ∈ C(Y ×U, B(Y ×U, R)), where B(V , ˜V ) is the space of linear bounded operators between Banach spaces V and ˜V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The following theorem and proposition claim that under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 minimizers of the function J, subject to equality constraints e, can be critical points of (1) [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 (Lagrange Multipliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let X := Y × U and V ⊆ X be an open subset such that J and e are Frech´et differentiable on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Assume x = (y, u) ∈ V to be a minimizer of J subject to the constraint e(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = 0, and e′(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) ∈ B(X, Q) to be surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then, there exists a Lagrange multiplier p ∈ Q∗ such that (x, p) is an unconstrained stationary point of the Lagrangian L in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, in order to find a stationary point (y, u, p) of L, one has to solve the following optimality system [12]: (2) � � � � � Ly(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)(¯y) = Jy(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)(¯y) + ⟨p, ey(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)(¯y)⟩Q∗Q = 0, ∀¯y ∈ Y, Lu(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)(¯u) = Ju(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)(¯u) + ⟨p, eu(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)(¯u)⟩Q∗Q = 0, ∀¯u ∈ U, Lp(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)(¯p) = ⟨¯p, e(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)⟩Q∗Q = 0, ∀¯p ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In the Lagrangian formulation Q∗ is said the adjoint space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The above result easily implies the following useful proposition [38], where we derive another system of three equations that we will use in the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 (Optimality System).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Suppose Xad = Y × U and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 holds, then for some p ∈ Q∗ a minimizer x = (y, u) of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 where e′(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) is surjective must satisfy (3) � � � � � Ly(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = Jy(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) + ey(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)∗p = 0, in Y ∗, Lu(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = Ju(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) + eu(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)∗p = 0, in U ∗, Lp(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = e(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = 0, in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 4 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL In (3), the first equation is called the adjoint equation, the second one is the gradient equation and, as we have already seen, the state equation is the third one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We remark that we will always consider Linear-Quadratic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 (Linear-Quadratic Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Consider a Banach space Z and α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let the Observation map O : Y → Z be a linear and bounded operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Consider an element zd(µ) ∈ Z, which is the so-called desired solution profile (the desired observed output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let J be a quadratic objective functional of the form (4) J(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = 1 2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α 2 n(u(µ), u(µ)), where m : Z × Z → R and n : U × U → R are symmetric and continuous bilinear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let e be affine, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' there exist A(µ) ∈ B(Y, Q), B(µ) ∈ B(U, Q) and f(µ) ∈ Q such that (5) e(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = A(µ)y + B(µ)u − f(µ), ∀ � y(µ), u(µ) � ∈ Y × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then an OCP(µ)s with the above properties is said a Linear-Quadratic Optimal Control Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For Linear-Quadratic OCP(µ)s Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 implies that a solution (y, u) to Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 must satisfy, for some p ∈ Q∗ [12], (6) � � � � � m(Oy, O¯y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) + ⟨A∗(µ)p, ¯y⟩Y ∗Y = m (O¯y, zd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) , ∀¯y ∈ Y, αn(u, ¯u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) + ⟨B∗(µ)p, ¯u⟩U ∗U = 0, ∀¯u ∈ U, ⟨¯p, A(µ)y + B(µ)u⟩Q∗Q = ⟨¯p, f(µ)⟩Q∗Q, ∀¯p ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this context, if (y, u, p) is a saddle point of L [56], then (y, u) minimizes J over all zeroes of e [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Moreover, under some precise hypotheses existence and uniqueness of a saddle point can be provided using Brezzi Theorem [9, 10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, a possible strategy to prove well-posedness of an Linear-Quadratic OCP(µ)s can be to demonstrate that a stationary point of (6) is a saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' At this purpose, System (6) can also be recast in a saddle-point structure [7, 12, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In order to derive this structure, assume x ∈ X := Y × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We define M(µ) ∈ B (Z, Z∗) , N(µ) ∈ B (U, U ∗) as the unique operators that satisfy the following relations: ⟨M(µ)z, ¯z⟩Z∗Z = m(z, ¯z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ), ⟨N(µ)u, ¯u⟩U ∗U = n(u, ¯u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ), ∀z, ¯z ∈ Z, ∀u, ¯u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This directly implies that m(Oy, O¯y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = ⟨O∗M(µ)Oy, ¯y⟩Y ∗Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Using Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3) and a matrix notation as follows [12]: (7) E(µ) = � A(µ) B(µ) � , D(µ) = � O∗M(µ)O 0 0 αN(µ) � , E∗(µ) = � A∗(µ) B∗(µ) � , defining also ¯g(µ) = O∗M(µ)zd, the optimality system (6) for Linear-Quadratic OCP(µ)s can be written in a more compact form as (8) � D(µ) E∗(µ) E(µ) 0 � � x p � = � ¯g(µ) f(µ) � in X∗, in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For Linear-Quadratic Problems, a saddle point of L is a stationary point [56], so it satisfies (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For Linear-Quadratic problems the solution to system (8), and hence to (6), is a saddle point of L when D(µ) is self-adjoint [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this case Brezzi Theorem gives us well-posedness [9, 10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' [12] If Y is reflexive so that D(µ) = D∗(µ), then (x, p) = (y, u, p) is a saddle point of L if and only if it solves the system (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We assume that Y, U are reflexive, A(µ) is weakly coercive, the operator B(µ) is not null, αN(µ) is coercive with constant α > 0 and m(z, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) ≥ 0, ∀z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Considering Linear-Quadratic OCP(µ)s and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6, it follows that E(µ) is inf-sup stable and D(µ) is coercive over the kernel of E(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Consequently, the well-posedness of the system (8) is assured by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 (Brezzi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' [9, 10, 12] Let X be a reflexive Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then the equivalence of the following statements holds: SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 5 (1) D(µ) ∈ B (X, X∗) , E(µ) ∈ B(X, Q) with the following properties: D(µ) is weakly coercive over the kernel of E(µ), E(µ) is inf-sup stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (2) The system (8) has a unique solution (x, p) ∈ X × Q∗ for all ¯g(µ) ∈ X∗, f(µ) ∈ Q, which satisfies for some constant C > 0 ∥x∥X + ∥p∥Q∗ ≤ C (∥¯g(µ)∥X∗ + ∥f(µ)||Q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (3) The operator S(µ) := � D(µ) E∗(µ) E(µ) 0 � is an isomorphism in X∗ × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 (Notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' From now on, we will always involve Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the sake of notation, there we will denote the various bilinear forms defined by A(µ), B(µ) and their adjoints ones in the following unique way: ⟨A(µ)y, p⟩QQ∗ := a(y, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) ⟨B(µ)u, p⟩QQ∗ := b(u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Unsteady Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We briefly recall results on well-posedness for time-dependent Linear- Quadratic OCP(µ)s based on [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider saddle-point formulation in order to prove well- posedness by using tools of the previous Sections in the case of null initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Differently from the steady case, here we will make some more technical assumptions, which will be fulfilled by both “Graetz-Poiseuille” and “Propagating Front in a Square” problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Consider two separable Hilbert spaces Y and H satisfying Y �→ H �→ Y ∗ and, moreover, other two Hilbert spaces U and Z ⊇ Y , where Y and U are the usual state and control spaces, and Z is the space of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We endow them with the standard norms inherited from their respectively scalar products: (·, ·)Y , (·, ·)Z, (·, ·)U and (·, ·)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We define the following Hilbert spaces: Y = L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Y ), Y∗ = L2 (0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Y ∗) , U = L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' U) Z := L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Z) ⊇ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' with respective norms, for instance in the case of Y and U given by (9) ∥y∥2 Y := T � 0 ∥y∥2 Y dt, and ∥u∥2 U := T � 0 ∥u∥2 Udt and similarly for the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Furthermore, let us define the Hilbert space Yt with its scalar product (·, ·)Yt: Yt := � y ∈ Y s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∂y ∂t ∈ Y∗ � , (y, z)Yt := T � 0 (y, z)Y dt + T � 0 �∂y ∂t , ∂z ∂t � Y ∗dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Our aim is to solve the following unconstrained Linear-Quadratic Parametric Parabolic OCP(µ): Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 (Parametric Parabolic OCP(µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For a given µ ∈ P find the pair (y(µ), u(µ)) ∈ Yt × U that satisfies (10) � � � � � � � � � � � � � � � ∂y(µ) ∂t + A(µ)y(µ) + B(µ)u(µ) − f(µ) = 0, in Ω × (0, T), ∂y(µ) ∂n = 0, on ΓN × (0, T), y(µ) = l, on ΓD × (0, T), y(µ)(0) = y0, in Ω, and minimizes min (y(µ),u(µ))∈Yt×U J(y, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = 1 2m (Oy(µ) − zd(µ), Oy(µ) − zd(µ)) + α 2 n(u(µ), u(µ)), where m : Yt × Yt → R and n : U × U → R are symmetric and continuous bilinear forms, zd(µ) ∈ Z is the observed desired solution profile and α > 0 is the fixed penalization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In our test case we will always take y0 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 6 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL Also in this case, we denote y := y(µ) and u := u(µ) omitting the parameter dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We can state the weak formulation of (10) as � � � � � � � T � 0 �∂y ∂t , q � Y∗Y dt + T � 0 ⟨A(µ)y, q⟩Y∗Ydt + T � 0 ⟨B(µ)u, q⟩Y∗Ydt − T � 0 ⟨f(µ), q⟩Y ∗Y dt = 0, ∀q ∈ Yt, y(0) = y0, in Ω, where f(µ) ∈ Y∗ gathers all forcing, boundary and, eventually, lifting terms of the state equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Nevertheless, for the sake of notation, we will consider a : Yt × Yt → R and b : U × Yt → R the bilinear forms defined as a(y, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = ⟨A(µ)y, q⟩Y∗Y and b(u, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = ⟨B(µ)u, q⟩Y∗Y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For a proper definition of the adjoint variable, it is opportune to take q ∈ Yt rather than q ∈ Y [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us define the state-control product space X = Yt × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then we define the operators E,D and ¯g similarly as made in the steady case in order to make the formulation more compact [50]: (11) D(µ) : X × X → R, D(x, ¯x, µ) =m(Oy, O¯y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) + αn(u, ¯u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' E(µ) : X × Yt → R, E(x, q, µ) = T � 0 �∂y ∂t , q � Y∗Y dt + T � 0 a(y, q, µ)dt + T � 0 b(u, q, µ)dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ¯g(µ) ∈ X ∗, T � 0 ⟨¯g(µ), ¯x⟩dt =m (O¯y, zd(µ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Denoting p := p(µ) and considering Q∗ = Yt [50], the Lagrangian and objective functionals are, respectively: (12) L(x, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = J(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)+E(x, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)− T � 0 ⟨f(µ), p⟩Y ∗Y dt, J(x, µ) = 1 2D(x, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)− T � 0 ⟨¯g(µ), x⟩dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As made in the steady case, the minimization of Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 means to seek the solution of the following system: given µ ∈ D, find (y, u, p) = (x, p) ∈ X × Yt which solve (13) � � � � � Ly(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)[¯y] = 0, ∀¯y ∈ Yt, Lu(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)[¯u] = 0, ∀¯u ∈ U, Lp(y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)[¯p] = 0, ∀¯p ∈ Yt, and satisfy boundary and initial conditions in Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 with p(T) = 0 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The saddle-point structure of steady Linear-Quadratic OCP(µ)s (8) can be derived in the parabolic case, too (here expressed in the weak formulation) [50]: (14) � � � � � � � � � � � � � � � � � D(x, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) + E(w, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = T � 0 ⟨¯g(µ), w⟩dt, ∀w ∈ X, E(x, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = T � 0 ⟨f(µ), q⟩Y ∗Y dt, ∀q ∈ Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The equivalence between the optimality system and saddle-point formulation for Linear-Quadratic Parabolic OCP(µ)s is straighforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For well-posedness the following assumption is needed [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The bilinear forms n(·, · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ), m(·, · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ), b(·, · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ), and a(·, · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) satisfy the following features: (1) m(·, · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) is positive definite, continuous, and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (2) n(·, · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) is coercive, continuous, and symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (3) there exists Ca > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' a(w, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) ≥ Ca(µ)∥w∥2 Y , ∀w ∈ Yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (4) there exists ca > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' |a(w, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)| ≤ ca(µ)∥w∥Y ∥p∥Y , ∀w, p ∈ Yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 7 (5) there exists cb > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' |b(v, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)| ≤ cb(µ)∥v∥U∥p∥Y , ∀v ∈ U and ∀p ∈ Yt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Finally, one can prove the well-posedness of Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 (for more details, we refer to [50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 (Well-posedness of Parabolic OCP(µ)s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' [50] Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 the saddle- point formulation (14) satisfies the hypothesis (1) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7, hence the solution is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For both steady and unsteady problems, we will consider the Identity operator restricted to our observation domain Ωobs as the Observation function O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, Z = Y is assumed and our desired state will be denoted by yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Truth Discretization In this Section we firstly pursue a numerical method for the solution of an OCP: a discretization of the optimality sistem (6) will be given following an one shot or all-at-once approach [23, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Secondly, we will consider SUPG stabilization for Advection-Dominated equations in case of high P´eclet number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' An optimize-then-discretize approach is followed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' at first we derive optimality conditions as system (6) and then we discretize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, we obtain a discretized system: (15) � � � � � LyN (yN , uN , pN ) = JyN (yN , uN ) + eyN (yN , uN )∗pN = 0 in � Y N �∗ LuN (yN , uN , pN ) = JuN (yN , uN ) + euN (yN , uN )∗pN = 0 in � U N �∗ LpN (yN , uN , pN ) = e(yN , uN ) = 0 in QN , where LyN , LuN , LpN are the discretizations of partial derivatives of L and Y N , U N , QN are the approximation of Y, U, Q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us start our discussion from the steady case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' From now on we will always assume to work with Y, U, Q Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We employ a FEM discretization, which will be named as the high-fidelity or truth approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider Ωh as a quasi-uniform mesh on the domain Ω, for which the parameter h indicates the mesh size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' maximum diameter of an element of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Th is a regular triangularization on Ω and Ωh := int � � K∈Th K � , where K is a triangle of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We define the FEM spaces Y N = Y ∩ XN ,r, U N = U ∩ XN ,r and � QN �∗ = Q∗ ∩ XN ,r, where XN ,r = � vN ∈ C0(¯Ω) : vN |K ∈ Pr(K), ∀K ∈ Th � and Pr(K) represents the space of polynomials of degree at most equal to r defined on a triangle K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As we will remark later, we will always use the same triangulation Th and a P1-FEM approximation for state, control and adjoint variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The dimensions of Y N , U N , QN are all equal to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The overall dimension of the discrete problem is Ntot = 3 · N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the sake of simplicity, we assume Q∗ h = Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Moreover, we indicate with XN = Y N × U N ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' From now on we will refer to the same symbol yd to also indicate the FEM discretization version of the desired state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The discretization of a Linear-Quadratic OCP of Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 reads as min (yN ,uN )∈Y N ×U N J � yN , uN � = 1 2m � yN − yd, yN − yd � + α 2 n(uN , uN ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' e � yN , uN � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Moreover, the operators m and n will be the L2 product on Ωobs and on Ω, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the saddle-point system, we define the operators ¯gN : XN → R, f N : Y N → R, DN : XN → � XN �∗ , and EN : XN → � QN �∗ as just the usual restrictions (16) � ¯gN , ¯xN � (XN )∗XN = � ¯g, ¯xN � X∗X , � DN xN , ¯xN � (XN)∗XN = � DxN , ¯xN � X∗X , ⟨f N , ¯pN ⟩(Y N )∗Y N = � f, ¯pN � Q∗∗Q∗ , � EN xN , ¯pN � (Y N )∗Y N = � ExN , ¯pN � Q∗∗Q∗ , for all xN ∈ XN , ¯pN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 8 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL We highlight the algebraic structure of the discretize optimality system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We define the basis of the finite spaces XN and Y N as below: (17) � ϕj ∈ XN �2N j=1 , � ψk ∈ Y N �N k=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As a result, we can rewrite a pair � xN , pN � ∈ XN × Y N in the following way: � �xN = 2N � j=1 xjϕj, pN = N � k=1 pkψk � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, we can define D ∈ R2N ·2N , E ∈ RN ·2N , ¯g ∈ R2N and f ∈ RN as follows: (18) Dij = ⟨DN ϕi, ϕj⟩(XN )∗XN , Elm = ⟨EN ϕl, ψm⟩(Y N )∗Y N , ¯gk = � ¯gN , ϕk � (XN )∗XN , fn = � f N , ψn � (Y N )∗Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Finally, we can build the following saddle point system, with a block structure: (19) � D ET E 0 � � x p � = � ¯g f � , where (x)i = xi, i = 1, · · · 2N and (p)k = pk, k = 1, · · · N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For this purpose, let us denote with y, u and p the vectors of coefficients of yN , uN and pN , expressed in terms of the nodal basis (17) by splitting components of XN in those of Y N and U N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We express with yd the vector with the components of the discretized desired state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' the Galerkin projection of yd on Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Moreover, let us indicate the stiffness matrix derived from the bilinear form a(·, ·) with K, KT is the stiffness matrix related to a∗ and the mass matrix is denoted with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In addition, we call B, BT is the mass matrix related to the forms b and b∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We have that: D = � M 0 0 αM � , E = � K B � , x = � y u � , ¯g = � Myd 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' and the optimality system shows this block structure: (20) � � M 0 KT 0 αM BT K B 0 � � � � y u p � � = � � Myd 0 f � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' SUPG stabilization for Advection-Dominated OCP(µ)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this Section we illustrate Advection-Dominated OCP(µ)s and the SUPG technique applied to an optimize-then-discretize approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' From now, we recall the dependence on parameters of our operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us start from our definition of an Advection-Diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 (Advection-Diffusion Equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us consider the following problem: (21) L(µ)y := −ε(µ)∆y + b(µ) · ∇y = f(µ) in Ω ⊂ R2, with suitable boundary conditions on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us suppose that: the diffusion coefficient ε : Ω → R belongs to L∞(Ω) and depends on the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We assume there exists a constant ¯ε > 0 such that ε(x) ≥ ¯ε, ∀x ∈ Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' the advection field b : Ω → R2 belongs to (L∞(Ω))2 and depends on the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We suppose that 0 ≥ div b(x) ≥ −˜k, holds for all x ∈ Ω, with ˜k ∈ R+ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' f(µ) : Ω → R is an L2(Ω)-function that can depend on the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this case, (21) is an Advection-Diffusion problem and the operator L(µ)y := −ε(µ)∆y +b(µ)· ∇y is said the Advection-Diffusion operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' From (21), we can easily derive the weak formulation of an Advection-Diffusion problem: (22) find y ∈ Y s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' a (y, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = F (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) ∀q ∈ Q∗, SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 9 where (23) a (y, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) := � Ω ε(µ)∇y∇q + b(µ) · ∇yq dx, F(q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) := � Ω f(µ)q dx, y ∈ Y, q ∈ Q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' From a numerical point of view, when the advection term b(µ) · ∇u “dominates” the diffusive one −ε(µ)∆u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' when |b(µ)| ≫ ε(µ), the approximated solution can show instability phenomena along the direction of the advection field [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In order to give an indicator of the instability, let us consider the regular triangulation Th related to FEM discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For any element K ∈ Th, we can then define the local P´eclet number as [42, 38]: (24) PeK(x) := |b(x)|hK 2ε(x) , ∀x ∈ K, where hK is the diameter of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 (Advection-Dominated problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Considering Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 we are dealing with an Advection-Dominated problem if PeK(x) > 1, ∀x ∈ K, ∀K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' To solve the issue of the instability, we will exploit the SUPG method [11, 25, 26, 42], which is a strongly consistent stabilization technique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' is consistent for weak PDEs and its order of accuracy can be greater than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us now consider the Advection-Diffusion operator (21): for the sake of simplicity, we define it on H1 0(Ω) and we do not indicate the parameter dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The operator L can be split into its symmetric and skew-symmetric parts [42], defined as: (25) symmetric part: LSy = −ε∆y − 1 2(div b)y, ∀y ∈ H1 0(Ω), skew-symmetric part: LSSy = b · ∇y + 1 2(div b)y, ∀y ∈ H1 0(Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' L = LS +LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Symmetric and skew-symmetric parts can be directly derived using the formulae: (26) LS = L + L∗ 2 , LSS = L − L∗ 2 , where L∗ is the adjoint operator related to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Now, let us analyze our OCP problem (6): we follow the optimize-then-discretize approach in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The discretized state equation is described as follows, where the control is distributed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' it acts on the whole domain Ω: (27) as � yN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' qN � + bs � uN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' qN � = Fs(qN ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∀qN ∈ � QN �∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' with (28) as � yN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' qN � := a � yN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' qN � + � K∈Th δK � LyN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' hK |b| LSSqN � K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (29) bs � uN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' qN � := − � Ω uN qN − � K∈Th δK � uN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' hK |b| LSSqN � K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' and (30) Fs(qN ) := F � qN � + � K∈Th δK � f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' hK |b| LSSqN � K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' where � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' · � K indicates the usual L2(K)-product,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' f collects all the forcing and lifting terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' and δK denotes a positive dimensionless stabilization parameter related to an element K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In principle, since δK is local, it can be different for each K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Considering the adjoint equation, we can see that it is also an Advection-Dominated equation, but with an advective term with opposite sign with respect to the state one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As a matter of fact, from (26) we obtain that L∗ = LS − LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The SUPG method leads to the discretized adjoint equation (31) a∗ s � zN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' pN � + � yN − yd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' zN � s = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∀zN ∈ Y N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 10 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL with (32) a∗ s � zN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' pN � := a∗ � zN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' pN � + � K∈Th δa K � (LS − LSS)pN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' hK |b| (−LSS) zN � K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' � yN − yd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' zN � s := � Ωobs (yN − yd)zN dx + � K∈Th|Ωobs δa K � yN − yd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' hK |b| (−LSS) zN � K ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' where a∗ is the adjoint form of a and δa K is the parameter related to the stabilized adjoint bilinear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As in this work we consider δK = δa K in numerical simulations, from now on we will always denote both stabilization parameter with δK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Instead, the discretized gradient equation is not affected by the SUPG and it remains untouched: (33) b∗� vN , pN � + αn � uN , vN � = 0, ∀vN ∈ U N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' With this setting it follows a nonsymmetric system for the computation of the numerical solution, but we gain the strongly-consistency of the method for the optimality system if y, u, p are regular [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' To summarize,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' the SUPG optimality system for a steady OCP is the following: (34) discretized adjoint equation: a∗ s � zN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' pN � + � yN − yd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' zN � s = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∀zN ∈ Y N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' discretized gradient equation: b∗� vN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' pN � + αn � uN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' vN � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∀vN ∈ U N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' discretized state equation: as � yN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' qN � + bs � uN ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' qN � = Fs(qN ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∀qN ∈ � QN �∗ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' referring to (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' the discretized algebraic system reads as: (35) � � Ms 0 KT s 0 αM BT Ks Bs 0 � � � � y u p � � = � � Msyd 0 fs � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' where Ms is the stabilized mass matrix related to m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' M is the not-stabilized mass matrix related to n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Ks and KT s are the stiffness matrices related to as and a∗ s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Bs is the stabilized mass matrix related to bs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' BT is the block linked to b∗ and fs is the vector whose components are the coefficients of the stabilized force term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Every block is derived as in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We indicate with |∥ · ∥| the energy norm related to the bilinear form a belonging to Advection- Diffusion equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (36) |∥w∥|2 := ε∥∇w∥2 L2(Ω) + 1 2 ���(div b) 1 2 w ��� 2 L2(Ω) , ∀w ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, we define the SUPG norm on H1 0(Ω) as (37) ∥w∥2 SUP G := |∥w∥|2 + � K∈Th δK � LSSw, hK |b| LSSw � K , ∀w ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Considering that (38) holds true, it is immediate to see that the SUPG bilinear form (28) is coercive with respect to the SUPG norm [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Finally, we can illustrate an estimate of the error for the adjoint and the state variables of the solution of an OCP [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 (Error for state and adjoint variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let m, r ≥ 1 and (y, u, p) be the solution of (6) with y ∈ Hm+1(Ω), p ∈ Hr+1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Furthermore, let yN , uN , pN be the numerical solution of (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' If δK satisfies (38) 0 < δK ≤ hK εη2 inv and δK = � � � δ1 hK ε ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' PeK(x) ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' δ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' PeK(x) > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' where δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' δ2 > 0 are chosen constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' and ηinv is defined as the following inverse constant |yN |1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='K ≤ ηinvh−1 K ∥yN ∥L2(K) and ∥∆yN ∥L2(K) ≤ ηinvh−1 K ∥∇yN ∥L2(K) ∀yN ∈ Y N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 11 with | · |1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∥ · ∥K seminorm and L2-norm on K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' then there exists C > 0 such that (39) ��y − yN �� SUP G ≤ C � hm � ε1/2 + h1/2� |y|Hm+1(Ω) + ��uN − u �� L2(Ω) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∀h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ��p − pN �� SUP G ≤ C � hr � ε1/2 + h1/2� |p|Hr+1(Ω) + ��yN − y �� L2(Ω) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∀h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' SUPG for Time-Dependent Advection-Dominated OCP(µ)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We briefly discuss the SUPG technique employed with time-dependent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Referring to (13), the main challenge comes from the fact that the time derivative should also enter into stabilization framework to ensure consistency [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' However, other approaches have been proposed: in [45], for instance, the time- derivative is not stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Nevertheless, our discussion follows works inherent to Graetz-Poiseuille and Propagating Front in a Square problems without optimal control [37, 54], where stabilization is used for time derivative, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This adds nonsymmetric terms to the discretized state and adjoint equations for time derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' To the best of our knowledge SUPG for Parabolic OCPs in an optimize-then-discretized approach is still a novelty element in literature from a theoretical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' However, we refer to [17, 20, 27] for SUPG applied to general Parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We firstly discretize the equation in time, considering each discrete time as a steady-state Advection- Diffusion equation, in a space-time approach, and then stabilized it with the SUPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The time interval (0, T) is divided in Nt sub-intervals of equal length ∆t := ti − ti−1, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , Nt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' On the other hand, all terms involving time-derivative go through a time discretization equivalent to a classical implicit Euler approach [3, 23, 46, 50, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The backward Euler method is used to discretize the state equation forward in time, instead the adjoint equation is discretized backward in time using the forward Euler method, which is equivalent to the backward Euler with respect to time T − t, for t ∈ (0, T) [16, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The global dimension of the discrete spaces is Ntot = 3 · N · Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We recall that Y, U, Q are Hilbert Spaces and that Y N ≡ (QN )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the state equation, the stabilized term added to the form related to the time derivative of the state ∂y ∂t and the bilinear form a is the following [27, 37, 54]: s � yN (t), qN � = � K∈Th δK �∂yN (t) ∂t + (LS + LSS) yN (t), hK |b| LSSqN � K , where yN (t) ∈ Y N for each t ∈ (0, T) and qN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Instead, the stabilized term added to the form related to the time derivative of the adjoint ∂p ∂t and the bilinear form a∗ is: s∗ � zN , pN (t) � = � K∈Th δK � −∂pN (t) ∂t + (LS − LSS) pN (t), −hK |b| LSSzN � K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We can write the discretized state formulation using a backward Euler approach as follows: (40) for each i ∈ {1, 2, · · · , Nt}, find yN i ∈ Y N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∀qN ∈ Y N , 1 ∆tms � yN i (µ) − yN i−1(µ), qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � + as � yN i (µ), qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � + bs � uN i , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � = Fs � qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � , given the initial condition yN 0 which satisfies (41) � yN 0 , qN � L2(Ω) = � y0, qN � L2(Ω) , ∀qN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The stabilized term ms above is defined as: (42) ms � yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � = � yN , qN � L2(Ω) + � K∈Th δK � yN , hK |b| LSSqN � K and it is related to the time discretization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' instead, as and Fs are defined as in the steady case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Similarly we can derive the same for the adjoint forms applying a forward Euler method: (43) for each i ∈ {Nt − 1, Nt − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=', 1}, find pN i ∈ Y N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 1 ∆tm∗ s � pN i (µ) − pN i+1(µ), zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � + a∗ s � zN , pN i (µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � = − � yN i − ydi, zN � s ∀zN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 12 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL The stabilized term m∗ s above is defined as: (44) m∗ s � pN , zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � = � pN , zN � L2(Ω) − � K∈Th δK � pN , hK |b| LSSzN � K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Now we give a look at the discretization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As in the steady case, yi ∈ Y N , ui ∈ U N and pi ∈ Y N , for 1 ≤ i ≤ Nt, represent the column vectors including the coefficients of the FEM dis- cretization for state, control and adjoint, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, we define y = � yT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , yT Nt �T , u = � uT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , uT Nt �T and p = � pT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , pT Nt �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The vector f s = � f T s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , f T sNt �T indicates the com- ponents of the stabilized forcing term, yd = � yT d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , yT dNt �T is the vector made of discrete time components of our desired state solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' instead, y0 = � yT 0 , 0T , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , 0T �T indicates the vector of ini- tial condition for the state, where 0 is the zero vector in RN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The block matrix system is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' State equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We recall that Ks and Bs are the matrices associated to the stabilized bilinear forms as and bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Using the backward Euler along time, one has to solve (45) Msyi + ∆tKsyi + ∆tBsui = Msyi−1 + fsi∆t for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , Nt} , where Ms is the stabilized mass matrix relative to the FEM discretization of ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore the related block matrix subsystem is � ���� Ms + ∆tKs 0 −Ms Ms + ∆tKs 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 0 0 −Ms Ms + ∆tKs � ���� � �� � As y+∆t � �� Bs 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 0 0 Bs � �� � �� � Cs u = Msy0 + ∆tf s, where Ms is a block diagonal matrix in RN ·Nt ×RN ·Nt whose element on the main diagonal are [Ms, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , Ms].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then everything can be recast in a more compact form as (46) Asy+∆tCsu = Msy0 + ∆tf s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Gradient equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We recall that BT indicates the mass matrix related to the b∗ form and hence at every time step we have to solve the equation (47) α∆tMui+∆tBT pi = 0, ∀i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , Nt} , which translates into the following block system: ∆t · α � ���� M M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' M � ���� � �� � M � ���� u1 u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' uNt � ���� +∆t � ���� BT 0 · · BT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' BT � ���� � �� � CT � ���� p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' pNt � ���� = � ���� 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 0 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In a vector notation we have (48) α∆tMu+∆tCT p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Adjoint equation: we have to solve the first equation of the optimality system (6) at each time step as follows, considering M T s the matrix formulation of m∗ s: M T s pi = M T s pi+1 + ∆t � −M T s yi − KT s pi + M T s ydi � for i ∈ {Nt − 1, Nt − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 13 As did in previous steps, we derive the following block system: � ���� M T s + ∆tKT s −M T s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' M T s + ∆tKT s −M T s M T s + ∆tKT s � ���� � �� � AT s p + � ����� ∆tM T s y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∆tM T s yNt � ����� = � ����� ∆tM T s yd1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∆tM T s ydNt � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then, defining MT s as the diagonal matrix in RN ·Nt × RN ·Nt which diagonal entries are [M T s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , M T s ], the adjoint system to be solved is: ∆tMT s y + AT s p = ∆tMT s yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In the end, we solve system (49) via an one shot approach: (49) � � ∆tMT s 0 AT s 0 α∆tM ∆tCT As ∆tCs 0 � � � � y u p � � = � � ∆tMT s yd 0 Msy0 + ∆tf s � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ROMs for advection-dominated OCP(µ)s FEM simulations can be expensive in terms of computational time and memory storage: this issue is obviously more evident in case of high-dimensional discrete spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Moreover, when we talk about parametrized PDEs, one can require to repeat the simulations for several values of the parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' To overcome these difficulties, we will use ROMs approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The basic idea of ROMs is to create a low-dimensional space, called the reduced space, exploiting the parameter dependence of the problem at hand, such that it is a good approximation of the discrete initial space [8, 22, 41, 40, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us consider a generic Parametrized OCPs described by the optimality conditions (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We can define the set of the parametric solutions of the optimality system with respect to the functional space W = Y × U × Q∗ for steady OCP(µ)s and W = Yt × U × Yt for the unsteady ones as (50) M := {(y(µ), u(µ), p(µ)) solution of (6) | µ ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The extension to space-time formulation for time-dependent problem is straightforward [6, 50] and requires small modifications, thus, we will exclusively refer to the steady framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 (Smoothness of the solution manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The continuous solution manifold M is smooth with respect to the parameter µ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let WN ⊂ W be our FEM approximation of the continuous space W, we call WN := Y N × U N × � QN �∗ the high-fidelity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then, for stabilized problems we define the discrete parametric solution manifold as (51) MN := �� yN (µ), uN (µ), pN (µ) � FEM solution of the (35) | µ ∈ P � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Starting from MN , ROM techniques create a reduced space of low dimension N denoted with WN, via a linear combination of snapshots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' high-fidelity evaluations of the optimal solution � yN (µ), uN (µ), pN (µ) � computed in properly chosen parameters values µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Obviously we have that WN ⊂ WN and we denote WN = Y N × U N × � QN�∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Here, Y N, U N and (QN)∗ are the reduced spaces for the state, the control and the adjoint variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The snapshots are collected by a POD algorithm using a partitioned approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This strategy is followed due to good results shown in literature [28, 35, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' After having built these reduced function spaces, a standard Galerkin projection is performed onto these ones [5, 38, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 14 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Offline-Online Procedure for ROMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ROM procedure is divided in two stages: offline phase: here the snapshots are collected by solving the high-fidelity system (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Secondly, the low-dimensional bases are created and hence all reduced spaces Y N, U N and (QN)∗ are built and stored, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Moreover, all the µ-independent quantities are assembled and stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' It is potentially an expensive phase, which depends on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' online phase: here a parameter µ is chosen and all the previous store µ-independent quan- tities are combined with the just-computed µ-dependent ones to build the reduced block matrix system based on a Galerkin projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' To be convenient, this phase should be N-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Whereas in the offline phase stabilization is present due to stabilized snap- shots, for the online phase this cannot be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, we have two possibilities: if stabilization is performed also here, we talk about Online-Offline stabilization, otherwise we denote the setting as Only-Offline stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As already said, the online phase should be performed in a number of operations independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' A sufficient condition is to admit the separation of the variables depending on µ and the solution (y, u, p) in the affine decomposition [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We require that all the forms in (35) are affine in µ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 we describe the POD algorithm used in the offline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Now, we illustrate the explicit expression of the reduced solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us make clear the structure of the three reduced spaces in terms of their bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, we define (52) Y N = span {ηy n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , N}, U N = span {ηu n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , N} , (QN)∗ = span {ηp n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , N} , the reduced state, the reduced control and the reduced adjoint space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' After having built them, we consider an enriched space for state and adjoint variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, let us denote with {τn}2N n=1 = {ηy n}N n=1 ∪ {ηp n}N n=1 the basis functions for the space ZN, with ZN ≡ Y N ≡ (QN)∗, then we have ZN = span {τn, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , 2N} [15, 19, 28, 29, 36, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Proper Orthogonal Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this Section we briefly describe the Proper Orthog- onal Decomposition (POD) Galerkin algorithm [6, 22, 49, 50] for the construction of a discrete solution manifold and the relative reduced spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Since in the unsteady case we use a space-time structure, this procedure can be described making no distinction between time-dependency and steadiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Firstly, we make a sampling of P by choosing Ntrain of its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, let us define the set of the train samples as PNtrain: we have that obviously PNtrain ⊂ P and the cardinality is |PNtrain| = Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The set PNtrain is denoted as the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We should pursue that Ntrain is large enough so as to ensure that PNtrain is a good “approximation” of the parameter space P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' PNtrain is built through a Monte-Carlo sampling method with respect to a uniform density with support equal to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Starting from the sampling, the POD algorithm manipulates Ntrain snapshots for the state, the adjoint and the control variables: (53) �� yN (µj), uN (µj), pN (µj) ��Ntrain j=1 with µj ∈ PNtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' After this step, a compressing stage is performed: from (53) we build N basis functions by only considering the most important parametric information and throwing away the redundant ones, with N ≤ Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' A partitioned approach is used, which means that, after the deterministic sampling, SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 15 we perform the POD algorithm separately for all the three variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Namely, we find three N- dimensional reduced spaces Y N, U N and (QN)∗ that minimizes the following three quantities: � � � � 1 Ntrain � µj∈PNtrain min ¯y∈Y N ��yN � µj � − ¯y ��2 Y , � � � � 1 Ntrain � µj∈PNtrain min ¯u∈U N ��uN � µj � − ¯u ��2 U, � � � � 1 Ntrain � µj∈PNtrain min ¯p∈(QN)∗ ��pN � µj � − ¯p ��2 Q∗, where obviously Y N ⊂ Y N , U N ⊂ U N and (QN)∗ ⊂ (QN )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us discuss the data compression procedure of the POD for the state variable y(µ) [6, 22, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As we are following a partitioned approach, the control and the adjoint variables follow the below discussion with usual modifications, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Firstly we collect a set of ordered parameters µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , µNtrain ∈ PNtrain, which the ordered snapshots yN (µ1) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , yN � µNtrain � are linked to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us define Cy ∈ RNtrain×Ntrain as the correlation matrix of the snapshots for the state variable as follows: (54) Cy ij := 1 Ntrain � yN (µi) , yN � µj �� Y , 1 ≤ i, j ≤ Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The next step is to find the pair eigenvalue-eigenvector (λy n, ey n), where ey n has norm equal to one, of the following problem: Cyey n = λy ney n, 1 ≤ n ≤ Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the sake of simplicity, we organise the eigenvalues λy 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , λy Ntrain in decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Consider the first N ones, specifically λy 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , λy N together with the related eigenvectors ey 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , ey N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We refer to the k-th component of the state eigenvector ey n ∈ RNtrain with the notation (ey n)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' After having finished the computation of the pair eigenvalue-eigenvector, the basis functions ηy n for the state equation are built through the following formula: (55) ηy n = 1 √ λy n Ntrain � k=1 (ey n)k yN (µk) , 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, our reduced spaces are built as (52) and, then, aggregated space technique is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As both N and Ntrain can be chosen by us, we should find sharp criteria in order to decide them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' A possibility can be to set them in based on a study of the eigenvalues, using the estimate [22, 39, 58]: (56) � � � � 1 Ntrain Ntrain � k=1 ∥yN (µk) − ΠN (yN (µk))∥2 Y = � � � � Ntrain � k=N+1 λy k, where ΠN : Y → Y N is a Galerkin projector of functions from Y onto Y N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (56) holds for the control and the adjoint in a partitioned approach, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The second member of equation (56) can be a measure of how well the FEM space is approximated by N reduced basis over the chosen training set of cardinality Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We summarise the whole POD procedure in the below Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 (Time-dependent problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' When we are dealing with time-dependent OCPs, the time instances are not separated in the POD procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, the space-time problem is studied as a steady one and each snapshot carries all the time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 16 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL Algorithm 1 POD algorithm for OCP problems in a partitioned approach Input: parameter domain P, FEM spaces Y N , U N and (QN )∗ and Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Output: reduced spaces Y N, U N and (QN)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Starting from the high-fidelity spaces Y N , U N and (QN )∗: 1: Sample Ptrain ⊂ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 2: for all µ ∈ Ptrain do 3: Solve the high-fidelity OCP system (34) (in this case a stabilized one);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 4: end for 5: Assemble the matrix Cy ij := 1 Ntrain � yN (µi) , yN � µj �� Y , 1 ≤ i, j ≤ Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Do the same for u and p variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 6: Compute its eigenvalues λy 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , λy Ntrain and the corresponding orthonormalised eigenvectors ey 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , ey Ntrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Do the same for u and p variables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 7: After having chosen N according to a certain criterion, define Y N = span {ηy n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , N}, where ηy n = 1 √ λy n �Ntrain k=1 (ey n)k yN (µk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Do the same for u and p variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 8: Define the aggregated space ZN = span � {ηy n}N n=1 ∪ {ηp n}N n=1 � and impose ZN ≡ Y N ≡ (QN)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Numerical Results In this Section we propose simulations regarding the Graetz-Poiseuille and the Propagating Front in a Square problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Regarding the steady case, the numerical experiments are coded through the RBniCS library [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' instead, the unsteady ones are implemented employing both RBniCS and multiphenics [1] libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' They are python-based libraries, built on FEniCS [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' When we will perform the Online-Offline stabilization procedure, we will always use the same stabilization parameter δK of the high-fidelity approximation also at the reduced level, both in steady and unsteady cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We will illustrate an analysis over relative errors between the FEM and the reduced solutions for all three variables, defined as (57) ey,N(µ) := ��yN (µ) − yN(µ) �� Y ∥yN (µ)∥Y , eu,N(µ) := ��uN (µ) − uN(µ) �� U ∥uN (µ)∥U , ep,N(µ) := ��pN (µ) − pN(µ) �� Q∗ ∥pN (µ)∥Q∗ , for the state, the control and the adjoint, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As we are dealing with parametrized OCPs, we will evaluate a simple average of (57) for µ uniformly distributed in a testing set Ptest ⊆ P of size Ntest for every dimension N = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' , Nmax of the reduced space obtained by our POD procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' More precisely, we will plot the base-10 logarithm of the average of (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For parabolic problems we will consider the sum of the errors with respect to each discretized instant of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Regarding the efficiency of ROMs, we use the speedup-index to compare the computational cost of the FEM solution with that of the reduced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This quantity is defined as: (58) speedup-index = computational time of the high-fidelity solution computational time of the reduced solution , which will be computed for each µ in the testing set with respect to the dimension N of the reduced spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As made with the relative error, we will consider the sample average of this quantity with respect to N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' however, for the sake of completeness, we will add its minumum and maximum value computed through the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For each test case, we will use the same Ptest to compute relative errors and the speedup-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The steady results are obtained with 16GB of RAM and Intel Core i7-7500U Dual Core, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7GHz for the CPU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' instead, the FEM and ROM parabolic simulations are run with 16GB of RAM and Intel Core i7 − 7700 Quad Core, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='60GHz for the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Numerical Experiments for the Graetz-Poiseuille Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The Graetz-Poiseuille prob- lem concerns the heat conduction in a straight duct, whose walls can be characterized by heat SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 17 exchange or maintained at a certain fixed temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This example is very well-known in the nu- merical Advection-Dominated literature [18, 37, 44, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We start by presenting the stationary case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We apply a distributed control in the whole domain and the parameter µ = µ1 > 0 is a physical component and characterizes the diffusion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The spatial coordinates of the system are denoted Ωobs Ωobs Ω Γ1 Γ2 Γ3 Γ4 Γ5 Γ6 (0,0) (1,0) (2,0) (2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2) (2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8) (2,1) (1,1) (0,1) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Geometry of the Graetz-Poiseuille Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' with (x0, x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The boundary of Ω is Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider Dirichlet boundary conditions (BC) on sides Γ1 := [0, 1] × {0}, Γ5 := [0, 1] × {1}, Γ6 := {0} × [0, 1] by imposing y = 0 and Γ2 := [1, 2] × {0} and Γ4 := [1, 2] × {1} by imposing y = 1, referring to Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We deal with homogeneous Neumann conditions on Γ3 := {2} × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The classic formulation of the problem is: (59) � � � � � � � � � � � � � � � − 1 µ1 ∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u, in Ω, y(µ) = 0, on Γ1 ∪ Γ5 ∪ Γ6, y(µ) = 1, on Γ2 ∪ Γ4, ∂y(µ) ∂ν = 0, on Γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Now we want to derive the optimality system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Ωobs := [1, 2]×[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8, 1]∪[1, 2]×[0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2] as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this case, the state belongs to the space: ˜Y := � v ∈ H1� Ω � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' it satisfies the BC in (59) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the sake of practice, it is better to introduce a lifting function Ry ∈ H1(Ω), such that it fulfills the BC in (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore we define the variable ¯y := y − Ry, with ¯y ∈ Y , where Y := � v ∈ H1 0 � Ω � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' ∂¯y ∂ν = 0, on Γ3 and ¯y = 0 on Γ \\ Γ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Nevertheless, without loss of generality, we will denote the new variable ¯y with y and we settle U := L2(Ω) and Q := Y ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, the adjoint variable p is null on Γ \\ Γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The mathematical formulation is described as follows (we omitted the dependence from µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Fixed α > 0, find the pair (y, u) ∈ Y × U that realizes (60) min (y,u)∈Y ×U J(y, u) = 1 2 � Ωobs � y − yd �2 dx + α 2 � Ω u2 dx such that e (y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) = 0, where e (y, u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) := a (y, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)+b (u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ)−⟨p, f(µ)⟩Y ∗Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As explained in Sections 2 and 3, we follow a Lagrangian approach and we use SUPG stabilization in a optimize-then-discretize framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We exploit P1-FEM approximation for the state, control and adjoint spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Here the stabilized forms as and a∗ s are, respectively: as � yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � := a � yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � + � K∈Th δK � K � 4x1(1 − x1)∂x0yN � � hK∂x0qN � , yN , qN ∈ Y N , a∗ s � zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � := a∗ � zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � + � K∈Th δK � K � 4x1(1 − x1)∂x0pN � � hK∂x0zN � , zN , pN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 18 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL We consider a parameter space P := � 104, 106� and a quite coarse mesh of size h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='029 for the FEM spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The training set Ptrain has cardinality Ntrain = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We choose δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 for all K ∈ Th and the penalization term is α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We pursue the convergence in the L2-norm of the state to the desired solution profile yd(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0, function defined on Ωobs of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We perform the POD algorithm for Nmax = 20 in a partitioned approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We illustrate the reduced solution for the state and adjoint variables in the best relative error scenario in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Namely, we plot the Only-Offline and Online-Offline Stabilized solutions for N = 1 and N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The values of N can be deduced by Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' From the gradient equation (34), we expect the distributed control u to be equal to the adjoint p up to the multiplicative constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (Top) Only-Offline stabilized state (left) and adjoint (right) for N = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (Bottom) Online-Offline stabilized state (left) and adjoint (right) for N = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' for the Graetz-Poiseuille Problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' P = � 104, 106� , µ1 = 105, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='029, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider the relative errors between the FEM and the reduced solution in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We use a testing set Ptest of 100 elements in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As previously cited, at N = 6 we reach the minima for all the three errors for the Online-Offline stabilization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' more precisely for the state we touch ey,6 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='65 · 10−9, for the adjoint ep,6 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='98 · 10−8 and the control eu,6 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='00 · 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In contrast with this situation, Only-Offline stabilization never falls under 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This implies that the best choice is to pursue the Online-Offline stabilization procedure for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' However, after N = 6 the errors begin slightly to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Our interpretation to this fact relies on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Despite the fact that this parameter space might be too large, however the coefficient which multiplies the diffusion operator is still absolutely low in value for every µ1 ∈ P, nearly 10−4 and 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, also thanks to SUPG stabilization and the distributed control action, the majority of snapshots can be very similar referring to the solution for µ1 = 105: this translates in very few bases to reach a good relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As a matter of fact, the eigenvalues λy 7, λu 7 are ≈ 10−15 and λp 7 ≈ 10−16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' by their decreasing order, all the subsequent eigenvalues are very close to zero machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Thus, recalling (55) it follows that all basis components with N ≥ 7 are affected by some rounding errors due to the orthonormalization procedure of the POD (for details see [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Finally, we take a look at the speedup-index in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' All the average values are better for the Only-Offline stabilized ROM procedure due to the fact that the stabilized forms are not taken into account in the online phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' However, the Online-Offline stabilized reduced solution shows very good behaviour, for instance, we have an average equal to 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 for N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Generally, in this case speedup-index takes average value around 2 · 102 order of magnitude for the first 20 basis elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9e-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1e+00-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1e-03 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5e-020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6e-02SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 19 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Relative errors between FEM and reduced solution for state (left), control (center) and adjoint (right), for Online-Offline and Only-Offline stabilization, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, Ntest = 100, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='029, P = � 104, 106� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Graetz-Poiseuille Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Only-Offline Stabilization Offline-Online Stabilization N min average max min average max 1 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 2 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 3 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 4 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 256, 7 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 5 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 6 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 7 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Speedup-index of the Graetz-Poiseuille Problem for Online-Offline and Only- Offline stabilizations with Ptest sampled from P = � 104, 106], Ntest = 100, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='029, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let us give a look to the unsteady version of Problem (59) with null initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The unsteady Graetz-Poiseuille problem without control has been presented in [37, 55], instead the OCP Graetz Problem under boundary control without Advection-dominancy is studied in [50, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Recalling Figure 1, for a fixed T > 0 we state the parabolic Graetz-Poiseuille Problem as follows: (61) � � � � � � � � � � � � � � � � � � � ∂y(µ) ∂t − 1 µ1 ∆y(µ) + 4x1(1 − x1)∂x0y(µ) = u, in Ω × (0, T), y(µ) = 0, on Γ1 ∪ Γ5 ∪ Γ6 × (0, T), y(µ) = 1, on Γ2 ∪ Γ4 × (0, T), ∂y(µ) ∂ν = 0, on Γ3 × (0, T), y(µ)(0) = 0, in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We do simulations in a space-time framework as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 for a prearranged number of time-steps Nt using a P1-FEM approximation for the high-fidelity solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The relative stabilized forms in (49) for derivatives along time for state and adjoint are, respectively: (62) ms � yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � = � yN , qN � L2(Ω) + � K∈Th δKhK � yN , ∂x0qN � K , yN , qN ∈ Y N , m∗ s � pN , zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � = � pN , zN � L2(Ω) − � K∈Th δKhK � pN , ∂x0zN � K , pN , zN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider a final time of T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 and a time step of ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1, hence we have Nt = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We choose a quite coarse mesh of h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='038 and the overall high-fidelity dimension is Ntot = 314820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' This means that a single FEM space for a fixed t has a dimension of N = 3498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider a initial condition of y0(x) = 0 for all x ∈ Ω referring to Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We want the state solution to converge FEM vs ROM averaged relative error - y (state) Online stab 101 Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Log-Error 10-1 10- Relative L 10-5 10-7 1 2 3 4 5 6 7 8 NFEM vs ROM averaged relative error - u (control) 101 Online stab Online not stab 10-1 10-5 10-7 1 2 3 4 5 6 7 8 NFEM vs ROM averaged relative error - p (adjoint) Online stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 101 10 10-3 10-5 10-7 1 2 3 4 5 6 7 8 N20 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL in the L2-norm to a desired solution profile yd(x, t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0, function defined for all t ∈ [0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0] and for all x in Ωobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Here the SUPG stabilization is implemented with parameters δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 for all K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' P := � 104, 106� and we choose a training set Ptrain of cardinality Ntrain = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then, we performed the POD algorithm with Nmax = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The penalization parameter is α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (Top) SUPG FEM solution for the state and (Bottom) for the adjoint at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1, t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5, t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Unsteady Graetz-Poiseuille Problem, µ1 = 105, Nt = 30, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='038, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As we will see in Figure 5 the performance of the Only-Offline stabilized reduced solutions are not so good in terms of accuracy, unlike the Online-Offline stabilized ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider a testing set of 100 elements in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As succeeded in the steady case in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1, after nearly N = 6 Online-Offline stabilized errors begin to fluctuate due to the nature of the eigenvalues of the correlation matrix (54) that are closed to zero machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For this reason we present the trend of error from 1 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' However, errors stays close to 10−7 for the state and the adjoint and 10−6 to the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For N = 6 we have ey,6 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='20 · 10−7, eu,6 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='10 · 10−6 and ep,6 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='18 · 10−7, instead for N = 20 we have ey,20 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='93 · 10−7, eu,20 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='25 · 10−7 and ep,20 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='21 · 10−7 for the Online-Offline stabilization ROM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Relative errors between the FEM and Only-Offline and Online-Offline stabi- lized solutions for the state (left), control (center) and adjoint (right), Unsteady Graetz- Poiseuille problem, Nt = 30, Ntest = 100, P = � 104, 106� , h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Finally, we can see the speedup-index for some value of N in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In both situation we can compute a huge number of reduced solutions in the time of a high-fidelity one: for the Offline-Online stabilization we have an average speedup-index of nearly 26000 for N = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' On the whole, average speedup-index has an order of magnitude of 2 · 104 for N ≤ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Numerical Experiments for Propagating Front in a Square Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this Section, we consider a problem studied in the Advection-Dominated form in [37, 55] from a numerical point of view and we will add a distributed control to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Let Ω be the unit square in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider the representation in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Also in this case, (x0, x1) are the coordinates of the square domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2e-021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2e-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='002 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5e-03FEM vs ROM averaged relative error - y (state) Online stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 100 Relative Log-Error 10-2 10-6 1 2 3 4 5 6 7 8 NFEM vs ROM averaged relative error - u (control) Online stab Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 101 Relative Log-Error 10-1 10-3 10-5 1 2 3 4 5 6 7 8 NFEM vs ROM averaged relative error - p (adjoint) Online stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 100 Relative Log-Error 10-2 10-4 10-6 1 2 3 4 5 6 7 8 NSUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 21 Only-Offline Stabilization Offline-Online Stabilization N min average max min average max 1 21588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 26588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 30971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 18968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 23588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 27062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 2 23821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 29723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 34817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 20757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 26018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 29929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 3 23571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 29468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 34349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 20547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 25698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 29662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 4 23062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 28880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 33702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 21385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 25380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 28883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 5 25762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 28767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 33488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 23021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 25882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 29388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 6 27003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 29707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 34544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 23236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 26054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 29677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 7 26658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 29481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 34277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 23206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 25879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 29505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Speedup-index of the Unsteady Graetz Problem for Online-Offline and Only- Offline stabilization with P = � 104, 106], α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, Nt = 30, Ntest = 100, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Γ1 Γ2 Γ3 Γ4 Γ5 Ω Ωobs (0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='25) (0,1) (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='75) (1,1) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='25,1) (1,0) (0,0) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Geometry of the Square Problem Referring to Figure 6, Γ1 := {0} × [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='25], Γ2 := [0, 1] × {0}, Γ3 := {1} × [0, 1], Γ4 := [0, 1] × {1}, Γ5 := {0} × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='25, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Ωobs := [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='25, 1] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='75, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Given µ = (µ1, µ2), the problem is formulated as (63) � � � � � � � − 1 µ1 ∆y(µ) + (cos µ2, sin µ2) · ∇y(µ) = u, in Ω, y(µ) = 1, on Γ1 ∪ Γ2, y(µ) = 0, on Γ3 ∪ Γ4 ∪ Γ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We assume that the Identity restricted to Ωobs as the Observation operator and Z := L2(Ωobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In our test cases, P := � 104, 105� × � 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='57 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In this case, we have that the domain of definition of our state y is ˜Y := � v ∈ H1� Ω � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' BC in (63) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Exactly as done in the previous paragraph, we define a lifting function Ry ∈ H1� Ω � such that satisfies BC in (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We define ¯y := y − Ry, even though we denote ¯y as y again for the sake of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider Y := H1 0(Ω), U = L2(Ω) and Q := Y ∗, hence the adjoint p is such that p = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We define the objective functional J exactly as in (60);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' instead, a and b are a (y, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) := � Ω 1 µ1 ∇y · ∇p + (cos µ2, sin µ2) · ∇yp dx, and b (u, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) := − � Ω up dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' and ⟨p, f(µ)⟩Y ∗Y = −a (Ry, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Then we follow usual discussions of Sections 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We exploit a P1-FEM approximation for the optimality system by using the usual SUPG stabi- lization technique, arriving to system (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Here, for the sake of completeness, we remark that the 22 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL stabilized forms as and a∗ s are, respectively: as � yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � := a � yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � + � K∈Th δK � K hK (cos µ2, sin µ2) · ∇yN (cos µ2, sin µ2) · ∇qN , a∗ s � zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � := a∗ � zN , pN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � + � K∈Th δK � K hK (cos µ2, sin µ2) · ∇pN (cos µ2, sin µ2) · ∇zN , for all yN , qN , zN , pN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As previously done, we build a training set Ptrain and a testing set Ptest with both cardinality ntrain = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The mesh size h is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='025 and therefore the overall dimension of the high-fidelity approximation is 12087, which implies that state, control and adjoint spaces have dimension equal N = 4029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The SUPG stabilization is implemented with parameters δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 for all K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The penalization parameter is α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01 and we pursue the state solution to be convergent in the L2-norm to a desired solution profile yd(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5, defined for all x in Ωobs of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In Figure 7 we observe state and adjoint FEM solutions for µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Instead, in Figure 8 we illustrate Only-Offline and Online-Offline reduced solution for the state and the adjoint variable with µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2) for N = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Numerical solution without stabilization and SUPG FEM solution with µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2) for state (left) and adjoint (right) variables in the Propagating Front in a Square Problem, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='025, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Only-Offline stabilized and Online-Offline stabilized reduced solutions with µ = (2 · 104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2) or state (left) and adjoint (right) variables in the Propagating Front in a Square Problem, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, N = 50, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='025, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0, P = � 104, 105] × � 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' These computational evidences and the analysis of the relative errors show that Online-Offline stabilization procedure is preferable in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In Figure 9, the trend is the same of all three variables, where errors continue to decrease along all N: we have ey,50 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='20·10−3, eu,50 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='67·10−4 and ep,50 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='16 · 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Concerning the speedup-index, the performance are quite good as seen in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the best approximation, we have that we can compute an average of 44 Online-Offline reduced solutions when we build the associated FEM one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Obviously, the Only-Offline stabilized one is slightly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' On the whole, speedup-index takes average value around 101, 102 order of magnitude for N ≤ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8e-029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='002 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='006 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4e-031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8e-029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='002 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='006 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4e-03SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 23 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Relative errors between FEM and reduced solutions with P = � 104, 105] × � 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='57] for the state (left), control (center) and adjoint (right) in the Propagating Front in a Square Problem, Ntest = 100, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='025, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Only-Offline Stabilization Offline-Online Stabilization N min average max min average max 1 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 10 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 20 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 30 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 40 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 50 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Speedup-index of the Propagating Front in a Square Problem for Online-Offline and Only-Offline stabilization with training set P = � 104, 105]× � 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='57], α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, Ntest = 100, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='025, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Now we study the unsteady case of the Propagating Front in a Square Problem for a fix T > 0: (64) � � � � � � � � � � � � � ∂y(µ) ∂t − 1 µ1 ∆y(µ) + (cos µ2, sin µ2) · ∇y(µ) = u, in Ω × (0, T), y(µ) = 1, on Γ1 ∪ Γ2 × (0, T), y(µ) = 0, on Γ3 ∪ Γ4 ∪ Γ5 × (0, T), y(µ)(0) = 0, in Ω, with initial value y0(x) = 0 for all x ∈ Ω referring to the domain in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We build a parabolic problem for a final time T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 and a time-step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1, hence Nt = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We choose a quite coarse mesh of size h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='036 and the overall dimension of the space-time system is Ntot = 174780, which means that a single FEM space has dimension N = 1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Again, our aim is to achieve in a L2-mean a desired solution profile yd(x, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5, defined for all t ∈ [0, 3] and x in Ωobs of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The penalization parameter is α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We set δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 for all K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Here we have that the stabilized forms in (49) for derivatives along time for state and adjoint equations are, respectively: ms � yN , qN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � = � yN , qN � L2(Ω) + � K∈Th δKhK � yN , (cos µ2, sin µ2) · ∇qN � K , yN , qN ∈ Y N , m∗ s � pN , zN ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' µ � = � pN , zN � L2(Ω) − � K∈Th δKhK � pN , (cos µ2, sin µ2) · ∇zN � K , pN , zN ∈ Y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We consider a parameter space equal to the steady case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' P := � 104, 105� × � 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='57 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Our training set has cardinality Ntrain = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In Figure 10 we show a representative stabilized FEM solution for µ = (2·104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2) for some instants of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We choose to perform a POD procedure with Nmax = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In Figure 11 one can see the relative errors of the three variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As previously said, Only- Offline procedure has not good error behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Instead, it is worth to note that in a Online-Offline FEM vs ROM averaged relative error - y (state) 100 10-2 Online stab Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 10-3 10 20 30 40 50 NFEM vs ROM averaged relative error - u (contro 100 Relative Log-Error 10-2 Online stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 10-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 10 20 30 40 50 NFEM vs ROM averaged relative error - p (adjoint) Online stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 101 100 10-1 10-2 10 20 30 40 50 N24 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' (Top) SUPG FEM state solution and (Bottom) SUPG FEM adjoint solution for µ = (2·104, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2) for time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 (left), t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 (center), t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0 (right), in the Parabolic Propagating Front in a Square Problem, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='036, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' stabilization context, errors between the FEM and the reduced solutions decrease as N grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The fact that we deal with a two-dimensional parameter space implies to require more N basis for a good approximation of the reduced solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We have ey,30 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='17 · 10−3, eu,30 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='59 · 10−3 and ep,30 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='62 · 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Therefore, also for this case test we can state that the SUPG stabilization is an efficient procedure for the ROMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Relative errors between the FEM and the Only-Offline and Online-Offline stabilized reduced solution for the state (left), control (center) and adjoint (right) solutions, respectively with P = � 104, 105� × � 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='57 � , Nt = 30, Ntest = 100, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Finally, we show the results about the speedup-index in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For N = 30, not only we have the best accuracy for the reduced problem, but we are able to computed, averagely, 3981 reduced solution in the interval of a FEM simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Speedup-index has an average order of magnitude of 103 overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Conclusions and Perspectives In this work, we presented the numerical experiments concerning Advection-Dominated OCPs in a ROM context with high P´eclet number, both in the steady and the unsteady cases, under SUPG stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Concerning ROMs, we can have two possibilities of stabilization: we can apply SUPG only to the offline phase or we can use it in both online and offline phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We analyzed relative 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8e-029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7e-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='002 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='006 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4e-03FEM vs ROM averaged relative error - y (state) 100 10-1 Relative L 10-2 Online stab Online not stab 5 10 15 20 25 30 NFEM vs ROM averaged relative error - u (control) 100 Relative Log-Error 10-2 Online stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Online not stab 5 10 15 20 25 30 NFEM vs ROM averaged relative error - p (adjoint) 100 10-1 10-2 Online stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Online not stab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 5 10 15 20 25 30 NSUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL 25 Only-Offline Stabilization Offline-Online Stabilization N min average max min average max 5 6136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='6 8659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='4 12654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 4588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 6605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 9914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 10 6018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 8278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 11989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='5 4282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 6231.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='1 8462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='8 2424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='7 3981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='3 6147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='9 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Speedup-index of the unsteady Propagating Front in a Square Problem for Online-Offline and Only-Offline stabilization with training set P := � 104, 105] × � 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='57], h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='036, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='01, Nt = 30, Ntest = 100, δK = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' errors between the reduced and the high fidelity solutions and of the speedup-index concerning the Graetz-Poiseuille and Propagating Front in a Square Problems, always under a distributed control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' A P1-FEM approximation for the state, control and adjoint spaces is used in a optimize-then- discretize framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Concerning parabolic problems, a space-time approach is followed and we applied in a suitable way the SUPG stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' For the ROM, we considered a partitioned approach for all three variables using the POD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In all the steady and unsteady experiments, the ROM technique performed excellently in a Online-Offline stabilization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Especially for parabolic problems, the speedup-index features large values thanks to the space-time formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Only-Offline stabilization technique performed very poorly in terms of errors, despite the little favorable speedup values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Thus, Online-Offline stabilization is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We also performed experiments inherent a geometrical parametrization and boundary control for the Graetz-Poiseuille Problem that are not shown in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Results were quite good for Online- Offline stabilization: we had some little oscillations regarding relative errors due to the complexity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' As a perspective, it might be interesting to create a strongly-consistent stabilization technique that flattens all the fluctuation for these two configurations, since, to the best of our knowledge, this topic is still a novelty in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Regarding the SUPG stabilization for parabolic OCPs in a optimize-then-discretize framework, it would be also worth to derive some theoretical results that gives us the accuracy of the numerical solution with respect to the time-step and the mesh-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In conclusion, as another goal it might be interesting to study the performance of new stabilization techniques for the online phases, such as the Online Vanishing Viscosity and the Online Rectification methods [4, 13, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Moreover, the extension of this setting to the uncertainty certification context will be the topic of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Acknowledgements We acknowledge the support by European Union Funding for Research and Innovation – Horizon 2020 Program – in the framework of European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' We also acknowledge the PRIN 2017 “Nu- merical Analysis for Full and Reduced Order Methods for the efficient and accurate solution of complex systems governed by Partial Differential Equations” (NA-FROM-PDEs) and the INDAM- GNCS project “Tecniche Numeriche Avanzate per Applicazioni Industriali”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' The computations in this work have been performed with RBniCS [2] library, developed at SISSA mathLab, which is an implementation in FEniCS [32] of several reduced order modelling techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' we acknowledge developers and contributors to both libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' References [1] multiphenics - easy prototyping of multiphysics problems in FEniCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' https://mathlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='it/multiphenics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' [2] RBniCS – reduced order modelling in FEniCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='rbnicsproject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' 26 SUPG ROMS FOR ADVECTION-DOMINATED PDES UNDER OPTIMAL CONTROL [3] Tu˘gba Akman, B¨ulent Karas¨ozen, and Zahire Kanar-Seymen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Streamline Upwind/Petrov-Galerkin solution of optimal control problems governed by time-dependent diffusion-convection-reaction equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' TWMS Journal of Applied and Engineering Mathematics, 7(2):221–235, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' [4] Shafqat Ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Stabilized reduced basis methods for the approximation of parametrized viscous flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' PhD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Thesis, SISSA, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' [5] Kendall Atkinson and Weimin Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Theoretical numerical analysis, volume 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Springer, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' [6] Francesco Ballarin, Gianluigi Rozza, and Maria Strazzullo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Chapter 9 - Space-time POD-Galerkin approach for parametric flow control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' In Emmanuel Tr´elat and Enrique Zuazua, editors, Numerical Control: Part A, volume 23 of Handbook of 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Weighted Reduced Order Methods for parametrized PDEs in uncertainty quantification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} +page_content=' Master’s thesis, University of Trieste and SISSA, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNA0T4oBgHgl3EQfBP9o/content/2301.01973v1.pdf'} diff --git a/G9FJT4oBgHgl3EQfty2e/vector_store/index.faiss b/G9FJT4oBgHgl3EQfty2e/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8c9034a52ca258d143db54258eadaafc310479df --- /dev/null +++ b/G9FJT4oBgHgl3EQfty2e/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e0192703b5db0d973d26f28def49bbe1e924b40c74370b5621604cc7f52c8649 +size 5570605 diff --git 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b/HdE3T4oBgHgl3EQfWgre/content/tmp_files/2301.04470v1.pdf.txt @@ -0,0 +1,1226 @@ +InstaGraM: Instance-level Graph Modeling +for Vectorized HD Map Learning +Juyeb Shin +Francois Rameau +Hyeonjun Jeong +Dongsuk Kum +Korea Advanced Institute of Science and Technology +{juyebshin, frameau, hyeonjun.jeong, dskum}@kaist.ac.kr +Abstract +The construction of lightweight High-definition (HD) +maps containing geometric and semantic information is of +foremost importance for the large-scale deployment of au- +tonomous driving. To automatically generate such type of +map from a set of images captured by a vehicle, most works +formulate this mapping as a segmentation problem, which +implies heavy post-processing to obtain the final vector- +ized representation. Alternative techniques have the abil- +ity to generate an HD map in an end-to-end manner but +rely on computationally expensive auto-regressive models. +To bring camera-based to an applicable level, we propose +InstaGraM, a fast end-to-end network generating a vec- +torized HD map via instance-level graph modeling of the +map elements. Our strategy consists of three main stages: +top-view feature extraction, road elements’ vertices and +edges detection, and conversion to a semantic vector rep- +resentation. After top-down feature extraction, an encoder- +decoder architecture is utilized to predict a set of vertices +and edge maps of the road elements. Finally, these ver- +tices along with edge maps are associated through an at- +tentional graph neural network generating a semantic vec- +torized map. Instead of relying on a common segmentation +approach, we propose to regress distance transform maps +as they provide strong spatial relations and directional in- +formation between vertices. Comprehensive experiments on +nuScenes dataset show that our proposed network outper- +forms HDMapNet by 13.7 mAP and achieves comparable +accuracy with VectorMapNet 5× faster inference speed. +1. Introduction +Autonomous vehicles heavily rely on HD maps to en- +sure reliable and accurate localization [14, 22, 48]. +This +kind of map contains various elements, such as road mark- +ings and traffic signs localized at a centimetric level. While +HD maps are crucial for deploying autonomous driving, +they remain costly to create and maintain. To cope with +these limitations, tremendous efforts have been deployed +toward online HD map generation from onboard sensor es- +timations [9,16,23,24,26–28,34–36,47,49]. Note that for +this mapping, sensors providing a full 360◦ field of view, +such as LiDAR and multi-camera systems, are preferred +to produce a complete map of the environment [5, 8, 46]. +Seminal works [24,34–36,47,49] formulate this problem as +a rasterized bird’s-eye-view (BEV) semantic segmentation. +While these approaches demonstrated their relevance, such +representation remains memory intensive and lacks struc- +tural relationships desirable for vehicular navigation. To al- +leviate these issues, recent works attempt to directly con- +struct vectorized HD maps that are lighter and readily ap- +plicable to autonomous driving platforms [16, 23, 26–28]. +Despite their compelling performances, the aforementioned +vectorized HD map networks remain too computationally +demanding for a realistic deployment. In this context, we +propose InstaGraM, Instance-level Graph Modeling for fast +HD map construction model that predicts map elements as +a set of structured polylines from a set of cameras mounted +on a vehicle. To reach this goal, our end-to-end network +is composed of three stages, as depicted in Figure 1. First, +we extract the CNN feature maps from each image – cap- +tured by the camera rig – and aggregate them into a single +top-down feature map using a 2D-to-BEV transformation +network [23, 34]. Then, given this top-down feature map, +we detect the road elements’ vertex points and edge maps +via CNNs. Finally, the vertices’ positions and their respec- +tive local edge maps are passed to a Graph Neural Network +(GNN) to predict their instance-level connections as an ad- +jacency matrix trained in a supervised manner. +Our contributions can be summarized as: +• We propose a novel graph modeling for vectorized +polylines of map elements that models geometric, se- +mantic and instance-level information as graph repre- +sentations. +• On top of the proposed graph modeling, we present +InstaGraM, an end-to-end vectorized HD map learning +network designed for real-time performance +1 +arXiv:2301.04470v1 [cs.CV] 10 Jan 2023 + +BEV feature projection +Map component detection +Association +Vectorized HD map +Input images +InstaGraM +Figure 1. We propose InstaGraM, a hybrid architecture of CNNs and a GNN for real-time HD map learning in bird’s-eye-view representa- +tion. Starting from the input surround images and camera parameters, a unified BEV representation is extracted by projecting and fusing +image features. InstaGraM extracts vertex locations and implicit edge maps of map elements, and final vectorized HD map elements are +generated throughout a GNN. +2. Related Work +Road Detection and Segmentation. Overall, extracting +high-level scene semantic information from onboard sen- +sors has always been of high interest for autonomous driv- +ing. +In particular, the detection [1, 21] and segmenta- +tion [17] of road elements (e.g., lanes, road marks, and road +signs) have been highly valuable for a large spectrum of +tasks, including localization [2,14], lane keeping [33], auto- +parking [18], and much more. To achieve this goal, early al- +gorithms for lane detection rely on hand-crafted binary seg- +mentation followed by lines or curves fitting strategies [3]. +While such approaches are often fast, they remain brittle to +adverse conditions (e.g., lighting, shadow, road wear) and +are limited to lane detection. For these reasons, these meth- +ods have been progressively substituted by deep learning- +based strategies offering more flexibility and robustness to +lane and road markings segmentation [25]. While the tech- +niques mentioned above provide rich information from the +camera’s viewpoint, they do not contain 3D information +about the observed elements in the scene. +Bird’s-eye-view Map Segmentation. +To offer a more +comprehensive 3D understanding of the vehicle’s surround- +ings, recent studies demonstrate that BEV semantic seg- +mentation is particularly well suited for autonomous driv- +ing [31]. When only road markings are considered, such +representation can easily be obtained via an Inverse Per- +spective Mapping (IPM) of the image-level segmentation +results [19] – assuming known camera’s intrinsics, eleva- +tion, and tilt. This IPM warping is valid under the planar +assumption but is violated for any object above the ground +level (e.g., cars and pedestrians) leading to severe perspec- +tive distortion in the resulting BEV. To avoid such stretch- +ing effect, [6, 29] predict the vehicles foot-print to respect +the planar assumption. To avoid using dedicated cars’ foot- +print datasets, Cam2Bev [37] attempts to correct this dis- +tortion directly from warped segmentation masks and [51] +proposes a GAN-based approach to directly transform front +facing views into BEV on which the segmentation can be +performed. +Alternatively, to deal with uneven road surfaces and non- +flat objects, some approaches utilize depth information to +warp the segmentation results adaptively [41]. Following a +similar philosophy, [36] proposes to use depth information +to combine the CNN features into the BEV space. Thus the +segmentation can be compute directly in this representation, +allowing a better integration of multi-camera system. Ho- +mographic or depth-aware warping strategies have the ad- +vantage of being intuitive, interpretable and to offer good +transferability to various camera setups. Despite these ad- +vantages, geometric warping solutions face multiple limita- +tions: they rely on strong prior, may suffer from perspective +distortions and require successive stages. To circumvent +these limitations, another solution consists in using neural +networks to learn the image(s) to BEV transformation im- +plicitly. One of the pioneering work employing this strat- +egy is VED [30] which directly employ a variational auto- +encoder to predict the BEV from an input image without +intermediate stages. +To better preserve spatial information and to ease the in- +tegration of cross-view information, follow up works rely +on more interpretable and elegant approaches to map the +transformation between the features in the camera view and +the BEV. One of the seminal work is [34] where a Multi- +layer Perceptron (MLP) is used to learn this mapping. After +the image features from multiple views are mapped onto +a unified BEV, the segmentation can be learned into this +final representation. This approach combines multiple ad- +vantages. Unlike its IPM counterpart, it does not require +any prior calibration and it is not affected by perspective +distortion (global receptive field). As a result, this strategy +has influenced numerous works proposing various improve- +ments such as multi-resolution features [38, 39] and learn- +able bi-directional projection [23]. More recently, to pro- +2 + +vide more expressive and data dependent mapping, the use +of transformer networks has grown [35,47]. The problem of +these approaches is their high memory requirement, to alle- +viate this issue recent works adopt deformable transformer +networks [9,24,49]. +Vectorized HD Map Detection. The previously introduced +literatures [24,34–36,47,49] predict map elements in a ras- +terized BEV space. The downside of this representation is +its lack of structural relations and instance-level informa- +tion. In order to provide a lighter and more suitable rep- +resentation for self-driving related downstream tasks [13], +recent works [16, 23, 26–28] propose estimating the vec- +torized HD map elements instead of a segmentation map. +InstaGraM belongs to this category. A representative work +is HDMapNet [23] which generates a vectorized HD map +by post-processing various BEV segmentation maps. De- +spite promising results, the heuristic post-processing re- +quires large amount of computations. In order to predict +a vectorized map in an end-to-end manner, VectorMap- +Net [28] proposes two successive transformer decoders; the +first decoder detects map elements via cross-attention be- +tween the BEV feature and element queries while the sec- +ond transformer adopts auto-regressive decoder to recur- +rently generate polylines. +However, detection from ele- +ment queries with cross-attention is known for its slow con- +vergence, thus requires longer train epochs [7, 50]. Auto- +regressive decoder in polyline generator of VectorMapNet +makes its computation heavy, which is not applicable for +real-time autonomous driving tasks. In contrast, our pro- +posed architecture does not require large amount of training +time nor heavy computation of recurrent model. +A work sharing similarities with our approach is Poly- +World [52] which predicts the buildings’ contours as a set +of polygons from satellite images. Similarly to our strategy, +this technique uses a CNN for vertex detection followed +by a GNN for association. In contrast with PolyWorld, we +adopt the interest point decoder from [11] predicting high- +resolution vertices’ coordinates. Furthermore, we leverage +the distance transform embedding for implicit directional +information between vertices to associate. Finally, our strat- +egy is designed for road elements detection requiring both +semantic and instance segmentation information. +3. Method +We propose an end-to-end network to compute a BEV +vectorized HD map from a set of cameras mounted on a +vehicle. +To represent road elements (i.e., lane dividers, +pedestrian crossing, and road boundaries), HD maps typi- +cally consist of 2D polyline vertices and their instance-level +adjacency connectivity. To obtain this vector representa- +tion, previous works rely on segmentation prediction and +heavy post-processing [23], or auto-regressive models [28] +known for their high computational cost. In contrast, we +propose a lighter pipeline based on a combination of CNNs +and a GNN able to predict a set of vertices and their adja- +cency directly. Our method is three-folded. First, similarly +to HDMapNet [23], we utilize an MLP-based approach to +build a unified BEV features map from the CNN features +extracted from each image captured from the camera rig – +via an EfficientNet [43]. From this BEV feature map, two +CNN decoders extract the vertices and edge maps of the +observed road elements. Finally, these vertices and their lo- +cal edge response are fed to an attentional GNN in order +to learn the semantic class and the connection between the +vertices. +3.1. Neural View Transform +The very first stage of our HD map estimation network is +extraction of the top-down BEV features map Fbev by com- +bining the CNN features [43] from each images captured +by the camera rig at a given time. For fair comparison with +baselines, we adopt simple Neural View Transform that re- +lates perspective view pixels and BEV pixels via a simple +MLP. For more information regarding the extraction of this +feature map, we refer to [23] that we have carefully repli- +cated for this stage. +3.2. Element Detector Heads +From the top-down feature map Fbev, we extract the ver- +tices and edges of the HD map elements using two CNN +decoders φV, φE respectively. These two components are +predicted in the rasterized BEV space RWbev×Hbev similar +to segmentation tasks. The vertex decoder φV adopts the in- +terest point decoder from [11] and extracts possible position +heatmap at every 8 × 8 local, non-overlapping grid in BEV +pixels. It computes X ∈ R +Wbev +8 +× +Hbev +8 +×65, the 65 channels +indicating possible position in the local grids with an addi- +tional ”no vertex” dustbin. After a channel-wise softmax, +the dustbin dimension is removed and the vertex heatmap is +reshaped from RWc×Hc×64 to RWbev×Hbev. In parallel with +the vertex decoder, the edge map decoder φE predicts the +distance transform map D ∈ RWbev×Hbev×3, the 3 channels +indicating the number of class categories of map elements. +This edge map of distance transform [4] implicitly provides +spatial relations between vertices and directional informa- +tion of map elements inspired from [16,26,27]. We further +demonstrate in section 4 that this distance transform repre- +sentation as an edge map plays a significant role in instance- +level association. We apply ReLU and a threshold after the +last Conv layer to predict the distance values from 0 to 10 +in the rasterized BEV image. +3.3. Association via Graph Neural Network +The two components extracted from element detector +heads are associated via a graph neural network, where all +vertices interact throughout an attention scheme [12, 44]. +3 + +Neural +View +Transform +Vertex Decoder ������������������������ +Edge Decoder ������������ℰ +Attentional +GNN +Backbone +CNN +CNN +Vertex map +Edge map +Self-attention layer +BEV feature extractor +Element detector heads +Association +ℒ������������ +ℒℰ +Input images +Vectorized HD map +Self-attention +Graph +embedding ������������������������ +Class score ������������������������ +(a). BEV feature extractor +(b). Element detector heads +(c). Association +Lane divider +Ped. crossing +Road boundary +⋯ +BEV feature ℱ������������������������������������ +× ������������ +Attentional GNN +Row +norm. +Col +norm. +Sinkhorn agrorithm +������������ +“Divider” +ℒ������������������������������������ +ℒ������������������������������������ +Adjacency matrix += 1 +Figure 2. Proposed InstaGraM architecture. The blocks at the top show the overall components of InstaGraM architecture and the bottom +blocks show the details of structure and training of each component. +This allows our network to reason about both point-level +and instance-level relations between map elements based on +various attributes including positions, implicit edge map of +distance values and class categories. +Graph Embeddings: We combine vertex positions and dis- +tance transform maps to form initial graph embeddings. We +first extract the position of each vertex in rasterized BEV co- +ordinate and their respective confidence from channel-wise +softmax in the vertex position heatmap, vi = (xi, yi, c). We +only extract one distinctive vertex position with maximum +confidence in each 8 × 8 grid cell, which is acting similar +to Non-Maximum Suppression. After extraction, a ith ver- +tex position vi is encoded by a sinusodial positional encod- +ing function γ to augment it into a high-dimensional vec- +tor [32]. This positional encoding is further supported by +an additional shallow MLP. To complement the positional +information of the vertex vi, we additionally include the +local directional information as the embedding of the dis- +tance transform patch corresponding to the same grid cell. +Then our initial graph consists of D-dimensional embed- +dings 0gi ∈ RD, combining both the vertex position and its +directional local information, can be formulated as: +(0)gi = MLPV(γ(vi)) + MLPE(di). +(1) +This enables us to associate multiple graph embeddings +based on their vertex and edge representation throughout +an attention scheme. +Attentional Message Passing from SuperGlue [40]: We +start from an initial graph (0)G with nodes containing +both vertex position and edge map embeddings as a high- +dimensional vector. +This initial graph has bidirectional +edges, connecting vertex i to all other vertices. +To fur- +ther enhance the nodes and find the final edges of the ver- +tices, we pass the initial graph to the attentional graph neu- +ral network and propagate this graph through message pass- +ing [40,45]. Our objective is to find final bidirectional edges +of the vertices as an instance-level information of map ele- +ments. We feed our initial graph to attentional graph neu- +ral network that aggregates graph embeddings via a mes- +sage passing consists of MLP and Multi-head Self Attention +(MSA): +(0)G = [(0)g1;(0) g2; . . . ;(0) gN ] +(l)G = (l−1)G + MLP([(l−1)G∥MSA((l)G)]), l = 1, . . . , L +(2) +Self-attention and aggregation in Equation 2 provides in- +teraction between all the graph embeddings based on their +spatial and directional appearance embedded in g1. Con- +cretely, each vertex node attend to all other nodes to find +the next possible vertices that would appear in the map. Af- +ter L layers of attentional aggregation, class scores li ∈ R3 +and graph matching embeddings fi ∈ RD are obtained: +li = MLPcls((L)gi) +fi = MLPmatch((L)gi) +(3) +Adjacency Matrix: We predict optimal edges by comput- +ing score matrix ˆS ∈ RN ×N between nodes of the graph +(L)G. The adjacency score between nodes i and j can be +4 + +������������5 +������������3 +������������8 +������������1 +������������4 +������������10 +������������9 +������������7 +������������6 +������������2 +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ ������������������������������������ +∅ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������ +������������������������������������ +∅ +������������5 +������������3 +������������8 +������������1 +������������4 +������������10 +������������9 +������������7 +������������6 +������������2 +Forward match +ℳ +Backward match +ℳ������������ +Adjacency matrix ̂������������ +Lane divider +Ped. crossing +Road boundary +Figure 3. We model polylines of map elements by a graph with +bidirectional edges. Attentional graph neural network and opti- +mal matching in InstaGraM computes symmetric adjacency matrix +ˆA ∈ R(N +1)×(N +1) with both forward and backward connec- +tions. With augmented dustbin vertex ∅, adjacency matrix pro- +vides instance-wise prediction of the map elements. Note that for +vertex that does not have any matched vertex (i.e. v2 above) only +has connection to dustbin vertex in both forward and backward, +thus colored in purple. +computed as cosine similarity of embedding vectors +ˆSij =< fi, fj >, ∀{i, j} ∈ N × N, +(4) +where < ·, · > is an inner product of two embeddings. +Following SuperGlue, we augment this score matrix to +¯S ∈ R(N +1)×(N +1) with dustbin node for vertices that +might not have any match, i.e. a vertex at the end of an +element instance. The Sinkhorn algorithm [10,42] that iter- +atively normalizes exp (¯S) along rows and columns is used +to compute final adjacency matrix of the graph. Adjacency +matrix ˆA ∈ R(N +1)×(N +1) with instance-level edges can +be computed throughout this optimal matching with aug- +mented score ˆS. +3.4. Losses +We design the whole network to be differentiable, allow- +ing us to train it in a fully supervised manner with combina- +tions of losses at multiple branches. For supervision of el- +ement detector heads, cross-entropy with softmax loss and +L2 loss are used for vertex location heatmap and distance +transform map respectively. +LV(X, Y) = +1 +Hc, Wc +Hc,Wc +� +h=1,w=1 +lp(xhw; yhw) +LE( ˆD, D) = 1 +N +� +dp∈D +∥dp − ˆdp∥2, +(5) +where Hc = Hbev +8 +and Wc = Wbev +8 +are the indexing dimen- +sion of 8 × 8 local cells. +The coordinates from the vertex location heatmap pre- +diction may not align perfectly to the ground truth vertex +coordinates, specifically in the early stage of training, re- +sulting in ambiguity of the ground truth adjacency and class +label. To address this we find the nearest pairs between the +ground truth vertices and predicted vertices to provide the +ground truth for the output of the graph neural network, ad- +jacency matrix and class predictions. The nearest ground +truth vertex σ(i) to the predicted vertex i is obtained that +minimizes the Chamfer distance cost with threshold D0: +σ = +arg min +D(vi,vσ(i)) / Etotal +10 +2 +10 +3 +0 +0.4 +0.8 +Rew + / Etotal +10 +2 +10 +3 +0 +0.4 +0.8 +Rew + / Etotal +10 +2 +10 +3 +0 +0.4 +0.8 +(a) +(b) +(c) +(1, 1) mode +(2, 1) mode +(1, 2) mode +(2, 2) mode +Figure 9. Time-averaged energy contained in the first four Fourier modes as functions of Rew +under (a) the (1, 1) type wall shear, (b) the (2, 1) type wall shear, (c) the (1, 2) type wall shear. +Note the Fourier mode decomposition is only applied when the flow is in turbulent state for 2-D +cases (refer to the phase diagram of flow states in figure 3). +Here, the Fourier basis (ˆum,n, ˆvm,n) is chosen as +ˆum,n(x, y) = 2 sin(mπx) cos(nπy) +(3.3) +ˆvm,n(x, y) = −2 cos(mπx) sin(nπy) +(3.4) +The instantaneous amplitude of the Fourier mode is then calculated as +Am,n +x +(t) = ⟨u(x, y, t), ˆum,n(x, y)⟩ = +� +i +� +j +u(xi, yi, t)ˆum,n(xi, yi) +(3.5) +Am,n +y +(t) = ⟨v(x, y, t), ˆvm,n(x, y)⟩ = +� +i +� +j +v(xi, yi, t)ˆvm,n(xi, yi) +(3.6) +where ⟨u, ˆu⟩ and ⟨v, ˆv⟩ denote the inner product of u and ˆu, v and ˆv, respectively. The +energy in each Fourier mode is calculated as Em,n(t) = +� +[Am,n +x +(t)]2 + [Am,n +y +(t)]2, the +total energy is calculated as Etotal = � +m,n⟨Em,n⟩, and ⟨· · · ⟩ denotes the time average. +In figure 9, we plot the time-averaged energy as functions of Rew for various types of +wall shear when the flow is in the turbulent state for 2-D cases. Here, we consider m and +n = 1, 2, namely the first four Fourier modes. From figure 9(a) we can see that under the +(1, 1) type wall shear, the (1, 1) Fourier mode is indeed dominant. Similarly, under the +(1, 2) type wall shear, the (1, 2) Fourier mode is the dominant flow mode (see figure 9c). +However, under the (2, 1) type wall shear, despite the energy percentage in the (2, 1) +mode being much larger compared to that in the absence of wall shear, the (2, 1) mode +does not contain the highest percentage of energy. We can see from figure 9(b) that the +(1, 1) Fourier mode contains more energy than the expected (2, 1) mode. To explain the +discrepancy, we check the snapshots of the flow fields and the heat flux fields (see figures +2b and 7b) and observe that some hot (or cold) plumes lose their energy before reaching +the cold top (or hot bottom) wall. The plumes then fall (or turn back up) to form small +rolls, thus substructures emerge inside the left-side big roll (Chen et al. 2019). When +the unstable small rolls inside the left-side big roll shrink their size, the Fourier mode +decomposition that captures flows in the bulk region implies the (1, 1) mode (i.e., one +big roll in the whole cell) prevails. +3.2. Stabilizing thermal turbulence via wall movement +The original objective of imposing the (m, n) type wall shear is to adjust the internal +flow mode and control heat transfer properties. While we found that by increasing the +wall shear strength, the thermal turbulence is relaminarized, and more surprisingly, the +heat transfer efficiency of the convection cell in the laminar state is higher than that in + +16 +A. Xu, B.-R. Xu and H.-D. Xi +the turbulent state. In the previous section, we have explained that the enhancement of +heat transfer efficiency at the laminar regime is due to the formation of more stable and +stronger convection channels. Below, we further discuss the origin of thermal turbulence +laminarization. We start by examining the turbulent kinetic energy (TKE) equation of +incompressible thermal convection, which is written as +∂∥ +∂t + uj∂j∥ = −u′ +iu′ +j∂jui ++ ∂j +Ç +−p′u′ +j + +… +Pr +Ra∂j∥ − 1 +2u′ +iu′ +iu′ +j +å +− +… +Pr +Ra(∂ju′ +i)2 + T ′v′ +(3.7) +In the above, the subscripts i and j are dummy indices. ∥ = u′ +iu′ +i/2 denotes the TKE and +the superscript (′) denotes the fluctuation part of an instantaneous flow variable. The +term −u′ +iu′ +j∂jui represents shear-produced TKE, and the term T ′v′ represents buoyancy- +produced TKE. Because the flow remains laminar in a smaller range of Rew for 3-D +cases, we mainly discuss the results for 2-D cases below. In figure 10, we show the shear- +produced, the buoyancy-produced, and the total TKE production under the (1, 1) type +wall shear as an example. With increasing the wall shear strength, the shear-produced +TKE is increasingly concentrated near the top-left and bottom-right corners of the +convection cells (see figures 10a and 10d), where rising hot (or falling cold) plumes impact +the cold (or hot) boundary layers. Compared to the shear-produced TKE, the buoyancy- +produced TKE is more intense (see figures 10b and 10e) and contributes a dominant part +of the total TKE production (see figures 10c and 10f); meanwhile, with increasing the wall +shear strength, the buoyancy-produced TKE becomes weaker. Previously, in the absence +of wall shear, Xia et al. (2003) quantitatively described that the TKE largely comes from +the buoyant motions of thermal plumes based on the particle image velocimetry (PIV) +results. With the aid of direct numerical simulation, T ′v′ is directly obtained in the +whole convection cell, we now provide direct evidence that thermal plumes are mainly +responsible for the TKE production. +After analyzing the TKE production, we now turn to the TKE dissipation. In figures +11(a) and 11(b), we show the TKE dissipation under the (1, 1) type wall shear as an +example. We can see that intense TKE dissipation occurs in the top-right and bottom- +left corners, namely, in the regions of plumes detachment. Meanwhile, with increasing +the wall shear strength, the TKE dissipation becomes weaker. It should be noted +that here we considered the dissipation term of +� +Pr/Ra(∂ju′ +i)2 in the TKE equation, +which is known as pseudo-dissipation by Pope (2000). Previously, Zhang et al. (2017) +and Bhattacharya et al. (2018) analyzed the statistics of TKE dissipation in terms of +1 +2 +� +Pr/Ra(∂ju′ +i + ∂iu′ +j)2 in the canonical RB convection without wall shear. We checked +that the numerical differences between +� +Pr/Ra(∂ju′ +i)2 and +1 +2 +� +Pr/Ra(∂ju′ +i + ∂iu′ +j)2 +are indeed very small. We then plot the volume-averaged TKE and the volume-averaged +TKE dissipation as functions of Rew for various types of wall shear in figures 11(c) and +11(d). With the increase of wall shear strength, the volume-averaged TKE is indeed +decreasing, eventually, the TKE vanishes, and the thermal turbulence is relaminarized. +However, the decreased TKE and the corresponding thermal turbulence laminarization +are not caused by the viscous dissipation, which is evident from figure 10(d) that the +volume-averaged TKE dissipation is also decreasing. +The above analysis suggests that the plume plays a key role in thermal turbulence +production and dissipation. To identify the mechanism responsible for the thermal tur- +bulence laminarization, we then analyze the spatial and temporal distributions of plumes. +In figures 12(a-c), we show the typical snapshots of instantaneous plume field under the + +Wall sheared thermal convection +17 +(a) +(b) +(c) +(d) +(e) +(f ) +TKE production +-0.02 -0.01 0 0.01 0.02 +Figure +10. +(a,d) +The +shear-produced +turbulent +kinetic +energy +(TKE), +(b,e) +the +buoyancy-produced TKE, and (c,f ) the total TKE production, under the (1, 1) type wall shear +for (a-c) Rew = 200, (d-f ) Rew = 500. +0 0.001 0.002 0.003 0.004 +(a) +(b) +Rew +TKE +10 +2 +10 +3 +10 +-4 +10 +-3 +10 +-2 +(c) +Rew +TKE dissipation +10 +2 +10 +3 +0 +0.0003 +0.0006 +TKE dissipation +(d) +(1, 1) type +(2, 1) type +(1, 2) type +(1, 1) type +(2, 1) type +(1, 2) type +Figure 11. The TKE dissipation for (a) Rew = 200 and (b) Rew = 500 under the (1, 1) type +wall shear; (c) the volume-averaged TKE, and (d) the volume-averaged TKE dissipation as +functions of Rew for various types of wall shear. + +0 +0.001 +0.002 +0.003 +0.0040.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.818 +A. Xu, B.-R. Xu and H.-D. Xi +Rew +Plume Area +10 +2 +10 +3 +0.05 +0.1 +0.15 +(1, 1) type +(2, 1) type +(1, 2) type +(a) +(b) +(c) +(d) +Figure 12. Typical snapshots of plume field at Rew = 100 for (a) the (1, 1) type wall shear, +(b) the (2, 1) type wall shear, (c) the (1, 2) type wall shear. (d) Time-averaged plume area in +the cell as functions of Rew under three types of wall shear. +three types of wall shear. Here, the criteria to quantitatively identify thermal plumes are +similar to those used in (Huang et al. 2013; van der Poel et al. 2015; Zhang et al. 2017), +which is +|T (x, t) − ⟨T (x)⟩| > c⟨Trms(x)⟩, +√ +Pr · Ra|v(x, t)T (x, t)| > cNu +(3.8) +Here, c is an empirical constant and its value can be chosen as 0.8 ⩽ c ⩽ 1.2, and we +adopt the value of c = 1. This criterion assumes that plumes occur in regions of local +temperature maximum (or minimum), as well as regions where local convective heat flux +is larger than the spatial and temporal averaged one. We can see from figures 12(a-c), +this empirical criterion can extract the plume structures reasonably well in the sheared +convection. We also calculate the time-averaged plume area in the cell and plot the +plume areas as functions of Rew. From figure 12(d), we can see that with the increase of +wall shear strength, plume areas generally decrease under all three types of wall shear. +Because thermal plumes are mainly responsible for TKE production, a reduced number +of plumes indicates reduced TKE production. +We then examine the flow field during the laminarization process, as shown in figure +13, and the corresponding video can be viewed in the supplementary movie 2. Initially, an +instantaneous flow field obtained at Rew = 200 for the 2-D case and Rew = 100 for the 3- +D case (i.e., the turbulent state) is used to start the simulation, in which the shear effects +are relatively weak and the flow is buoyancy dominated. We can see plumes self-organize +into the LSC, and large magnitudes of velocity vectors appear near the region where +plumes erupt (see figure 13a for the 2-D case). When the wall shear strength increases to +Rew = 2000 for the 2-D case, the plumes have less chance to detach from the boundary +layers near the top and bottom walls, and they will be swept along the walls (see figure +13b). Because the organization of plume motions leads to the LSC in the turbulent RB + +Wall sheared thermal convection +19 +convection cell (Xi et al. 2004), suppressing plume detachment will weaken the LSC. In +addition, hot (or cold) plumes are forced to sweep to the cold top (or hot bottom) wall +(see figures 13c and 13d) and thermal plumes exchange heat near the walls, while the +temperature in the bulk region of the cell is more uniform and well-mixed (see figure +13e). Eventually, one regular big roll is formed, and hot and cold fluids flow along the +wall, which is completely influenced by the external wall shear (see figure 13f). We can +also see from figures 13(g-l) that, the turbulence relaminarization process in similar for +both 2-D and 3-D cases; however, it is noteworthy that due to prominent shear instability +effects in 3-D, the turbulence relaminarization is rare and occurs in a smaller range of +wall shear Reynolds number for 3-D cases. +3.3. Expenditure of mechanical energy due to external wall shear +We manipulated the internal flow modes via imposing external wall shear, and the cor- +responding heat transfer efficiency enhancement requires the expenditure of mechanical +energy. To evaluate whether such mechanical energy expenditure is worthy or not, we +calculate the ratio between the enhanced heat flux δQ (which is further normalized by +heat flux Q0 in the absence of wall shear) and the required mechanical energy Ws due +to wall shear (which is further normalized by energy dissipation due to viscosity W0 in +the absence of wall shear) as +η = δQ/Q0 +Ws/W0 +(3.9) +Here, δQ = Quw − Q0. Generally, the heat flux Quw is calculated as +Quw = +Æ� L +0 +Å +−κ∂T +∂n +ã +dx +∏ +t +(3.10) +In the above, κ denotes thermal conductivity of the fluids, and ⟨· · · ⟩t denotes the time +average. To impose the wall shear, an additional external mechanical energy Ws is +required, which is calculated as +Ws = +≠� +l +����µduw +dn · uw +���� dl +∑ +t +(3.11) +Here, the integration +� +l(· · · ) is performed along all the shear wall. In the absence of wall +shear, the energy dissipation due to viscosity in the convection cell is +W0 = +∞� +V +µ +2 +� +i,j +Å ∂ui +∂xj ++ ∂uj +∂xi +ã +dV +∫ +t +(3.12) +The ratio between enhanced heat flux and imposed mechanical energy can be regarded as +a metric that describes the efficiency of facilitating heat transport via external shearing. +From figure 14, we can see that for the (1, 1) and (2, 1) types of wall shear, the efficiency η +decreases monotonously with the increase of Rew. Recall that from figures 4(a) and 4(c), +we found the Nu increases monotonously with the increase of Rew, thus the enhanced +Nu requires a larger expenditure of mechanical energy at a larger Rew. For the (1, 2) +type wall shear, the efficiency η is negative at Rew ⩽ 200, because the Nu is weakened in +that range, as shown in figures 4(a) and 4(c); at larger Rew, the efficiency η is positive, +yet it does not exhibit monotonous behavior with the increase of Rew. Among the three +types of wall shear, the (2, 1) type results in the highest efficiency η, because the flow is +more coherent in the corresponding (2, 1) flow mode. At the largest Rew, all three types +of wall shear exhibit very small values of efficiency η, implying heat transfer enhancement + +20 +A. Xu, B.-R. Xu and H.-D. Xi +(d) +(e) +(f ) +Temperature +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +(g) +(h) +(i) +(j) +(k) +(l) +(a) +(b) +(c) +Figure 13. Turbulence relaminarization process: time evolution of instantaneous flow fields +(temperature contours and velocity vectors in 2-D, volume rendering of temperature field in +3-D). (a-f ) Snapshots for the the 2-D case at t = 0, 2, 4, 8, 22 and 354 tf, respectively, with +wall shear strength Rew = 2000. (g-l) Snapshots for the 3-D case at t = 0, 2, 4, 8, 11 and 158 +tf, respectively, with Rew = 3000. Initially, an instantaneous flow field obtained at Rew = 200 +for the 2-D case and Rew = 100 for the 3-D case, is used to start the simulation, respectively. + +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8Wall sheared thermal convection +21 +Rew +(�Q/Q0) / (Ws/W0) +10 +2 +10 +3 +10 +4 +-1 +0 +1 +(1, 1) type +(2, 1) type +(1, 2) type +Figure 14. The ratio between enhanced heat flux and imposed mechanical energy as a +function of shear Reynolds number for various types of wall shear in the 2-D cases. +comes at a very high price. We deduce that further increase the wall shear strength, the +efficiency η would approach the limit of zero (but non-negative) value because the heat +transfer increases monotonously at large Rew. +4. Conclusions +In this work, we have performed direct numerical simulations of thermal convection +under three different (m, n) types of wall shear. The (m, n) type wall shear is imposed to +facilitate m rolls in the horizontal direction and n rolls in the vertical direction. Under the +(1, 1) type, the (2, 1) type, and the (1, 2) type wall shear, we can observe the single-roll, +the horizontally stacked double-roll, and the vertically stacked double-roll flow modes, +respectively, are generally prevail flow modes in the convection cell. With the increase of +Rew, we generally found enhanced heat transfer efficiency and global flow strength for +all three types of wall shear. However, even with the same magnitude of flow strength, +the heat transfer efficiency of the convection cell varies significantly under different types +of wall shear. Specifically, the (2, 1) type wall shear results in the largest magnitude of +heat transfer efficiency, and the (1, 2) type wall shear results in the smallest one, which +is consistent with our expectation that facilitating the horizontally stacked double-roll +flow modes is efficient for heat transfer, yet facilitating the vertically stacked double-roll +is inefficient for heat transfer. +The original objective of imposing the wall shear was to manipulate flow mode to +control heat transfer efficiency. While it is found that by increasing the wall shear +strength, the thermal turbulence is relaminarized, and more surprisingly, the heat transfer +efficiency of the convection in the laminar state is higher than that in the turbulent state. +By examining the flow field and the convective heat flux field, we found the enhancement +of heat transfer efficiency at the laminar regime is due to the formation of more stable +and stronger convection channels. +We explained the origin of thermal turbulence laminarization in the sheared convection +cell. Analysis of the shear-produced TKE (i.e., −u′ +iu′ +j∂jui) and the buoyancy-produced +TKE (i.e., T ′v′) provide direct evidence that thermal plumes are mainly responsible for +the TKE production. We then quantitatively measure the changes in plume areas under +the wall shear and found that plumes are swept away by the wall shear once they are +detached from the top cold and bottom hot walls, and such a reduced number of thermal +plumes decreases the TKE production in the bulk cell. +We evaluated whether the mechanical energy expenditure by wall shear is worthy or + +22 +A. Xu, B.-R. Xu and H.-D. Xi +not. We used the ratio between the enhanced heat flux and the required mechanical +energy to quantify the efficiency of facilitating heat transport via external shearing. We +found that at a larger Rew, although the heat transfer efficiency increases, it comes at a +price of a larger expenditure of mechanical energy. +Finally, we emphasize that in turbulent wall-sheared thermal convection, the heat +transfer may not always monotonically increase with increasing shear. For example, in +the RB system with a Couette-type wall shear, Blass et al. (2020, 2021) found with +increasing wall shear, the heat transfer first decreases (due to the breakup of the thermal +convection rolls) and then increases. In our study, the heat transfer enhancement is a +consequence of the moving adiabatic side walls advecting fluid in the vertical direction, +thus facilitating the formation of stable and strong convection channels between the top +cold wall and the bottom hot wall. +5. Acknowledgments +This work was supported by the National Natural Science Foundation of China (NSFC) +through Grant Nos. 12272311 and 12125204, and the 111 project of China (No. B17037). +6. Declaration of interests +The authors report no conflict of interest. +7. Supplementary movies +Supplementary movies are available at https://doi.org/10.1017/jfm.xxxxxx. +8. Appendix A. Flow and heat transfer patterns in the canonical RB +convection +In figure 15, we show the typical instantaneous temperature and flow fields, as well as +vertical convective heat flux field for the canonical RB convection in the absence of wall +shear. In the 2-D case, we can see that there exists a well-defined LSC, together with +counter-rotating corner rolls (figure 15a). The LSC is in the form of a tilted ellipse, sitting +along a diagonal of the flow cell with two secondary corner vortices that exist along the +other diagonal. Strong positive heat flux occurs in regions of rising hot (or falling cold) +plumes (see figure 15d). In the 3-D case, the very confined cell with Γ⊥ = 1/8 exhibits +similar flow and heat transfer pattern to that of the 2-D case, with persistent LSC [see +figures 15b and 15e]. When the cell aspect ratio Γ⊥ increases to 1/4, the LSC is less +stable and its shape becomes distorted [see figures 15c and 15f]. +9. Appendix B. Determination of flow states via time recordings and +its power spectrum density +In figure 16, we give examples of temperature series at the location of (0.25, 0.5) in the +2-D convection cell under (1, 1) type wall shear. We also show the power spectrum density +(PSD) of the corresponding temperature series. At Rew = 100 and 500, the temperature +fluctuates randomly around 0.5 [see figure 16(a) and 16(c)], and the corresponding PSD +[see figure 16(b) and 16(d)] exhibit continuous spectra; thus, we determine the flow +states as the turbulent state. At Rew = 1000, the fluctuation of the temperature series +is within a smaller range [see figure 16(e) and its inset], and the corresponding PSD + +Wall sheared thermal convection +23 +(a) +(b) +Temperature +(d) +(e) +(f ) +(c) +Heat flux +Figure 15. 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H., Verzicco, R., Lohse, D. & +Ahlers, G. 2009 Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport +in turbulent rotating Rayleigh-B´enard convection. Phys. Rev. Lett. 102 (4), 044502. +Zhu, X.-J., Stevens, R. J. A. M., Shishkina, O., Verzicco, R. & Lohse, D. 2019 Scaling +enabled by multiscale wall roughness in Rayleigh–B´enard turbulence. J. Fluid Mech. 869, +R4. + diff --git a/K9AyT4oBgHgl3EQfgPgL/content/tmp_files/load_file.txt b/K9AyT4oBgHgl3EQfgPgL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe27f6039d6afa842e0785ee3c2cc2cc41d9efea --- /dev/null +++ b/K9AyT4oBgHgl3EQfgPgL/content/tmp_files/load_file.txt @@ -0,0 +1,1619 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf,len=1618 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='00353v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='flu-dyn] 1 Jan 2023 This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 Wall sheared thermal convection: heat transfer enhancement and turbulence relaminarization Ao Xu1,2, Ben-Rui Xu1 and Heng-Dong Xi1,2† 1School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, PR China 2Institute of Extreme Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' accepted xx) We studied the flow organization and heat transfer properties in two-dimensional and three-dimensional Rayleigh-B´enard cells that are imposed with different types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The external wall shear is added with the motivation of manipulating flow mode to control heat transfer efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We imposed three types of wall shear that may facilitate the single-roll, the horizontally stacked double-roll, and the vertically stacked double-roll flow modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Direct numerical simulations are performed for fixed Rayleigh number (Ra) of Ra = 108 and fixed Prandtl number (Pr) of Pr = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3, while the wall shear Reynolds number (Rew) is in the range of 60 ⩽ Rew ⩽ 6000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Generally, we found enhanced heat transfer efficiency and global flow strength with the increase of Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' However, even with the same magnitude of global flow strength, the heat transfer efficiency varies significantly when the cells are under different types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' An interesting finding is that by increasing the wall shear strength, the thermal turbulence is relaminarized, and more surprisingly, the heat transfer efficiency in the laminar state is higher than that in the turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We found the enhanced heat transfer efficiency at the laminar regime is due to the formation of more stable and stronger convection channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We propose the origin of thermal turbulence laminarization is the reduced amount of thermal plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Because plumes are mainly responsible for turbulent kinetic energy production, when the detached plumes are swept away by the wall shear, the reduced number of plumes leads to weaker turbulent kinetic energy production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We also quantify the efficiency of facilitating heat transport via external shearing, and found that for larger Rew, the enhanced heat transfer efficiency comes at a price of a larger expenditure of mechanical energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Key words: B´enard convection, plumes/thermals, turbulent convection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Introduction Thermal convection occurs ubiquitously in nature and has wide applications in in- dustry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' A paradigm for the study of thermal convection is the Rayleigh-B´enard (RB) convection, which is a fluid layer heated from the bottom and cooled from the top (Ahlers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Lohse & Xia 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Chill`a & Schumacher 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xia 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The control parameters of the canonical RB system include the Rayleigh number (Ra, defined later † Email address for correspondence: hengdongxi@nwpu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='cn 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi in the paper) that describes the strength of the buoyancy force relative to the thermal and viscous dissipative effects, and the Prandtl number (Pr) that represents the thermo- physical fluid properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' One of the response parameters of the RB system is the Nusselt number (Nu) which characterizes the global heat transfer efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Various approaches have been designed to enhance the heat transfer efficiency of the convection cells, such as adding roughness to the walls (Ciliberto & Laroche 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Wagner & Shishkina 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Rusaou¨en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2019), introducing vibration forcing (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2020a), adding dispersed phase of particles or bubbles (Lakkaraju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Guzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Gvozdi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2022), confinement (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Chong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2022), rotation (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2009, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2020b), the addition of passive barriers (Liu & Huisman 2020), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Roche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2002) and Chill`a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2004) conjectured that the internal flow structure is correlated with global heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2005) compared the Nu in a leveled cell and a tilted cell, correspondingly, the large-scale circulation (LSC) plane sweeps azimuthally or is locked in a particular orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' They showed that the Nu is larger in the leveled cell, indicating that different flow structures can results in different values of Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi & Xia (2008) observed both the single-roll and the double-roll flow structures in the LSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' They examined the average Nu corresponding to a particular flow structure and found that the single-roll flow structure is more efficient for heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Weiss & Ahlers (2011) further confirmed the occurrence of a double-roll structure in the LSC, and the higher heat transfer efficiency of the single-roll state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' van der Poel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2011, 2012) numerically showed the coexistence of different turbulent structures also exists in simple two-dimensional RB cells with various cell aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' They also studied the effect of various velocity boundary conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', no-slip, stress-free, and periodic boundary conditions) on the heat transfer and flow topoplogy (van der Poel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2014), and they showed either the roll-like or the zonal flow can appear under different velocity boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Adopting Fourier mode decompositions, Xi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2016) presented direct evidence that the first Fourier mode is more efficient for heat transfer in a cylindrical cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2020) analyzed the coherent flow structure in two-dimensional square convection cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Results from both Fourier mode decomposition and proper orthogonal decomposition indicate that the single-roll flow mode and the horizontally stacked double- roll mode are efficient for heat transfer on average;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' in contrast, the vertically stacked double-roll mode is inefficient for heat transfer on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' A natural question arises on how to manipulate flow mode to control heat transfer efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In this work, we impose various types of wall shear to control the internal flow mode, which further leads to modification of heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Previously, Blass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2020, 2021) added a Couette type shear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', the top and bottom walls move in opposite directions with constant speed uw) to the RB system as an attempt to trigger the transition to the ultimate convection regime (Kraichnan 1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' With the increasing wall shear strength, they observed the variation of flow states from a buoyancy-dominated regime to a shear- dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the buoyancy dominated regime, the flow structure is similar to that in the canonical RB convection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' in the transitional regime, the rolls are increasingly elongated with increasing shear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' in the shear dominated regime, there are large-scale meandering rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2022) further added the Couette type shear to convection cells that have rough walls, and the moving rough plates introduce an external shear to strengthen the LSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' As a result, the interaction between the LSC and secondary flows within cavities are increased, and more thermal plumes are triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In this work, our motivation of imposing wall shear is to facilitate various flow modes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', the single-roll, the horizontally stacked double-roll, and the vertically stacked double-roll modes) in the Wall sheared thermal convection 3 convection cell to further control heat transfer efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Specifically, we will add the (m, n) type of wall shear to the RB system, and such types of wall shear are expected to facilitate m rolls in the horizontal direction and n rolls in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The use of shear modulated boundary conditions essentially leads to mixed convection, which has received considerable attention due to its importance in many engineering applications such as cooling of electronic devices, coating, and float glass production (Hunt 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Shankar & Deshpande 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In section 2, we present numerical details for the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In section 3, general flow and heat transfer features are presented, and heat transfer enhancement under various types of wall shear is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' An interesting finding is thermal turbulence relaminarization under the imposed wall shear, and we then discuss the possible mechanism behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In addition, we quantify the efficiency of facilitating heat transport via external shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In section 4, the main findings of the present work are summarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Numerical method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Direct numerical simulation of incompressible thermal convection We consider incompressible thermal convection under the Boussinesq approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The temperature is treated as an active scalar, and its influence on the velocity field is realized through the buoyancy term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' all the transport coefficients are assumed to be constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The governing equations can be written as ∇ · u = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='1) ∂u ∂t + u · ∇u = − 1 ρ0 ∇P + ν∇2u + gβ(T − T0)ˆy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2) ∂T ∂t + u · ∇T = α∇2T (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3) where u is the fluid velocity, P and T are pressure and temperature of the fluid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' β, ν, and α are the thermal expansion coefficient, kinematic viscosity, and thermal diffusivity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The zero subscripts (0) refer to the reference values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' g is the gravity acceleration value, and ˆy is the unit vector parallel to the gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' With the following non-dimensional group x∗ = x/H, t∗ = t/ » H/(gβ∆T ), u∗ = u/ � gβ∆T H, P ∗ = P/(ρ0gβ∆T H), T ∗ = (T − T0)/∆T (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4) Then, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3) can be rewritten in dimensionless form as ∇ · u∗ = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5) ∂u∗ ∂t∗ + u∗ · ∇u∗ = −∇P ∗ + … Pr Ra∇2u∗ + T ∗ˆy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6) ∂T ∗ ∂t∗ + u∗ · ∇T ∗ = … 1 PrRa∇2T ∗ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7) Here, H is the cell height and ∆T is the temperature difference between heating and cooling walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the following, for convenience, we will drop the superscript star (∗) to denote a dimensionless variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The dimensionless parameters of the Rayleigh number (Ra), the Prandtl number (Pr) and the cell aspect ratio (Γ∥ in the plane parallel to the 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi LSC’s circulation plane and Γ⊥ in the plane perpendicular to the LSC) are defined as Ra = gβ∆T H3 να , Pr = ν α, Γ∥ = L H , Γ⊥ = W H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8) where L is cell length and W is the cell width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We adopt the spectral element method (Patera 1984) implemented in the open- source Nek5000 solver (version v19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='0) as the numerical tool for the direct numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the Nek5000 solver, the effective grid number equals the product of spectral element number and polynomial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We set the spectral elements for the velocity with polynomial order N and the spectral elements for the pressure with polynomial order N − 2 (to avoid spurious pressure modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Similar to previous turbulent flow simulations (Kooij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018), we fix the polynomial order as N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The viscous term is treated implicitly with the second-order backward difference scheme, while the convection term and other terms are treated with an explicit second-order extrapolation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The discretized system is solved with preconditioned conjugate gradient (PCG) iteration, and Jacobi preconditioning is adopted for the linear velocity system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' A pressure correction step follows the solution of the discretized system, which is also solved with PCG iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' and the linear pressure system is solved by the multilevel overlapping Schwarz method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' As for the energy equation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', temperature governed by convection- diffusion type equation), the transient term is treated implicitly with the second-order backward difference scheme, and the convection term is treated with an explicit second- order extrapolation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' For the Navier-Stokes and convection-diffusion equations, the temporal derivative applies a Courant-Friedrichs-Lewy constraint of max(|u|∆t/∆x) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' More numerical details of the spectral element method and validation of the Nek5000 solver can be found in (Fischer 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Deville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Kooij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' To verify the results obtained from the Nek5000 solver, we also performed a set of simulations at wall shear Reynolds number (Rew, defined later in the paper) of 100 using an in-house solver based on the lattice Boltzmann (LB) method (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu & Li 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The results from the open-source Nek5000 solver and the in-house LB solver are shown to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Simulation settings As illustrated in figure 1, the dimensions H, L and W correspond to the y, x and z in the Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The top and the bottom of the horizontal walls are kept at constant low and high temperatures of Tcold and Thot, respectively, while the vertical sidewalls are adiabatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' For the velocity at the walls, we designed the (m, n) type wall shear to facilitate the flow structure with m rolls in the x-direction and n rolls in the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Specifically, we consider three types of wall shear boundary conditions: the (1, 1) type wall shear that may facilitate the single-roll flow mode (see figures 1a and 1d), the (2, 1) type wall shear that may facilitate the horizontally stacked double-roll mode (see figures 1b and 1e), and the (1, 2) type wall shear that may facilitate the vertically stacked double-roll mode (see figures 1c and 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the (1, 1) type wall shear, the velocity boundary conditions are (i) at 0 ⩽ x ⩽ L and y = 0, we have u = (−uw, 0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (ii) at 0 ⩽ x ⩽ L and y = H, we have u = (uw, 0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (iii) at x = 0 and 0 ⩽ y ⩽ H, we have u = (0, uw, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (iv) at x = L and 0 ⩽ y ⩽ H, we have u = (0, −uw, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Similar mathematical formulations for the velocity boundary conditions under the (2, 1) type and the (1, 2) type wall shear can be easily written (not present here for clarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' When an external wall shear is introduced, an additional control parameter of wall shear Reynolds number (Rew = Huw/ν) is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, uw is the wall shear velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Simulation results are provided for fixed Rayleigh number of Ra = 108, fixed Prandtl number of Pr = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3 Wall sheared thermal convection 5 (a) (b) (c) (d) (e) (f ) Y X Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Schematic illustration of the shear convection cells in (a-c) two-dimension (2-D) and (d-f ) three-dimension (3-D), for (a, d) the (1, 1) type wall shear, (b, e) the (2, 1) type wall shear, and (c, f ) the (1, 2) type wall shear boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' [corresponds to the working fluids of water at 31◦C (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2017)] and fixed aspect ratio of Γ∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the 3-D cases, we consider aspect ratios of Γ⊥ = 1/8 and 1/4 such that the LSC is confined in the x− y plane, enabling easy manipulation of the flow mode via wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The wall shear Reynolds number is in the range of 60 ⩽ Rew ⩽ 6000 for 2-D cases, and Rew = 100 and 3000 for 3-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the simulation, after the initial transient stage, we run at least 5000 tf for 2-D cases and 800 tf for 3-D cases to obtain the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, tf denotes free-fall time units: tf = � H/(gβ∆T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We check whether the grid spacing ∆g and time interval ∆t are properly resolved by comparing them with the Kolmogorov and Batchelor scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The Kolmogorov length scale can be estimated as ηK = (ν3/⟨εu⟩)1/4, the Batchelor length scale can be estimated as ηB = ηKPr−1/2 (Batchelor 1959;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Silano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2010), and the Kolmogorov time scale can be estimated as τη = � ν/⟨ε⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the canonical RB convection, we adopted spectral elements of 64 × 64 for 2-D cases, 32 × 32 × 5 for 3-D cases with Γ⊥ = 1/8, and 32 × 32 × 9 for 3-D cases with Γ⊥ = 1/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' the corresponding effective grid number is listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the wall-sheared thermal convection, we adopted a finer distributed spectral element of 96 × 96 for 2-D cases, 44×44×7 for 3-D cases with Γ⊥ = 1/8, and 44×44×13 for 3-D cases with Γ⊥ = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We estimate the global kinetic energy dissipation rate as ⟨εu⟩ = RaPr−2(Nu−1)ν3/H4 in the canonical RB convection (Shraiman & Siggia 1990), and ⟨εu⟩ = � Pr/Ra⟨(∂ju′ i)⟩V,t in the wall-sheared convection (Pope 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, the subscripts i and j are dummy indices, and ⟨· · · ⟩V,t denotes the spatial and temporal average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' As shown in table 1, the maximum grid spacing (∆g)max is less than (or comparable) to the Kolmogorov and Batchelor length scales for 2-D cases (or 3-D cases);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' the maximum time interval (∆t)max is far less than the Kolmogorov time scale for all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Thus, adequate spatial and temporal resolution is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Each simulation was conducted with 48 Message Passing Interface (MPI) processes on an in-house cluster that required around 12,000 core hours for 2-D cases and 50,000 core hours for 3-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' T = Tcold 个 aT aT = 0 = 0 an an T= hot uT = Tcold aT =0 an aT =0 an Ww Y6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi Wall shear type Rew Γ⊥ Effective grid number (∆g)max/ηK (∆g)max/ηB (∆t)max/τη 0 512 × 512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='0019 0 1/8 256 × 256 × 40 0.' metadata={'source': 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+page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='0016 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' A posteriori check of spatial and temporal resolutions of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The columns from left to right indicate the following: imposed wall shear type (’-’ denotes convection without wall shear), the wall shear Reynolds number Rew, cell aspect ratio Γ⊥ in the plane perpendicular to the LSC (’-’ denotes 2-D cases), effective grid number (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', the product of spectral element number and polynomial order), the ratio of maximum grid spacing over the Kolmogorov length scale, the ratio of maximum grid spacing over the Batchelor length scale, the ratio of maximum time interval over the Kolmogorov time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Note that not all the simulations in this work are listed in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Results and discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Global flow and heat transfer features Typical snapshots of temperature field and flow field under the three types of wall shear are shown in figures 2, and the corresponding video can be viewed in the supplementary movie 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, the Ra is fixed as Ra = 108 and the Pr is fixed as Pr = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' At small wall shear strength of Rew = 100, the convection is still buoyancy-dominated, plumes detach from thermal boundary layers and further self-organize into the LSC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' meanwhile, the flow structure in the convection cell is influenced by the imposed wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' For the convenience of comparison, we also provide the flow and heat transfer patterns in the canonical RB convection without wall shear (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The single-roll flow structure appears under the (1, 1) type wall shear (see figures 2a and 2g), whilst the corner rolls are suppressed compared to that without wall shear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' the horizontally stacked double-roll flow structure appears under the (2, 1) type wall shear (see figures 2b and 2h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Wall sheared thermal convection 7 the vertically stacked double-roll flow structure appears under the (1, 2) type wall shear (see figures 2c and 2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' At large wall shear strength of Rew = 4000 in 2-D and Rew = 3000 in 3-D, the convection is shear-dominated, and the flow structures inside the convection cell are completely influenced by the external wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' For example, under the (1, 1) type wall shear (see figures 2d and 2j), the hot (or cold) fluids near the bottom (or top) wall are swept away by the LSC in the clockwise direction, and rise (or fall) along the left (or right) vertical wall, while the fluids in the bulk region are well-mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Similar observation can be found for the flow structure under the (2, 1) type wall shear (see figures 2e and 2k), while the cold fluids also fall along the vertical mid-plane of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' As for the flow structure under the (1, 2) type wall shear, in the 2-D case (see figure 2f), the top and bottom subregions are completed separated without heat transfer between each other, acting as a ”thermal barrier” exists at the half height of the cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' however, in the 3-D case (see figure 2l), we did not observe a complete separation of hot and cold fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We infer the differences in flow structure between 2-D and 3-D configurations are due to the flow state: in the steady laminar flow (as in the 2-D case), the rising hot fluids and falling cold fluids can remain stable boundaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' while in the turbulent flow (as in the 3-D case), the hot and cold fluids are more mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We also checked the flow field within the Γ⊥ = 1/8 cell, where the flow is in a laminar state, and we indeed found a separation of hot and cold fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' With simulations of three different types of wall shear in the range of 60 ⩽ Rew ⩽ 6000, we can obtain the phase diagram of whether the flow is in the turbulent state or laminar state, as shown in figure 3(a) for 2-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, we placed numerical probers in the cell and analyzed the time recordings of local velocity and temperature series to determine the flow states (Heslot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Silano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We determined the flow is in the laminar state if the time recordings do not vary with time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', steady laminar state) or the power spectral density (PSD) of the time recordings exhibits characteristic peaks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', unsteady laminar state);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' otherwise, if the PSD of the time recording exhibit continuous spectra, the flow is in the turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In Appendix B, we give examples of temperature series and the corresponding PSD at the location of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5) in the 2-D convection cell under (1, 1) type wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The phase diagram of the flow states can be understood in terms of competition between buoyancy and shear effects, which can be quantified by the Richardson number as Ri = Ra/(Re2 wPr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In figure 3(b), we redraw the phase diagram of the flow states at different Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' For lower Rew (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', higher Ri at fixed Ra and Pr), the flow is buoyancy dominated and possesses the key features of turbulent convection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' for higher Rew (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', lower Ri), the flow is shear dominated and enters a laminar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Turbulent laminarisation is counterintuitive and it is recently found in pipe flow by amplifying wall shear (K¨uhnen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Scarselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' It should also be noted that when the Rew further increases, the wall shear would introduce flow instability and the flow would transit to a turbulent state again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' However, our numerical tests show that the flow can remain laminar for a wide range of Rew in 2-D cases, a further transition to shear turbulence may occur at a much higher Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We also found that the shear instability is prominent in 3-D cases, particularly when Γ⊥ is larger, thus the flow remains laminar in a smaller range of wall shear Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' For example, at Rew = 3000, the flow is laminar in convection cells with Γ⊥ = 1/8 under all three types of wall shear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' while the flow is laminar only under the (1, 1) type wall shear in the Γ⊥ = 1/4 cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We then examine the global response parameters of Nusselt number (Nu) and Reynolds number (Re) on the control parameter Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, the heat transfer efficiency is calcu- lated as Nu = √ Ra · Pr⟨vT ⟩V,t + 1, the global flow strength is calculated as Re = � ⟨∥u∥2⟩V,tH/ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The measured Nu and Re as a function of Rew for various types of 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi (a) (b) (c) (d) (e) (f ) Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8 (g) (h) (i) (j) (k) (l) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Typical instantaneous temperature field (contours in 2-D and volume rendering in 3-D) and flow field (streamlines in 2-D) at (a-c) Rew = 100, (d-f ) Rew = 4000, (g-i) Rew = 100 and Γ⊥ = 1/4, (j -l) Rew = 3000 and Γ⊥ = 1/4, under (left-column) the (1, 1) type wall shear, (middle-column) the (2, 1) type wall shear, (right-column) the (1, 2) type wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='80Wall sheared thermal convection 9 � � � � � � � �� � � � Rew 10 2 10 3 10 4 (1, 2) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='C (2, 1) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (1, 1) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' turbulent state laminar state � � � � � � � � � � � � Ri 10 0 10 1 10 2 10 3 (1, 2) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='C (2, 1) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (1, 1) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (a) (b) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Phase diagram of the flow states (a) at different Rew and (b) at different Ri in 2-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' wall shear in 2-D cells are shown in figures 4(a) and 4(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Generally, with the increase of Rew, we can observe enhanced heat transfer efficiency and global flow strength for all three types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' However, at Rew ⩽ 200 for the (1, 2) type wall shear, the flow structure gradually changes from an LSC that spans the whole cell to the vertically stacked double-roll mode, leading to a decreased Nu value (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' To clearly visualize the relative changes of Nu and Re after imposing the wall shear, we further plot (Nu − Nu0)/Nu0 and (Re − Re0)/Re0 as a function of Rew in figures 4(c) and 4(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, Nu0 and Re0 are the Nusselt and Reynolds numbers in the absence of wall shear, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Among the three types of wall shear, at the same Rew, the (2, 1) type wall shear results in the largest magnitude of heat transfer efficiency up to 568%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' and the (1, 2) type wall shear results in the smallest one around 179%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The trend is consistent with our expectation that facilitating the horizontally stacked double-roll flow modes is efficient for heat transfer, yet facilitating the vertically stacked double-roll is inefficient for heat transfer (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' On the other hand, as Rew increases, all three types of wall shear exhibit a similar trend of increasing global flow strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The results indicate that in even with the same magnitude of flow strength, the heat transfer efficiency of the convection cell still varies significantly under different types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In addition, we provide tabulated value of Nusselt and Reynolds numbers for 3-D cases in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We can conclude heat transfer enhancement can also be found in 3-D configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Figure 5 shows the scaling of the global quantities in 2-D cells, such as Nu and Re, on one of the control parameters Ra (for 106 ⩽ Ra ⩽ 109), whilst the control parameters Rew is fixed as Rew = 100 and Pr is fixed as Pr = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We also provide Nu and Re in the canonical RB convection without shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Previously, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2017) provided tabulated values of the Nu and Re versus the Ra at Pr = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Our simulation results on the canonical RB convection are in good agreement with that reported by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The data shown in figure 5 indicates that in the buoyancy dominated regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', when Ra is larger at fixed Rew), the increase of Nu and Re gradually approaches the power-law relations of Nu ∝ Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='30 and Re ∝ Ra0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='59, consistent with previous results reported in the canonical RB convection (Ciliberto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' van der Poel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Huang & Xia 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Overall, the global heat transfer and momentum quantities reveal that the simulated system possesses the key features of turbulent convection in the buoyancy dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the shear dominated regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', when Ra is smaller at fixed Rew), the scaling behavior of Nu and Re with Ra deviate significantly from that of the canonical RB convection, suggesting heat transfer and momentum exchange are not solely governed by the boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi Rew Nu 10 2 10 3 10 4 50 100 150 200 Rew Re 10 2 10 3 10 4 2000 4000 6000 (a) (b) (c) (d) (1, 1) type (2, 1) type (1, 2) type Rew (Nu - Nu0) / Nu0 (%) 10 2 10 3 10 4 200 0 200 400 600 Rew (Re - Re0) / Re0 (%) 10 2 10 3 10 4 200 0 200 400 600 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 (1, 1) type (2, 1) type (1, 2) type Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (a) Nusselt number, (b) Reynolds number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (c) values of Nu/Nu0 − 1, and (d) values of Re/Re0 − 1 as a function of Rew for various types of wall shear in the 2-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, Nu0 and Re0 are the Nusselt and Reynolds numbers in the absence of wall shear, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The insets magnify Rew in the range 100 ⩽ Rew ⩽ 500 shown in (c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Wall shear type Γ⊥ Rew Nu Re (Nu − Nu0)/Nu0 (Re − Re0)/Re0 1/8 0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='80 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='39 1/4 0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='38 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='32 (1, 1) type 1/8 100 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='18 280.' 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global flow strength in the 3-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The columns from left to right indicate the following: imposed wall shear type (’-’ denotes convection without wall shear), cell aspect ratio (Γ⊥) in the plane perpendicular to the LSC, wall shear strength (Rew), Nusselt number (Nu), Reynolds number (Re), heat transfer enhancement (Nu − Nu0)/Nu0, global flow strength enhancement (Re − Re0)/Re0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, Nu0 and Re0 are the Nusselt and Reynolds numbers in the absence of wall shear at the same Γ⊥, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Wall sheared thermal convection 11 Ra Nu 10 6 10 7 10 8 10 9 10 1 10 2 Ra Re 10 6 10 7 10 8 10 9 10 1 10 2 10 3 10 4 (a) (b) (1, 1) type (2, 1) type (1, 2) type without shear (1, 1) type (2, 1) type (1, 2) type without shear Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (a) Nusselt number, (b) Reynolds number as a function of Rayleigh number for various types of wall shear in the 2-D cases, when the wall shear Reynolds number is fixed as Rew = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We further quantitatively investigate the influence of different types of wall shear on the temperature distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Figure 6 shows the probability density functions (PDFs) of the normalized temperature (T − µT )/σT in the bulk region of the 2-D cell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4L ⩽ x ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6L and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4H ⩽ y ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6H), where µT and σT are the mean and standard deviation of the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the absence of wall shear, the PDFs of temperature in the bulk show a stretched exponential behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the (1, 1) type wall shear, the temperature in the bulk is well-mixed and the PDFs are symmetric at different Rew (see figure 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' With imposed external wall shear, the PDFs at different Rew collapse and they deviate significantly from that in the absence of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The narrowed PDF tails imply that fewer plumes pass through the bulk region and the temperature fluctuation is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the (2, 1) type wall shear, the PDF is negatively skewed at smaller Rew (see figure 6b), which is due to cold plumes descend through the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' However, as Rew increases, the skewness of the temperature PDFs decreases, and their tails become narrower, implying that temperature is better mixed and fewer cold fluids pass through the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the (1, 2) type wall shear, the PDFs are symmetric (see figure 6c) due to the top-down symmetry of the convection cell, both hot and cold plumes pass through the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' As the strength of the wall shear increases, the heads of the PDFs gradually exhibit a bi-modal shape (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', the inset shown in figure 6c), suggesting the top cold and bottom hot subregions are gradually separated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' meanwhile, all the tails of the PDFs exhibit Gaussian shape and their profiles collapse for different Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The collapse of PDF indicates a similar flow pattern in the bulk region because the functional form of the temperature PDF is determined by the coherence of plumes (Solomon & Gollub 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xia & Lui 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We now investigate how the local heat transfer properties are influenced by different types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In figure 7, we show the vertical convective heat flux field v · δT , where the temperature fluctuation is δT = T − (Thot + Tcold)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' At small wall shear strength of Rew = 100 (see figures 7a-c for 2-D cases and 7g-i for 3-D cases), the heat is mainly transported by the moving thermal plumes and the magnitudes of vertical convective heat flux are relatively weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the (1, 1) type wall shear (see figures 7a and 7g), plumes that carry heat mainly go up and down near the sidewalls;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' under the (2, 1) type wall shear (see figures 7b and 7h), plumes can also penetrate vertically in the bulk region of the cell, thus forming additional convection channels between the cold top wall and the hot bottom wall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' under the (1, 2) type wall shear (see figures 7c and 7i), plumes that penetrate the bulk region of the cell exhibit horizontal motion at the mid-height of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' At large wall shear strength of Rew = 4000 (see figures 7d-f for 2-D 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi PDF 10 0 10 10 5 10 4 10 3 10 2 10 1 10 0 (a) (c) (b) PDF 10 0 10 10 5 10 4 10 3 10 2 10 1 10 0 PDF 10 0 10 10 5 10 4 10 3 10 2 10 1 10 0 � � / T T T � � � 2 0 2 10 1 10 0 Rew = 0 Rew = 100 Rew = 200 Rew = 500 Rew = 0 Rew = 100 Rew = 200 Rew = 500 Rew = 0 Rew = 100 Rew = 200 Rew = 500 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The probability density functions (PDFs) of the temperature measured in the bulk region of the 2-D cells (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4L ⩽ x ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6L and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4H ⩽ y ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6H) under (a) the (1, 1) type wall shear, (b) the (2, 1) type wall shear, (c) the (1, 2) type wall shear, when the flow is in turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The dotted-dashed line represents a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The inset in (c) magnifies the head of the PDF at Rew = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' cases) and Rew = 3000 (see figures 7j-l for 3-D cases), the vertical convective heat flux forms much more stable and regular convection channels, and their magnitudes are much stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' It should be noted that there are small regions of negative convective heat flux immediately adjacent to the regions of large positive convective heat flux, which is known as counter-gradient local heat transport (Gasteuil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Huang & Zhou 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The counter-gradient local heat transport essentially describes that both the LSC and the corner flows may contribute to heat transport in the ’wrong’ direction: hot (or cold) plumes can be brought back to the hot (or cold) plate by either the corner flows or the LSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The counter-gradient local heat transport is ubiquitous and can be found in 2D and 3D systems, either turbulent or laminar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' An interesting finding is that under the (1, 2) type wall shear in the 2-D case (see figures 7f), there exists strong negative vertical convective heat flux along the right vertical wall, which is opposite to the temperature gradient of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the external wall shear, hot (or cold) fluids are forced to form a circulation in the bottom (or top) subregion of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' When the hot (or cold) fluids fall (or rise) along the right vertical wall, they do not exchange heat with the other and do not lose their thermal energy at all, thus hot (or cold) fluids are swept back to the hot (or cold) walls and exhibit counter-gradient heat transport behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Previously, Blass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2020, 2021) observed that by adding the Couette type shear, the increase of heat transfer efficiency is due to elongated streaks generating vertical cross- stream motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' While in our work, adding the (m, n) type wall shear mainly facilitates a more coherent flow structure and forms more stable and stronger convection channels, particularly in the cases of laminar flows when the wall shear strength is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Wall sheared thermal convection 13 (g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='04Heat flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02Heat flux (j) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02Heat flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='04Heat flux (a) (b) (c) (d) (e) (f ) (i) (h) (l) (k) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Snapshots of vertical convective heat flux field at (a-c) Rew = 100, (d-f ) Rew = 4000, (g-i) Rew = 100 and Γ⊥ = 1/4, (j -l) Rew = 3000 and Γ⊥ = 1/4, under (left column) the (1, 1) type, (middle-column) the (2, 1) type and (right-column) the (1, 2) type wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='814 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi PDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='03 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='09 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 (a) (c) (b) PDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='03 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='09 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 PDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='03 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='09 10 3 10 2 10 1 10 0 10 1 10 2 10 3 10 4 v T � � Rew = 0 Rew = 100 Rew = 200 Rew = 500 Rew = 0 Rew = 100 Rew = 200 Rew = 500 Rew = 0 Rew = 100 Rew = 200 Rew = 500 103 101 10-1 10-3 103 101 10-1 10-3 103 101 10-1 10-3 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The PDFs of the heat flux measured in the whole cells for 2-D cases under (a) the (1, 1) type wall shear, (b) the (2, 1) type wall shear, (c) the (1, 2) type wall shear, when the flow is in turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In figure 8, we further plot the PDFs of the vertical convective heat flux v · δT in the whole cell for 2-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' All the PDFs have longer positive tails and shorter negative tails, implying strong upward convective heat transfer, yet there exist counter-gradient convective heat transfer (Huang & Zhou 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the wall shear, the strength of the upward convective heat transfer is enhanced with the increase of wall shear strength in the whole cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' meanwhile, we checked the PDFs of convective heat flux in the bulk region (not shown here for clarity), and found that their shapes are much narrower, implying that heat exchange is weak in the bulk region, and hotter (or colder) fluids tend to flow upward (or downward) along the sidewalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Such strong counter-gradient convective heat transfer is consistent with our qualitative observation shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In this work, we designed the (m, n) type wall shear to facilitate the flow structure with m rolls in the x-direction and n rolls in the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the imposed wall shear, to quantitatively evaluate whether the expected flow structure is dominated or not, we per- form Fourier mode decomposition on the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The Fourier mode decomposition is a powerful tool to extract coherent structure in turbulent convection (Petschel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Chandra & Verma 2011, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Chong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Specifically, the instantaneous velocity field (u, v) is projected onto the Fourier basis (ˆum,n, ˆvm,n) as u(x, y, t) = � m,n Am,n x (t)ˆum,n(x, y) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='1) v(x, y, t) = � m,n Am,n y (t)ˆvm,n(x, y) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2) Wall sheared thermal convection 15 Rew / Etotal 10 2 10 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8 Rew / Etotal 10 2 10 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8 Rew / Etotal 10 2 10 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8 (a) (b) (c) (1, 1) mode (2, 1) mode (1, 2) mode (2, 2) mode Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Time-averaged energy contained in the first four Fourier modes as functions of Rew under (a) the (1, 1) type wall shear, (b) the (2, 1) type wall shear, (c) the (1, 2) type wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Note the Fourier mode decomposition is only applied when the flow is in turbulent state for 2-D cases (refer to the phase diagram of flow states in figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, the Fourier basis (ˆum,n, ˆvm,n) is chosen as ˆum,n(x, y) = 2 sin(mπx) cos(nπy) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3) ˆvm,n(x, y) = −2 cos(mπx) sin(nπy) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4) The instantaneous amplitude of the Fourier mode is then calculated as Am,n x (t) = ⟨u(x, y, t), ˆum,n(x, y)⟩ = � i � j u(xi, yi, t)ˆum,n(xi, yi) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5) Am,n y (t) = ⟨v(x, y, t), ˆvm,n(x, y)⟩ = � i � j v(xi, yi, t)ˆvm,n(xi, yi) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6) where ⟨u, ˆu⟩ and ⟨v, ˆv⟩ denote the inner product of u and ˆu, v and ˆv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The energy in each Fourier mode is calculated as Em,n(t) = � [Am,n x (t)]2 + [Am,n y (t)]2, the total energy is calculated as Etotal = � m,n⟨Em,n⟩, and ⟨· · · ⟩ denotes the time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In figure 9, we plot the time-averaged energy as functions of Rew for various types of wall shear when the flow is in the turbulent state for 2-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, we consider m and n = 1, 2, namely the first four Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' From figure 9(a) we can see that under the (1, 1) type wall shear, the (1, 1) Fourier mode is indeed dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Similarly, under the (1, 2) type wall shear, the (1, 2) Fourier mode is the dominant flow mode (see figure 9c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' However, under the (2, 1) type wall shear, despite the energy percentage in the (2, 1) mode being much larger compared to that in the absence of wall shear, the (2, 1) mode does not contain the highest percentage of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We can see from figure 9(b) that the (1, 1) Fourier mode contains more energy than the expected (2, 1) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' To explain the discrepancy, we check the snapshots of the flow fields and the heat flux fields (see figures 2b and 7b) and observe that some hot (or cold) plumes lose their energy before reaching the cold top (or hot bottom) wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The plumes then fall (or turn back up) to form small rolls, thus substructures emerge inside the left-side big roll (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' When the unstable small rolls inside the left-side big roll shrink their size, the Fourier mode decomposition that captures flows in the bulk region implies the (1, 1) mode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', one big roll in the whole cell) prevails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Stabilizing thermal turbulence via wall movement The original objective of imposing the (m, n) type wall shear is to adjust the internal flow mode and control heat transfer properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' While we found that by increasing the wall shear strength, the thermal turbulence is relaminarized, and more surprisingly, the heat transfer efficiency of the convection cell in the laminar state is higher than that in 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi the turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the previous section, we have explained that the enhancement of heat transfer efficiency at the laminar regime is due to the formation of more stable and stronger convection channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Below, we further discuss the origin of thermal turbulence laminarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We start by examining the turbulent kinetic energy (TKE) equation of incompressible thermal convection, which is written as ∂∥ ∂t + uj∂j∥ = −u′ iu′ j∂jui + ∂j Ç −p′u′ j + … Pr Ra∂j∥ − 1 2u′ iu′ iu′ j å − … Pr Ra(∂ju′ i)2 + T ′v′ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7) In the above, the subscripts i and j are dummy indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' ∥ = u′ iu′ i/2 denotes the TKE and the superscript (′) denotes the fluctuation part of an instantaneous flow variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The term −u′ iu′ j∂jui represents shear-produced TKE, and the term T ′v′ represents buoyancy- produced TKE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Because the flow remains laminar in a smaller range of Rew for 3-D cases, we mainly discuss the results for 2-D cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In figure 10, we show the shear- produced, the buoyancy-produced, and the total TKE production under the (1, 1) type wall shear as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' With increasing the wall shear strength, the shear-produced TKE is increasingly concentrated near the top-left and bottom-right corners of the convection cells (see figures 10a and 10d), where rising hot (or falling cold) plumes impact the cold (or hot) boundary layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Compared to the shear-produced TKE, the buoyancy- produced TKE is more intense (see figures 10b and 10e) and contributes a dominant part of the total TKE production (see figures 10c and 10f);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' meanwhile, with increasing the wall shear strength, the buoyancy-produced TKE becomes weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Previously, in the absence of wall shear, Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2003) quantitatively described that the TKE largely comes from the buoyant motions of thermal plumes based on the particle image velocimetry (PIV) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' With the aid of direct numerical simulation, T ′v′ is directly obtained in the whole convection cell, we now provide direct evidence that thermal plumes are mainly responsible for the TKE production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' After analyzing the TKE production, we now turn to the TKE dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In figures 11(a) and 11(b), we show the TKE dissipation under the (1, 1) type wall shear as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We can see that intense TKE dissipation occurs in the top-right and bottom- left corners, namely, in the regions of plumes detachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Meanwhile, with increasing the wall shear strength, the TKE dissipation becomes weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' It should be noted that here we considered the dissipation term of � Pr/Ra(∂ju′ i)2 in the TKE equation, which is known as pseudo-dissipation by Pope (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Previously, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2017) and Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2018) analyzed the statistics of TKE dissipation in terms of 1 2 � Pr/Ra(∂ju′ i + ∂iu′ j)2 in the canonical RB convection without wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We checked that the numerical differences between � Pr/Ra(∂ju′ i)2 and 1 2 � Pr/Ra(∂ju′ i + ∂iu′ j)2 are indeed very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We then plot the volume-averaged TKE and the volume-averaged TKE dissipation as functions of Rew for various types of wall shear in figures 11(c) and 11(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' With the increase of wall shear strength, the volume-averaged TKE is indeed decreasing, eventually, the TKE vanishes, and the thermal turbulence is relaminarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' However, the decreased TKE and the corresponding thermal turbulence laminarization are not caused by the viscous dissipation, which is evident from figure 10(d) that the volume-averaged TKE dissipation is also decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The above analysis suggests that the plume plays a key role in thermal turbulence production and dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' To identify the mechanism responsible for the thermal tur- bulence laminarization, we then analyze the spatial and temporal distributions of plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In figures 12(a-c), we show the typical snapshots of instantaneous plume field under the Wall sheared thermal convection 17 (a) (b) (c) (d) (e) (f ) TKE production 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='02 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (a,d) The shear-produced turbulent kinetic energy (TKE), (b,e) the buoyancy-produced TKE, and (c,f ) the total TKE production, under the (1, 1) type wall shear for (a-c) Rew = 200, (d-f ) Rew = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='004 (a) (b) Rew TKE 10 2 10 3 10 4 10 3 10 2 (c) Rew TKE dissipation 10 2 10 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='0006 TKE dissipation (d) (1, 1) type (2, 1) type (1, 2) type (1, 1) type (2, 1) type (1, 2) type Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The TKE dissipation for (a) Rew = 200 and (b) Rew = 500 under the (1, 1) type wall shear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (c) the volume-averaged TKE, and (d) the volume-averaged TKE dissipation as functions of Rew for various types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='0040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='818 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi Rew Plume Area 10 2 10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='15 (1, 1) type (2, 1) type (1, 2) type (a) (b) (c) (d) Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Typical snapshots of plume field at Rew = 100 for (a) the (1, 1) type wall shear, (b) the (2, 1) type wall shear, (c) the (1, 2) type wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (d) Time-averaged plume area in the cell as functions of Rew under three types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' three types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Here, the criteria to quantitatively identify thermal plumes are similar to those used in (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' van der Poel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2017), which is |T (x, t) − ⟨T (x)⟩| > c⟨Trms(x)⟩, √ Pr · Ra|v(x, t)T (x, t)| > cNu (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8) Here, c is an empirical constant and its value can be chosen as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8 ⩽ c ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2, and we adopt the value of c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' This criterion assumes that plumes occur in regions of local temperature maximum (or minimum), as well as regions where local convective heat flux is larger than the spatial and temporal averaged one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We can see from figures 12(a-c), this empirical criterion can extract the plume structures reasonably well in the sheared convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We also calculate the time-averaged plume area in the cell and plot the plume areas as functions of Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' From figure 12(d), we can see that with the increase of wall shear strength, plume areas generally decrease under all three types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Because thermal plumes are mainly responsible for TKE production, a reduced number of plumes indicates reduced TKE production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We then examine the flow field during the laminarization process, as shown in figure 13, and the corresponding video can be viewed in the supplementary movie 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Initially, an instantaneous flow field obtained at Rew = 200 for the 2-D case and Rew = 100 for the 3- D case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', the turbulent state) is used to start the simulation, in which the shear effects are relatively weak and the flow is buoyancy dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We can see plumes self-organize into the LSC, and large magnitudes of velocity vectors appear near the region where plumes erupt (see figure 13a for the 2-D case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' When the wall shear strength increases to Rew = 2000 for the 2-D case, the plumes have less chance to detach from the boundary layers near the top and bottom walls, and they will be swept along the walls (see figure 13b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Because the organization of plume motions leads to the LSC in the turbulent RB Wall sheared thermal convection 19 convection cell (Xi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2004), suppressing plume detachment will weaken the LSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In addition, hot (or cold) plumes are forced to sweep to the cold top (or hot bottom) wall (see figures 13c and 13d) and thermal plumes exchange heat near the walls, while the temperature in the bulk region of the cell is more uniform and well-mixed (see figure 13e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Eventually, one regular big roll is formed, and hot and cold fluids flow along the wall, which is completely influenced by the external wall shear (see figure 13f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We can also see from figures 13(g-l) that, the turbulence relaminarization process in similar for both 2-D and 3-D cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' however, it is noteworthy that due to prominent shear instability effects in 3-D, the turbulence relaminarization is rare and occurs in a smaller range of wall shear Reynolds number for 3-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Expenditure of mechanical energy due to external wall shear We manipulated the internal flow modes via imposing external wall shear, and the cor- responding heat transfer efficiency enhancement requires the expenditure of mechanical energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' To evaluate whether such mechanical energy expenditure is worthy or not, we calculate the ratio between the enhanced heat flux δQ (which is further normalized by heat flux Q0 in the absence of wall shear) and the required mechanical energy Ws due to wall shear (which is further normalized by energy dissipation due to viscosity W0 in the absence of wall shear) as η = δQ/Q0 Ws/W0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='9) Here, δQ = Quw − Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Generally, the heat flux Quw is calculated as Quw = Æ� L 0 Å −κ∂T ∂n ã dx ∏ t (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='10) In the above, κ denotes thermal conductivity of the fluids, and ⟨· · · ⟩t denotes the time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' To impose the wall shear, an additional external mechanical energy Ws is required, which is calculated as Ws = ≠� l ����µduw dn · uw ���� dl ∑ t (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='11) Here, the integration � l(· · · ) is performed along all the shear wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the absence of wall shear, the energy dissipation due to viscosity in the convection cell is W0 = ∞� V µ 2 � i,j Å ∂ui ∂xj + ∂uj ∂xi ã dV ∫ t (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='12) The ratio between enhanced heat flux and imposed mechanical energy can be regarded as a metric that describes the efficiency of facilitating heat transport via external shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' From figure 14, we can see that for the (1, 1) and (2, 1) types of wall shear, the efficiency η decreases monotonously with the increase of Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Recall that from figures 4(a) and 4(c), we found the Nu increases monotonously with the increase of Rew, thus the enhanced Nu requires a larger expenditure of mechanical energy at a larger Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' For the (1, 2) type wall shear, the efficiency η is negative at Rew ⩽ 200, because the Nu is weakened in that range, as shown in figures 4(a) and 4(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' at larger Rew, the efficiency η is positive, yet it does not exhibit monotonous behavior with the increase of Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Among the three types of wall shear, the (2, 1) type results in the highest efficiency η, because the flow is more coherent in the corresponding (2, 1) flow mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' At the largest Rew, all three types of wall shear exhibit very small values of efficiency η, implying heat transfer enhancement 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi (d) (e) (f ) Temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8 (g) (h) (i) (j) (k) (l) (a) (b) (c) Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Turbulence relaminarization process: time evolution of instantaneous flow fields (temperature contours and velocity vectors in 2-D, volume rendering of temperature field in 3-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (a-f ) Snapshots for the the 2-D case at t = 0, 2, 4, 8, 22 and 354 tf, respectively, with wall shear strength Rew = 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (g-l) Snapshots for the 3-D case at t = 0, 2, 4, 8, 11 and 158 tf, respectively, with Rew = 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Initially, an instantaneous flow field obtained at Rew = 200 for the 2-D case and Rew = 100 for the 3-D case, is used to start the simulation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='8Wall sheared thermal convection 21 Rew (�Q/Q0) / (Ws/W0) 10 2 10 3 10 4 1 0 1 (1, 1) type (2, 1) type (1, 2) type Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The ratio between enhanced heat flux and imposed mechanical energy as a function of shear Reynolds number for various types of wall shear in the 2-D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' comes at a very high price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We deduce that further increase the wall shear strength, the efficiency η would approach the limit of zero (but non-negative) value because the heat transfer increases monotonously at large Rew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Conclusions In this work, we have performed direct numerical simulations of thermal convection under three different (m, n) types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The (m, n) type wall shear is imposed to facilitate m rolls in the horizontal direction and n rolls in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Under the (1, 1) type, the (2, 1) type, and the (1, 2) type wall shear, we can observe the single-roll, the horizontally stacked double-roll, and the vertically stacked double-roll flow modes, respectively, are generally prevail flow modes in the convection cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' With the increase of Rew, we generally found enhanced heat transfer efficiency and global flow strength for all three types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' However, even with the same magnitude of flow strength, the heat transfer efficiency of the convection cell varies significantly under different types of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Specifically, the (2, 1) type wall shear results in the largest magnitude of heat transfer efficiency, and the (1, 2) type wall shear results in the smallest one, which is consistent with our expectation that facilitating the horizontally stacked double-roll flow modes is efficient for heat transfer, yet facilitating the vertically stacked double-roll is inefficient for heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The original objective of imposing the wall shear was to manipulate flow mode to control heat transfer efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' While it is found that by increasing the wall shear strength, the thermal turbulence is relaminarized, and more surprisingly, the heat transfer efficiency of the convection in the laminar state is higher than that in the turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' By examining the flow field and the convective heat flux field, we found the enhancement of heat transfer efficiency at the laminar regime is due to the formation of more stable and stronger convection channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We explained the origin of thermal turbulence laminarization in the sheared convection cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Analysis of the shear-produced TKE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', −u′ iu′ j∂jui) and the buoyancy-produced TKE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', T ′v′) provide direct evidence that thermal plumes are mainly responsible for the TKE production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We then quantitatively measure the changes in plume areas under the wall shear and found that plumes are swept away by the wall shear once they are detached from the top cold and bottom hot walls, and such a reduced number of thermal plumes decreases the TKE production in the bulk cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We evaluated whether the mechanical energy expenditure by wall shear is worthy or 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We used the ratio between the enhanced heat flux and the required mechanical energy to quantify the efficiency of facilitating heat transport via external shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We found that at a larger Rew, although the heat transfer efficiency increases, it comes at a price of a larger expenditure of mechanical energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Finally, we emphasize that in turbulent wall-sheared thermal convection, the heat transfer may not always monotonically increase with increasing shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' For example, in the RB system with a Couette-type wall shear, Blass et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (2020, 2021) found with increasing wall shear, the heat transfer first decreases (due to the breakup of the thermal convection rolls) and then increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In our study, the heat transfer enhancement is a consequence of the moving adiabatic side walls advecting fluid in the vertical direction, thus facilitating the formation of stable and strong convection channels between the top cold wall and the bottom hot wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) through Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 12272311 and 12125204, and the 111 project of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' B17037).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Declaration of interests The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Supplementary movies Supplementary movies are available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='1017/jfm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='xxxxxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Flow and heat transfer patterns in the canonical RB convection In figure 15, we show the typical instantaneous temperature and flow fields, as well as vertical convective heat flux field for the canonical RB convection in the absence of wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the 2-D case, we can see that there exists a well-defined LSC, together with counter-rotating corner rolls (figure 15a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' The LSC is in the form of a tilted ellipse, sitting along a diagonal of the flow cell with two secondary corner vortices that exist along the other diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Strong positive heat flux occurs in regions of rising hot (or falling cold) plumes (see figure 15d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' In the 3-D case, the very confined cell with Γ⊥ = 1/8 exhibits similar flow and heat transfer pattern to that of the 2-D case, with persistent LSC [see figures 15b and 15e].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' When the cell aspect ratio Γ⊥ increases to 1/4, the LSC is less stable and its shape becomes distorted [see figures 15c and 15f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Determination of flow states via time recordings and its power spectrum density In figure 16, we give examples of temperature series at the location of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5) in the 2-D convection cell under (1, 1) type wall shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' We also show the power spectrum density (PSD) of the corresponding temperature series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' At Rew = 100 and 500, the temperature fluctuates randomly around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 [see figure 16(a) and 16(c)], and the corresponding PSD [see figure 16(b) and 16(d)] exhibit continuous spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' thus, we determine the flow states as the turbulent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' At Rew = 1000, the fluctuation of the temperature series is within a smaller range [see figure 16(e) and its inset], and the corresponding PSD Wall sheared thermal convection 23 (a) (b) Temperature (d) (e) (f ) (c) Heat flux Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Typical instantaneous (a-c) temperature field and (d-f ) vertical convective heat flux field for the canonical RB convection at Ra = 108 and Pr = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3: (a,d) 2-D case, (b,e) 3-D case at Γ⊥ = 1/8, (c,f ) 3-D case at Γ⊥ = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' exhibits characteristic peaks, suggesting the flow is quasi-periodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' thus, we determine the flow states as the laminar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' At Rew = 4000, the temperature series gradually approaches a steady value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 [see figure 16(f)], thus, we also determine the flow states as the laminar state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} 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flow reversals in turbulent convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' E 83 (6), 067303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='01 0.' metadata={'source': 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4000 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7 Frequency PSD 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='4 10 3 10 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='7 (a) (b) (c) (d) (e) (g) (f ) 0 2000 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='502 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' (Left-column) Temperature series at the location of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='5) in the 2-D convection cell under (1, 1) type wall shear, and (right-column) the power spectrum density (PSD) of the corresponding temperature series, at (a, b) Rew = 100, (c, d) Rew = 500, (e, f ) Rew = 1000, (g) Rew = 4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Note for the temperature series shown in (g), it eventually approaches a steady value, thus we did not calculate the corresponding PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Chandra, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' & Verma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2013 Flow reversals in turbulent convection via vortex reconnections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 110 (11), 114503.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' & Hof, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2018 Destabilizing turbulence in pipe flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 14 (4), 386–390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Lakkaraju, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Stevens, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Sugiyama, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' & Lohse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2012 Flow states in two-dimensional Rayleigh-B´enard convection as a function of aspect-ratio and Rayleigh number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Fluids 24 (8), 085104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' van der Poel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Verzicco, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' & Lohse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2009 Transitions between turbulent states in rotating Rayleigh-B´enard convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 103 (2), 024503.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Xia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='50 and Prandtl number Pr=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 676, 5–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Lam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' & Xia, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2004 From laminar plumes to organized flows: the onset of large-scale circulation in turbulent thermal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 503, 47–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' & Xia, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2008 Flow mode transitions in turbulent thermal convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Fluids 20 (5), 055104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Xi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Hao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2017 Accelerated lattice Boltzmann simulation using GPU and OpenACC with data management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 109, 577–588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Yang, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2022 Exploring the plume and shear effects in turbulent Rayleigh–B´enard convection with effective horizontal buoyancy under streamwise and spanwise geometrical confinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 940, A37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Stevens, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Clercx, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Shishkina, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=', Verzicco, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' & Lohse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 2019 Scaling enabled by multiscale wall roughness in Rayleigh–B´enard turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} +page_content=' 869, R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9AyT4oBgHgl3EQfgPgL/content/2301.00353v1.pdf'} diff --git a/KtE1T4oBgHgl3EQfYwQa/vector_store/index.pkl b/KtE1T4oBgHgl3EQfYwQa/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..617b613d27ebd9130574ec3fb39ee0aeeaa54743 --- /dev/null +++ b/KtE1T4oBgHgl3EQfYwQa/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e1d929702fa90ea1d823de8f57061bacb220d0b5bbd52a511f3823ebb36800c3 +size 62044 diff --git 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D. Cole,1,a) S. Ballmer,2 G. Billingsley,3 S. B. Cataño-Lopez,1 M. Fejer,4 +P. Fritschel,5 A. M. Gretarsson,6 G. M. Harry,7 D. Kedar,8 T. Legero,9 +C. Makarem,3 S. D. Penn,10 D. Reitze,3 J. Steinlechner,11,12 U. Sterr,9 +S. Tanioka,2 G.-W. Truong1, J. Ye8, J. Yu9 + +1Thorlabs Crystalline Solutions, Santa Barbara, CA, 93101, USA +2Department of Physics, Syracuse University, Syracuse, NY, 13244, USA +3LIGO Laboratory, California Institute of Technology, Pasadena, CA, 91125, USA +4Stanford University, Palo Alto, CA, 94309, USA. +5LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA, 02142, USA. +6Department of Physics, Embry-Riddle Aeronautical University, Prescott, AZ, 86301, USA +7Department of Physics, American University, Washington, DC, 20016, USA +8JILA, National Institute of Standards and Technology, University of Colorado Boulder, Boulder, CO, 80309, USA +9Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany +10Department of Physics, Hobart and William Smith Colleges, Geneva, NY, 14456, USA +11Maastricht University, 6200 MD Maastricht, The Netherlands +12Nikhef, 1098 XG Amsterdam, The Netherlands + +In this Perspective we summarize the status of technological development for large-area and low-noise substrate-transferred GaAs/AlGaAs +(AlGaAs) crystalline coatings for interferometric gravitational-wave (GW) detectors. These topics were originally presented in a workshop† +bringing together members of the GW community from the laser interferometer gravitational-wave observatory (LIGO), Virgo, and KAGRA +collaborations, along with scientists from the precision optical metrology community, and industry partners with extensive expertise in the +manufacturing of said coatings. AlGaAs-based crystalline coatings present the possibility of GW observatories having significantly greater +range than current systems employing ion-beam sputtered mirrors. Given the low thermal noise of AlGaAs at room temperature, GW detectors +could realize these significant sensitivity gains, while potentially avoiding cryogenic operation. However, the development of large-area +AlGaAs coatings presents unique challenges. Herein, we describe recent research and development efforts relevant to crystalline coatings, +covering characterization efforts on novel noise processes, as well as optical metrology on large-area (~10 cm diameter) mirrors. We further +explore options to expand the maximum coating diameter to 20 cm and beyond, forging a path to produce low-noise AlGaAs mirrors +amenable to future GW detector upgrades, while noting the unique requirements and prospective experimental testbeds for these novel +materials. + + + + + +_________________________ +†This Perspective serves as a summary of the AlGaAs Workshop, held at American University, Washington DC USA Aug. 15-17, 2022. +a) Author to whom correspondence should be addressed. Electronic mail: gcole@thorlabs.com. + + + + + + + + +2 + + +Thermal noise in high-reflectivity optical interference coatings is a limiting noise source in precision interferometric systems. +The pioneering theoretical work on thermal noise by Callen and Greene [1] was introduced to the GW community by Saulson +[2,3], as well as Braginsky and collaborators [4]. In 1998, Yuri Levin [5] identified coating thermal noise (CTN) as a potential +limiting noise source for gravitational wave detectors. Harry, et al. [6] measured the elastic loss in the Initial LIGO coatings +and confirmed that CTN would limit the sensitivity of Advanced LIGO [7]. A collaboration of Syracuse, Glasgow, Stanford, +and the LIGO Lab determined the source of the loss to be the high-index material [8] and designed the coating [9] used in +Advanced LIGO to make the first direct detection of these ripples in space-time [10]. +For the past 20 years, a concerted research effort has sought to reduce CTN by identifying coating materials exhibiting +both low levels of optical and elastic losses. The first decade of this work is summarized in Ref. 11. In the Advanced LIGO +interferometers, coating Brownian noise limits the achievable strain sensitivity in the most sensitive frequency band around +100 Hz [12]. Similarly, this noise source impacts the stability of ultrastable optical resonators, placing a limit on the minimum +linewidth achievable in lasers employed for cutting-edge optical atomic clocks [13,14]. This was initially explored in cavity- +stabilized laser systems owing to theoretical work by Numata [15], followed by measurements on cm-length reference cavities +by Notcutt and colleagues [16]. Exploratory efforts focusing on alternative materials with these same requirements were also +carried out with micrometer-scale systems in the burgeoning field of cavity optomechanics [17]. Early work in this field +ultimately led to the development of the substrate-transferred GaAs/AlGaAs (AlGaAs) crystalline coatings as described herein. +A key motivation of these efforts is the potential for significant performance enhancements in GW detectors, owing to the low +elastic losses and correspondingly low Brownian noise of these mirrors. As shown in Figure 1, in a model LIGO-based +interferometer employing crystalline mirrors, the achievable strain sensitivity at 100 Hz is 1.1×10-24 /√Hz, representing a 3.6× +improvement over the Advanced LIGO design target at the same frequency (4.0×10-24 /√Hz) [12]. This results in a significant +enhancement in the astrophysical reach for a binary neutron star merger, from 175 Mpc for the current design target to 600 Mpc +for this proposed upgrade with AlGaAs-based crystalline coatings—yielding a factor of ~40 increase in detection rates. + + + + + + +3 + + + +Fig. 1. Target strain sensitivity, as limited by fundamental noise sources, for a potential upgrade to the LIGO interferometers +using AlGaAs crystalline coatings and operating at room temperature. The CTN curve (red) is calculated using a mechanical +loss angle of 6.2×10-6, based on direct thermal noise measurements at MIT*1, and a beam radius of 5.5 cm on the end test +masses and 4.5 cm on the input test masses; these beam sizes are compatible with a 30 cm diameter AlGaAs coating. The test +masses are 100 kg fused silica (substrate noises), suspended with fused silica fibers (suspension thermal noise). The quantum +vacuum noise derives from 1.5 MW of laser power in each arm cavity (1.06 µm wavelength), combined with 10 dB of effective +vacuum squeezing at all frequencies, which is realized using a narrow linewidth filter cavity that appropriately rotates the +squeezing angle as a function of frequency. The noise due to residual gas in the vacuum system arises from damping of the +suspended test masses at frequencies below 50 Hz, and from scattering of the laser beams in the 4 km arms at frequencies above +50 Hz. The Newtonian noise is due to density perturbations in the earth close to the test masses, producing fluctuating +gravitational forces. The seismic noise, in contrast, is the earth vibration that couples through the seismic isolation and test +mass suspensions. + +In this Perspective we provide a detailed account of the status of AlGaAs-based crystalline coatings as a solution to the +coating Brownian noise problem. We begin with a brief historical overview of crystalline multilayers in cavity optomechanics +experiments from ~2007-2012. We then discuss the transition of this technology to precision metrology applications with the +development of centimeter-scale AlGaAs coatings for ultrastable optical reference cavities. Recent findings from key partners +at national metrology labs point to novel noise processes in these coatings at cryogenic temperatures. Exploring size scaling, + +*1 Personal communication, S. Gras and N. Demos, MIT. + +10 +-21 +Total +Coating Thermal +QuantumVacuum +SubstrateBrownian +Seismic +Substrate Thermo-Elastic +Newtonian +Residual Gas +SuspensionThermal +-22 +10 +10 +23 +10 +-24 +-25 +10 +10 +10° +Frequency [Hz] + + + + +4 + + +we cover preliminary results for crystalline mirrors at diameters up to 10 cm, with discussions relevant to expanding to 20 cm +and beyond, covering the optical properties of these single-crystal films in terms of their absorption, scatter, birefringence, and +surface uniformity. Given the opaque nature of AlGaAs coatings in the visible range, alternative lock acquisition schemes must +be defined; one potential solution is presented here. Next is an overview of experimental testbeds that would enable detailed +metrology of large-area crystalline mirrors. Finally, a brief overview of paths forward in terms of research and funding +requirements is presented. +AlGaAs-based monocrystalline multilayers were first pursued as high-performance micromechanical resonators for cavity +optomechanics experiments. This compound semiconductor material platform had historically been employed for microwave +devices, as well as for micro-cavity-based optoelectronics devices. It was not until 2008 that measurements of the intrinsic +elastic losses of AlGaAs were performed, revealing a unique combination of low optical and mechanical losses [18]. The +realization of low elastic losses, represented by the mechanical loss angle, ϕ (or conversely, the mechanical quality factor, +𝑄 = +1 +𝜙), was paramount to reaching the quantum regime in cavity optomechanics. Amorphous ion-beam sputtered (IBS) +multilayers, while capable of high reflectivity, exhibited Q values at the few thousand level (corresponding ϕ of a few ×10-4). +In comparison, crystalline Bragg mirrors, owing to their improved structural order, show a significant improvement, with +measured Q values in the range of ~ 20,000 to > 200,000 (ϕ at the low 10-5 level or below) [18-21]. The low Q values in IBS- +deposited optical coatings presented a major roadblock in these efforts. This can be understood by the 𝑄𝑓 product, with f being +the mechanical eigenfrequency of the resonator. This eigenfrequency must exceed the thermal decoherence rate, yielding the +condition: 𝑄𝑓 > 𝑘𝐵𝑇𝑏𝑎𝑡ℎ/ℎ (kB – Boltzmann constant, Tbath – system temperature, h – Planck’s constant), for the system to +survive at least a single oscillation before a thermal phonon causes decoherence. This is a necessary requirement to prepare and +detect nonclassical states of motion [22,23]. Ultimately, high-Q AlGaAs-based micromechanical devices have been +instrumental in studying fundamental aspects of quantum-limited interferometry [24,25]. +The low-noise potential of these suspended micromirrors motivated the development of centimeter-scale reflectors based +on substrate-transferred AlGaAs multilayers [26]. Production considerations for these novel coatings have been covered in +detail elsewhere [27]. Given lattice matching constraints in epitaxial (crystal) growth, direct deposition is not possible, thus +separate growth, microfabrication, and bonding is necessary to generate the coated optic. For low-loss macroscopic mirrors, +optical quality is paramount. Each stage of the production process has the potential for defects, with the epitaxial growth stage + + + + + + +5 + + +contributing the largest share of defects. Since 2012, crystalline coatings, typically 5-20 mm in diameter, transferred to planar +and curved bulk fused silica substrates have been realized. Other substrate materials have been successfully implemented +including Si and Al2O3 (sapphire) for cryogenic reference cavities, as well as SiC, diamond, and YAG, for high-power laser +systems. Optimized crystalline coatings with a radius of curvature as tight as 10 cm have demonstrated excess losses (scatter + +absorption) below 2 ppm, with absorption as low as ~0.5 ppm observed between 1 μm and 1.5 μm. More recently, excess losses +< 10 ppm have been demonstrated for mirrors operating near 4.5 μm [28,29]. The maturation of cm-scale crystalline coating +production in the past decade has put this technology on par with IBS in terms of optical losses in the near-infrared spectral +region, while exceeding the state-of-the-art in the mid-infrared (wavelengths from 2-5 μm). Standard mirrors are now +commercially available, finding applications in cavity-stabilized lasers and in cavity-enhanced spectroscopy [30]. +In time-and-frequency metrology, where high-finesse reference cavities employing crystalline coatings are becoming +ubiquitous, the noise of AlGaAs multilayers has been closely studied and compared against theory. Room temperature cavity- +stabilized lasers employing these coatings have been demonstrated to operate near the thermal noise floor [31,32], while turn- +key systems capable of a fractional frequency instability < 5×10-16 are commercially available [33]. As the metrology +community pushes optical oscillators to lower instabilities, the research focus has shifted to cryogenic systems. Progress on +cryogenic reference cavities has matured to the point where the dominant noise contribution is CTN from the amorphous +mirrors [34], making them ideal testbeds for probing low-temperature noise sources in AlGaAs coatings. +Two independent studies using silicon cavities with crystalline coatings at 1.5 μm have pioneered crystalline coating +characterization at cryogenic temperatures. In both systems, the expected contributions from technical noise, and spacer and +substrate thermal noise are well below the expected coating thermal noise (one cavity is 21 cm long and held at 124 K [35], the +other is a 6 cm long cavity operated at 4 K and 16 K [36]). Interestingly, both systems have revealed hitherto unknown noise +sources that can be manipulated by the polarization of the probe beam. Although it is well-known that coating birefringence +leads to a static frequency splitting between orthogonal polarizations of the TEM00 mode, the cryogenic testbeds additionally +observe dynamic frequency fluctuations of the two polarization components that are anti-correlated. Additionally, the +magnitude of this effect increases with the intracavity optical power. The resulting noise level is far above the coating thermal +noise floor (20-40 dB depending on the cavity and the temperature) and the power spectral density acquires a slope steeper than +1/𝑓. Both birefringent modes of the cavity exhibit similar levels of frequency noise for equivalent optical conditions. However, + + + + + + +6 + + +if the two modes are addressed simultaneously, the anti-correlated frequency fluctuations can be averaged and suppressed +[35,36], as in Figure 2. The residual noise after cancellation no longer scales with intracavity optical intensity, though it is still +above the expected thermal noise level. This noise source is coherent between a TEM10 and TEM00 beam, implying a longer +spatial correlation length than the spot size of ~1 mm. The source of this “global” noise is not yet understood and further +investigations are ongoing. It also remains to be seen whether these measured noise scalings persist at higher frequency, as +these cavities are optimized for the frequency range of 1 Hz and below, lower than the frequency band where coating thermal +noise is relevant for Advanced LIGO, roughly 30-300 Hz. + +Fig. 2. Potential non-Brownian CTN observed in ultrastable cryogenic Si reference cavities with AlGaAs coatings. (a) Optical +path length fluctuations of a 21-cm long Si cavity measured for the two polarization eigenmodes of the TEM00 resonance (blue, +orange). (b) Power spectral densities, Sd, of the length fluctuations of the cavity. The plot in this panel includes the birefringent +noise (purple), the noise of an individual polarization eigenmode (orange), as well as the average of the two polarizations (red). +The sum of technical noise and the measured Brownian thermal noise limit for the TEM00 mode (green) is included for +comparison. (c) and (d) Power spectral densities of the length fluctuations of a 6-cm long Si cavity at 16 K (c) and 4 K (d). +Shown are the frequency stability of the individual polarization eigenmodes (orange), the average of two polarization +eigenmodes (red), and the predicted Brownian thermal noise (green). As explained in the main text, while the average +polarization can be used to suppress the birefringence fluctuations, the residual noise remains above the expected CTN level +for these systems. Figure reproduced from Ref. 35. + +a) +b) +T = 124 K +Slow axis +T = 124 K +10 +-29 +Birefringent noise (fast - slow) +Optical path length change (fm) +0.3 +Average +Single polarization +0.2 +Fastaxis +10 +30 +Polarization-average +Predicted total noise +0.1 +(zH/zw) +10 +-31 +0.0 +10 +32 +0.1 +10 +-33 +0.2 +10 +34 +0.3 +W +10 +-35 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +10 °3 +10~1 +100 +Time (h) +Fourierfrequency(Hz) +c) +d) +T=16K +Single polarization +T=4K +Single polarization +10 +31 +Polarization-average +10 +-31 +Polarization-average +PredictedBrownianthermal noise +PredictedBrownianthermal noise +10 +-32 +10 +32 +(zH/zw) +(zH/zw) Ps +10 +-33 +10 +-33 +s +34 +10 +-34 +10*35 +10 +35 +10 +-36 +10 +-36 +10 2 +10 +100 +10 ~2 +10° +100 +Fourierfrequency(Hz) +Fourierfrequency(Hz) + + + + +7 + + +It is important to note that the observed birefringence in crystalline coatings is an “extrinsic” effect, as 100-oriented GaAs +is optically symmetric. The current conjecture is that non-uniform strain relaxation, upon cooling from the growth temperature +drives the symmetry breaking [37]. The coatings are grown at elevated temperature (~600 ºC) leaving a compressive residual +stress of ~100 MPa upon cooling. It is possible to tailor the strain by alloying the multilayer with In or P (e.g., GaAsP, InGaP, +InGaAs, etc.) [38]. Careful measurement of the optomechanical properties of these materials would be necessary. The observed +birefringence, measured from cavity mode splitting and corresponding to the accumulated difference in phase on reflection +between the fast and slow polarizations, 𝛥𝜃=𝜃𝑓−𝜃𝑠, are all within a similar range, roughly 2×10-3 radians, and are temperature +and substrate independent. Furthermore, thermal cycling does not affect the mode splitting in cryogenic coatings. +Beyond studies of the static and dynamic (fluctuating) components of birefringence, there is a need to investigate potential +thermally-induced effects. Owing to the anisotropic nature of AlGaAs coatings, radial thermal gradients will induce shear +strains in the crystal. Assuming the mirror is a flat, half-infinite disk and the coating face is parallel to the [001] crystal plane, +if we choose the 𝑥 and 𝑦 coordinate axes to lie in the [010] and [100] planes respectively, then the magnitude of the induced +birefringence is largest along lines at +/- 45º to the coordinate axes. In addition, the orientation of the principal axes varies with +azimuthal angle across the face of the mirror. Assuming a 100 µm diameter perfect absorber and a mirror irradiance at the level +of Advanced LIGO, the effect may be similar in magnitude to the static birefringence seen in AlGaAs mirrors; several point +absorbers could combine to impart a significant effect and must be investigated further. +Challenges with large-area mirror production are being tackled to extend the application of crystalline mirrors from +advanced reference cavities to GW-detection. Studies on 2” and 3” (50.8-76.2 mm) diameter test mirrors have yielded: i) mean +absorption < 1 ppm, ii) mean total integrated scatter (TIS): < 10 ppm, and iii) coating thickness variation (rms): < 100 ppm +[39,40]. Ongoing efforts involve the production 10 cm and ultimately 20 cm diameter test mirrors, the latter representing the +largest continuous crystalline coatings that can be produced, limited by the availability of base substrates for epitaxial growth. +In terms of the observed optical properties in large coatings, there appears to be a greater number of scattering centers compared +with the best IBS coatings, but the background total integrated scatter level away from larger (> 20 µm) scatterers is comparable. +These scattering centers may be caused by “oval defects” arising from spitting of the gallium source during deposition, +generating crystallites that locally disrupt the structure. It is not known whether these defects are absorbing; thus, distinguishing +pure scattering centers from local absorbers is an important task. Both are a source of optical loss but have different impact on + + + + + + +8 + + +interferometer. Similarly, a bidirectional reflectance distribution function analysis of larger point defects will be useful for +estimating the effect of rare but large scatterers versus frequent but small scatterers on interferometer scatter noise. +Recent results from Caltech include surface maps and scattering data from the first 10 cm diameter AlGaAs mirror +(transferred to a 10-mm thick planar synthetic fused silica substrate), see Panel a of Figure 3 for more details. Surface figure +measurements were made on the bare substrate before coating and on also the final coated mirror. After coating, the surface +appears to have gained 4 nm of astigmatism over an 80 mm diameter aperture. The radius of the coated substrate changed by +–784 m, with a corresponding sagitta change of 270 nm (convex) over an 80 mm diameter aperture. This could be due to non- +uniformity of the coating or stress imparted on the 10 mm thick substrate. Panels b and c of Figure 3 show the results of the +scattering measurements performed on this test structure. The mean TIS is somewhat higher than for the smaller samples +mentioned earlier. This is due to an increased number of relatively strong scatterers indicated by red points (Fig. 3c). Excluding +these large scattering centers, the mean total integrated scatter of this initial test mirror was approximately 2 ppm higher than +equivalent measurements on an Advanced LIGO ETM, with a point scatterer density of 86 cm-2. Additional 10 cm diameter +test mirrors are currently in production to ascertain the repeatability in optical performance of these first large mirrors. + + +Fig. 3. Optical characterization of a first 10 cm diameter test mirror with substrate-transferred AlGaAs coatings. a) Photographs +of the mirror backside (left), viewing the bond interface through the 10-mm thick fused silica substrate, and mirror frontside +(right) following substrate and etch stop removal, leaving only the GaAs/AlGaAs multilayer on the fused silica substrate. b) +Histogram of the TIS for apertures of 40 mm diameter (blue) and 80 mm diameter (red). The vertical dashed line shows the +mean TIS for the 40 mm diameter aperture. The legend gives the means and standard deviations of both histograms. c) Scatter +map obtained via an integrating sphere raster-scanned over the mirror surface. The thin blue circle shows the 40 mm diameter +inner aperture referenced in panel b). + +ppm +a) +c) +40 +30 +20 +10 2 +10 +(ww) +b) +Entries +55557 +10 4 +Mean +Φ80mm +37.19 +Y +RMS +135.6 +Entries +14073 +-10 +103 +Mean +Φ40 mm +27.40 +10 +Count +RMS +111.4 +10 2 +-20 +10 +-30 +HW +102 +-30 +-20 +-10 +0 +10 +20 +30 +10 +103 +40 +1 +TIS(ppm, 1°≤≤750) +X (mm) + + + + +9 + + +Given the lack of commercially available options, scaling to AlGaAs coating diameters beyond 20 cm will entail the +growth of custom GaAs boules for waferization. Sticking with traditional wafer geometries, 30 cm diameter would be an +obvious choice as a next step. In terms of multilayer epitaxy, production MBE systems have demonstrated sufficiently good +optical performance and uniformity [27,40] and would not require customization. A dedicated system will be necessary to +produce high-performance prototype and deliverable optics capable of meeting the strict optical specifications of GW-detectors +[41]. At the bonding stage, commercial vendors have demonstrated the production of silicon-on-insulator wafers up to 45-cm +diameter [42]. However, such systems are typically limited to a total bonded thickness of a few mm, while GW-relevant optics +exhibit high mass (~40 kg) and much greater thickness (~20 cm), entailing modified tooling and unique challenges in production. +If sufficiently large mirrors with the desired performance metrics can be produced, it will then be necessary to explore +impacts on the overall system operation. For instance, given the narrow bandgap of the high index GaAs layers, traditional +cavity arm locking with frequency-doubled 532 nm light is no longer an option [43]; thus, alternative locking schemes must be +developed. Given the long-wavelength transparency of AlGaAs, a dichroic coating with sufficient reflectance at both 1064 nm +as well as an auxiliary wavelength of 2128 nm may be used. The auxiliary 2128 nm beam would be generated from 1064 nm +beam, which is phase-locked to the main laser, using a degenerate optical parametric oscillator (DOPO) instead of a +conventional frequency doubling [44]. If low-noise dichroic AlGaAs coatings can be produced, then a locking system can be +realized with minimal risk. In principle such a coating design is possible, and based on recent optical loss measurements, at 4.5 +μm, absorption at the 1 ppm level or less is expected for wavelengths near 2 μm [28,29]. +Possible options for installing AlGaAs-coated input and end test masses (ITMs and ETMs) as an upgrade to the existing +4 km Advanced LIGO interferometers will require minimum coating diameters of 21-22 cm to exceed current requirements on +coating thermal noise. Thus, the use of 20 cm GaAs wafers for both test masses is not possible without radical modifications +and custom boules are needed. However, mixed mirror sizes could be employed to avoid this, with larger IBS-coated ITMs +focusing a smaller spot onto AlGaAs coated ETMs. Such a design is limited by the size of the beamsplitter but has the advantage +of keeping the power-recycling and signal-recycling cavity designs mostly unchanged (though it would require reshaping the +anti-reflective side of the ITM to form a lens). Similar “mixed mirror” solutions leveraging 20 cm diameter AlGaAs will be +evaluated for noise, alignment stability, resonances of higher order modes in the arms, sideband resonances in the arms, etc. +These designs have the potential to serve as technology demonstrators for next-generation instruments. + + + + + + +10 + + +As with cutting-edge ultrastable laser efforts, third-generation GW detectors such as the Einstein Telescope (ET) are +proposing to incorporate cryogenics. To maintain compatibility with available growth substrates, the ET low frequency +interferometer could potentially implement similar mixed mirror designs, including cooled ETMs with a 13 cm beam and 70 cm +diameter IBS coatings and ITMs having a 4 cm spot size with 20 cm diameter AlGaAs coatings (with or without cryogenic +cooling depending on the ultimate outcome of the cryogenic crystalline coating noise). With cryogenic cooling, this geometry +could employ ultrapure float-zone silicon substrates for the ITMs, which are currently available up to 20 cm diameter. +Open questions relevant to AlGaAs in future GW detectors may go beyond that which can be answered in table-top +experiments. These include: (a) an accurate wideband (frequency) measurement of coating noise; (b) successful production of +larger than ‘lab-scale’ mirrors, spanning substrate procurement, polishing, and bonding, to integration with relevant suspension +systems; and (c) investigations of large-area coating performance when integrated in a complex and sensitive system at high +laser power. Several platforms will be available within the gravitational-wave community in the near-term for such efforts: +i) +10 m prototype at the AEI in Hannover, Germany [45], operating at 1064 nm and room temperature. A key aim +is to investigate and overcome the standard quantum limit [25] and the use of low-noise AlGaAs-coated, 4.8 cm +diameter mirrors have been proposed. +ii) +Gingin prototype in Western Australia [46] will investigate high-power effects in silicon mirrors at a wavelength +near 2 μm. This is a three-phase project: (1) 7 m Fabry Perot cavity at 5 W laser power and 3 mm beam diameter +with interchangeable fused silica and silicon mirrors, (2) 72 m Fabry Perot cavity using silicon mirrors with 10 cm +diameter AlGaAs coatings and ~1 cm beam diameter, (3) 23 kW laser power in the arm cavities at 123 K. This +system can be used to explore the impact of point absorbers, wide angle scattering, thermal distortions and +birefringence of coated and uncoated silicon substrates, and possibly electro-optic and non-linear effects at high +laser power. +iii) +There are currently two cryogenic protoypes under development, the ET-pathfinder in Maastricht, Netherlands +[47] and a cryogenic upgrade to the 10 m prototype in Glasgow, UK [48], with the aim of testing technologies +for low temperature operation of future GW detectors. Parameters such as 1.5 μm and 2 μm laser wavelengths at +temperatures of ~120 K and ~15-20 K are planned, using Si substrates for the mirrors. These systems could be +ideal platforms to test large-area AlGaAs coatings beyond ongoing efforts with cm-scale reference cavities. + + + + + + +11 + + +These platforms will be instrumental in confirming the viability of AlGaAs coatings in these unique astronomical instruments. +We have outlined the historical background in the initial development of, as well as the current status and potential paths +forward for, AlGaAs-based crystalline coatings. These unique coatings exhibit promising optomechanical properties for +enhanced sensitivity in GW detection and thus demand further investigation. We end by clarifying that the cost and timeline to +realize the large-diameter production capabilities above (custom base wafers, epitaxy, and bonding) is comparable to the +development of other important subsystems such as seismic isolation [49] and quantum squeezing [50], which is an appropriate +comparison in that coating thermal noise is the limiting noise in the most sensitive frequency band of second-generation GW +detectors [7]. This can also be compared to potential budgetary savings realized by putting off or even eliminating the need to +develop cryogenics for future detectors [51]. A proposed timeline (available to LIGO, Virgo, and KAGRA members) has been +developed that would allow GW-relevant AlGaAs coatings to be realized within a span of 5 years. This will allow for AlGaAs +to be considered for upgrades to the Advanced LIGO detectors. +ACKNOWLEDGEMENTS +This research was supported by the National Science Foundation through the following grant awards: American University +(PHYS-2012017, PHYS-2011787), Caltech (PHY-1764464), HWS (PHY-1912699, PHY-2011688, PHY-2208079), M.I.T. +(PHY-1764464), Embry-Riddle Aeronautical University (PHY- 2110598), Syracuse University (PHY-2011723, PHY- +2207640). J. Steinlechner acknowledges the support of ETpathfinder (Interreg Vlaanderen-Nederland), E-TEST (Interreg +Euregio Meuse-Rhine), the Province of Limburg, and support by the NWO Talent Programme Vidi 2020, project number +VI.Vidi.203.062. D. Kedar and J. Ye acknowledge support from NSF PHY-1734006, NIST, and AFRL. T. Legero, U. Sterr +and J. Yu acknowledge support by the Project 20FUN08 NEXTLASERS, which has received funding from the EMPIR +programme co-financed by the Participating States and from the European Union’s Horizon 2020 Research and Innovation +Programme. A portion of this work was performed in the UCSB Nanofabrication Facility, an open access laboratory. +DATA AVAILABILITY +The data that support the findings of this study are available from the corresponding author upon reasonable request. + + + + + + + + +12 + + +REFERENCES +[1] H. B. Callen and R. F. Greene, “On a Theorem of Irreversible Thermodynamics,” Phys. Rev. 86, 702 (1952). +[2] P. R. Saulson, “Thermal noise in mechanical experiments”, Phys. Rev. D 42, 2437 (1990). +[3] G. I. González and P. R. Saulson, “Brownian motion of a mass suspended by an anelastic wire,” J. Acoust. Soc. Am. 96, +207 (1994) +[4] V. B. Braginsky, V. P. Mitrofanov, and V. I. Panov, Systems with Small Dissipation. University of Chicago Press 1985. +[5] Y. 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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Penn,10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Reitze,3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Steinlechner,11,12 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Sterr,9 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Tanioka,2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Truong1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Ye8, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Yu9 1Thorlabs Crystalline Solutions, Santa Barbara, CA, 93101, USA 2Department of Physics, Syracuse University, Syracuse, NY, 13244, USA 3LIGO Laboratory, California Institute of Technology, Pasadena, CA, 91125, USA 4Stanford University, Palo Alto, CA, 94309, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 5LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA, 02142, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 6Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Embry-Riddle Aeronautical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Prescott,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' AZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 86301,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' USA 7Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' American University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 20016,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' USA 8JILA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' National Institute of Standards and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' CO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 80309,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' USA 9Physikalisch-Technische Bundesanstalt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 38116 Braunschweig,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Germany 10Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Hobart and William Smith Colleges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Geneva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 14456,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' USA 11Maastricht University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 6200 MD Maastricht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The Netherlands 12Nikhef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 1098 XG Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The Netherlands In this Perspective we summarize the status of technological development for large-area and low-noise substrate-transferred GaAs/AlGaAs (AlGaAs) crystalline coatings for interferometric gravitational-wave (GW) detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' These topics were originally presented in a workshop† bringing together members of the GW community from the laser interferometer gravitational-wave observatory (LIGO), Virgo, and KAGRA collaborations, along with scientists from the precision optical metrology community, and industry partners with extensive expertise in the manufacturing of said coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' AlGaAs-based crystalline coatings present the possibility of GW observatories having significantly greater range than current systems employing ion-beam sputtered mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Given the low thermal noise of AlGaAs at room temperature, GW detectors could realize these significant sensitivity gains, while potentially avoiding cryogenic operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' However, the development of large-area AlGaAs coatings presents unique challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Herein, we describe recent research and development efforts relevant to crystalline coatings, covering characterization efforts on novel noise processes, as well as optical metrology on large-area (~10 cm diameter) mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' We further explore options to expand the maximum coating diameter to 20 cm and beyond, forging a path to produce low-noise AlGaAs mirrors amenable to future GW detector upgrades, while noting the unique requirements and prospective experimental testbeds for these novel materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' _________________________ †This Perspective serves as a summary of the AlGaAs Workshop, held at American University, Washington DC USA Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 15-17, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' a) Author to whom correspondence should be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Electronic mail: gcole@thorlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 2 Thermal noise in high-reflectivity optical interference coatings is a limiting noise source in precision interferometric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The pioneering theoretical work on thermal noise by Callen and Greene [1] was introduced to the GW community by Saulson [2,3], as well as Braginsky and collaborators [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In 1998, Yuri Levin [5] identified coating thermal noise (CTN) as a potential limiting noise source for gravitational wave detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Harry, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' [6] measured the elastic loss in the Initial LIGO coatings and confirmed that CTN would limit the sensitivity of Advanced LIGO [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' A collaboration of Syracuse, Glasgow, Stanford, and the LIGO Lab determined the source of the loss to be the high-index material [8] and designed the coating [9] used in Advanced LIGO to make the first direct detection of these ripples in space-time [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' For the past 20 years, a concerted research effort has sought to reduce CTN by identifying coating materials exhibiting both low levels of optical and elastic losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The first decade of this work is summarized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In the Advanced LIGO interferometers, coating Brownian noise limits the achievable strain sensitivity in the most sensitive frequency band around 100 Hz [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Similarly, this noise source impacts the stability of ultrastable optical resonators, placing a limit on the minimum linewidth achievable in lasers employed for cutting-edge optical atomic clocks [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This was initially explored in cavity- stabilized laser systems owing to theoretical work by Numata [15], followed by measurements on cm-length reference cavities by Notcutt and colleagues [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Exploratory efforts focusing on alternative materials with these same requirements were also carried out with micrometer-scale systems in the burgeoning field of cavity optomechanics [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Early work in this field ultimately led to the development of the substrate-transferred GaAs/AlGaAs (AlGaAs) crystalline coatings as described herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' A key motivation of these efforts is the potential for significant performance enhancements in GW detectors, owing to the low elastic losses and correspondingly low Brownian noise of these mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' As shown in Figure 1, in a model LIGO-based interferometer employing crystalline mirrors, the achievable strain sensitivity at 100 Hz is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='1×10-24 /√Hz, representing a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='6× improvement over the Advanced LIGO design target at the same frequency (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='0×10-24 /√Hz) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This results in a significant enhancement in the astrophysical reach for a binary neutron star merger, from 175 Mpc for the current design target to 600 Mpc for this proposed upgrade with AlGaAs-based crystalline coatings—yielding a factor of ~40 increase in detection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Target strain sensitivity, as limited by fundamental noise sources, for a potential upgrade to the LIGO interferometers using AlGaAs crystalline coatings and operating at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The CTN curve (red) is calculated using a mechanical loss angle of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='2×10-6, based on direct thermal noise measurements at MIT*1, and a beam radius of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 cm on the end test masses and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 cm on the input test masses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' these beam sizes are compatible with a 30 cm diameter AlGaAs coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The test masses are 100 kg fused silica (substrate noises), suspended with fused silica fibers (suspension thermal noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The quantum vacuum noise derives from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 MW of laser power in each arm cavity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='06 µm wavelength), combined with 10 dB of effective vacuum squeezing at all frequencies, which is realized using a narrow linewidth filter cavity that appropriately rotates the squeezing angle as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The noise due to residual gas in the vacuum system arises from damping of the suspended test masses at frequencies below 50 Hz, and from scattering of the laser beams in the 4 km arms at frequencies above 50 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The Newtonian noise is due to density perturbations in the earth close to the test masses, producing fluctuating gravitational forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The seismic noise, in contrast, is the earth vibration that couples through the seismic isolation and test mass suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In this Perspective we provide a detailed account of the status of AlGaAs-based crystalline coatings as a solution to the coating Brownian noise problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' We begin with a brief historical overview of crystalline multilayers in cavity optomechanics experiments from ~2007-2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' We then discuss the transition of this technology to precision metrology applications with the development of centimeter-scale AlGaAs coatings for ultrastable optical reference cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Recent findings from key partners at national metrology labs point to novel noise processes in these coatings at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Exploring size scaling, 1 Personal communication, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Gras and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Demos, MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 10 21 Total Coating Thermal QuantumVacuum SubstrateBrownian Seismic Substrate Thermo Elastic Newtonian Residual Gas SuspensionThermal 22 10 10 23 10 24 25 10 10 10° Frequency [Hz] 4 we cover preliminary results for crystalline mirrors at diameters up to 10 cm, with discussions relevant to expanding to 20 cm and beyond, covering the optical properties of these single-crystal films in terms of their absorption, scatter, birefringence, and surface uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Given the opaque nature of AlGaAs coatings in the visible range, alternative lock acquisition schemes must be defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' one potential solution is presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Next is an overview of experimental testbeds that would enable detailed metrology of large-area crystalline mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Finally, a brief overview of paths forward in terms of research and funding requirements is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' AlGaAs-based monocrystalline multilayers were first pursued as high-performance micromechanical resonators for cavity optomechanics experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This compound semiconductor material platform had historically been employed for microwave devices, as well as for micro-cavity-based optoelectronics devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' It was not until 2008 that measurements of the intrinsic elastic losses of AlGaAs were performed, revealing a unique combination of low optical and mechanical losses [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The realization of low elastic losses, represented by the mechanical loss angle, ϕ (or conversely, the mechanical quality factor, 𝑄 = 1 𝜙), was paramount to reaching the quantum regime in cavity optomechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Amorphous ion-beam sputtered (IBS) multilayers, while capable of high reflectivity, exhibited Q values at the few thousand level (corresponding ϕ of a few ×10-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In comparison, crystalline Bragg mirrors, owing to their improved structural order, show a significant improvement, with measured Q values in the range of ~ 20,000 to > 200,000 (ϕ at the low 10-5 level or below) [18-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The low Q values in IBS- deposited optical coatings presented a major roadblock in these efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This can be understood by the 𝑄𝑓 product, with f being the mechanical eigenfrequency of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This eigenfrequency must exceed the thermal decoherence rate, yielding the condition: 𝑄𝑓 > 𝑘𝐵𝑇𝑏𝑎𝑡ℎ/ℎ (kB – Boltzmann constant, Tbath – system temperature, h – Planck’s constant), for the system to survive at least a single oscillation before a thermal phonon causes decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This is a necessary requirement to prepare and detect nonclassical states of motion [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Ultimately, high-Q AlGaAs-based micromechanical devices have been instrumental in studying fundamental aspects of quantum-limited interferometry [24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The low-noise potential of these suspended micromirrors motivated the development of centimeter-scale reflectors based on substrate-transferred AlGaAs multilayers [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Production considerations for these novel coatings have been covered in detail elsewhere [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Given lattice matching constraints in epitaxial (crystal) growth, direct deposition is not possible, thus separate growth, microfabrication, and bonding is necessary to generate the coated optic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' For low-loss macroscopic mirrors, optical quality is paramount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Each stage of the production process has the potential for defects, with the epitaxial growth stage 5 contributing the largest share of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Since 2012, crystalline coatings, typically 5-20 mm in diameter, transferred to planar and curved bulk fused silica substrates have been realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Other substrate materials have been successfully implemented including Si and Al2O3 (sapphire) for cryogenic reference cavities, as well as SiC, diamond, and YAG, for high-power laser systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Optimized crystalline coatings with a radius of curvature as tight as 10 cm have demonstrated excess losses (scatter + absorption) below 2 ppm, with absorption as low as ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 ppm observed between 1 μm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' More recently, excess losses < 10 ppm have been demonstrated for mirrors operating near 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 μm [28,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The maturation of cm-scale crystalline coating production in the past decade has put this technology on par with IBS in terms of optical losses in the near-infrared spectral region, while exceeding the state-of-the-art in the mid-infrared (wavelengths from 2-5 μm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Standard mirrors are now commercially available, finding applications in cavity-stabilized lasers and in cavity-enhanced spectroscopy [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In time-and-frequency metrology, where high-finesse reference cavities employing crystalline coatings are becoming ubiquitous, the noise of AlGaAs multilayers has been closely studied and compared against theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Room temperature cavity- stabilized lasers employing these coatings have been demonstrated to operate near the thermal noise floor [31,32], while turn- key systems capable of a fractional frequency instability < 5×10-16 are commercially available [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' As the metrology community pushes optical oscillators to lower instabilities, the research focus has shifted to cryogenic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Progress on cryogenic reference cavities has matured to the point where the dominant noise contribution is CTN from the amorphous mirrors [34], making them ideal testbeds for probing low-temperature noise sources in AlGaAs coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Two independent studies using silicon cavities with crystalline coatings at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 μm have pioneered crystalline coating characterization at cryogenic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In both systems, the expected contributions from technical noise, and spacer and substrate thermal noise are well below the expected coating thermal noise (one cavity is 21 cm long and held at 124 K [35], the other is a 6 cm long cavity operated at 4 K and 16 K [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Interestingly, both systems have revealed hitherto unknown noise sources that can be manipulated by the polarization of the probe beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Although it is well-known that coating birefringence leads to a static frequency splitting between orthogonal polarizations of the TEM00 mode, the cryogenic testbeds additionally observe dynamic frequency fluctuations of the two polarization components that are anti-correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Additionally, the magnitude of this effect increases with the intracavity optical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The resulting noise level is far above the coating thermal noise floor (20-40 dB depending on the cavity and the temperature) and the power spectral density acquires a slope steeper than 1/𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Both birefringent modes of the cavity exhibit similar levels of frequency noise for equivalent optical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' However, 6 if the two modes are addressed simultaneously, the anti-correlated frequency fluctuations can be averaged and suppressed [35,36], as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The residual noise after cancellation no longer scales with intracavity optical intensity, though it is still above the expected thermal noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This noise source is coherent between a TEM10 and TEM00 beam, implying a longer spatial correlation length than the spot size of ~1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The source of this “global” noise is not yet understood and further investigations are ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' It also remains to be seen whether these measured noise scalings persist at higher frequency, as these cavities are optimized for the frequency range of 1 Hz and below, lower than the frequency band where coating thermal noise is relevant for Advanced LIGO, roughly 30-300 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Potential non-Brownian CTN observed in ultrastable cryogenic Si reference cavities with AlGaAs coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' (a) Optical path length fluctuations of a 21-cm long Si cavity measured for the two polarization eigenmodes of the TEM00 resonance (blue, orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' (b) Power spectral densities, Sd, of the length fluctuations of the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The plot in this panel includes the birefringent noise (purple), the noise of an individual polarization eigenmode (orange), as well as the average of the two polarizations (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The sum of technical noise and the measured Brownian thermal noise limit for the TEM00 mode (green) is included for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' (c) and (d) Power spectral densities of the length fluctuations of a 6-cm long Si cavity at 16 K (c) and 4 K (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Shown are the frequency stability of the individual polarization eigenmodes (orange), the average of two polarization eigenmodes (red), and the predicted Brownian thermal noise (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' As explained in the main text, while the average polarization can be used to suppress the birefringence fluctuations, the residual noise remains above the expected CTN level for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Figure reproduced from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' a) b) T = 124 K Slow axis T = 124 K 10 -29 Birefringent noise (fast - slow) Optical path length change (fm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='3 Average Single polarization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='2 Fastaxis 10 30 Polarization-average Predicted total noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='1 (zH/zw) 10 -31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='0 10 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='1 10 -33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='2 10 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='3 W 10 -35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='0 10 °3 10~1 100 Time (h) Fourierfrequency(Hz) c) d) T=16K Single polarization T=4K Single polarization 10 31 Polarization-average 10 -31 Polarization-average PredictedBrownianthermal noise PredictedBrownianthermal noise 10 -32 10 32 (zH/zw) (zH/zw) Ps 10 -33 10 -33 s 34 10 -34 10*35 10 35 10 -36 10 -36 10 2 10 100 10 ~2 10° 100 Fourierfrequency(Hz) Fourierfrequency(Hz) 7 It is important to note that the observed birefringence in crystalline coatings is an “extrinsic” effect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' as 100-oriented GaAs is optically symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The current conjecture is that non-uniform strain relaxation, upon cooling from the growth temperature drives the symmetry breaking [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The coatings are grown at elevated temperature (~600 ºC) leaving a compressive residual stress of ~100 MPa upon cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' It is possible to tailor the strain by alloying the multilayer with In or P (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=', GaAsP, InGaP, InGaAs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=') [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Careful measurement of the optomechanical properties of these materials would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The observed birefringence, measured from cavity mode splitting and corresponding to the accumulated difference in phase on reflection between the fast and slow polarizations, 𝛥𝜃=𝜃𝑓−𝜃𝑠, are all within a similar range, roughly 2×10-3 radians, and are temperature and substrate independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Furthermore, thermal cycling does not affect the mode splitting in cryogenic coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Beyond studies of the static and dynamic (fluctuating) components of birefringence, there is a need to investigate potential thermally-induced effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Owing to the anisotropic nature of AlGaAs coatings, radial thermal gradients will induce shear strains in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Assuming the mirror is a flat, half-infinite disk and the coating face is parallel to the [001] crystal plane, if we choose the 𝑥 and 𝑦 coordinate axes to lie in the [010] and [100] planes respectively, then the magnitude of the induced birefringence is largest along lines at +/- 45º to the coordinate axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In addition, the orientation of the principal axes varies with azimuthal angle across the face of the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Assuming a 100 µm diameter perfect absorber and a mirror irradiance at the level of Advanced LIGO, the effect may be similar in magnitude to the static birefringence seen in AlGaAs mirrors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' several point absorbers could combine to impart a significant effect and must be investigated further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Challenges with large-area mirror production are being tackled to extend the application of crystalline mirrors from advanced reference cavities to GW-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Studies on 2” and 3” (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='8-76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='2 mm) diameter test mirrors have yielded: i) mean absorption < 1 ppm, ii) mean total integrated scatter (TIS): < 10 ppm, and iii) coating thickness variation (rms): < 100 ppm [39,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Ongoing efforts involve the production 10 cm and ultimately 20 cm diameter test mirrors, the latter representing the largest continuous crystalline coatings that can be produced, limited by the availability of base substrates for epitaxial growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In terms of the observed optical properties in large coatings, there appears to be a greater number of scattering centers compared with the best IBS coatings, but the background total integrated scatter level away from larger (> 20 µm) scatterers is comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' These scattering centers may be caused by “oval defects” arising from spitting of the gallium source during deposition, generating crystallites that locally disrupt the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' It is not known whether these defects are absorbing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' thus, distinguishing pure scattering centers from local absorbers is an important task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Both are a source of optical loss but have different impact on 8 interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Similarly, a bidirectional reflectance distribution function analysis of larger point defects will be useful for estimating the effect of rare but large scatterers versus frequent but small scatterers on interferometer scatter noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Recent results from Caltech include surface maps and scattering data from the first 10 cm diameter AlGaAs mirror (transferred to a 10-mm thick planar synthetic fused silica substrate), see Panel a of Figure 3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Surface figure measurements were made on the bare substrate before coating and on also the final coated mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' After coating, the surface appears to have gained 4 nm of astigmatism over an 80 mm diameter aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The radius of the coated substrate changed by –784 m, with a corresponding sagitta change of 270 nm (convex) over an 80 mm diameter aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This could be due to non- uniformity of the coating or stress imparted on the 10 mm thick substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Panels b and c of Figure 3 show the results of the scattering measurements performed on this test structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The mean TIS is somewhat higher than for the smaller samples mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This is due to an increased number of relatively strong scatterers indicated by red points (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Excluding these large scattering centers, the mean total integrated scatter of this initial test mirror was approximately 2 ppm higher than equivalent measurements on an Advanced LIGO ETM, with a point scatterer density of 86 cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Additional 10 cm diameter test mirrors are currently in production to ascertain the repeatability in optical performance of these first large mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Optical characterization of a first 10 cm diameter test mirror with substrate-transferred AlGaAs coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' a) Photographs of the mirror backside (left), viewing the bond interface through the 10-mm thick fused silica substrate, and mirror frontside (right) following substrate and etch stop removal, leaving only the GaAs/AlGaAs multilayer on the fused silica substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' b) Histogram of the TIS for apertures of 40 mm diameter (blue) and 80 mm diameter (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The vertical dashed line shows the mean TIS for the 40 mm diameter aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The legend gives the means and standard deviations of both histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' c) Scatter map obtained via an integrating sphere raster-scanned over the mirror surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The thin blue circle shows the 40 mm diameter inner aperture referenced in panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' ppm a) c) 40 30 20 10 2 10 (ww) b) Entries 55557 10 4 Mean Φ80mm 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='19 Y RMS 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='6 Entries 14073 10 103 Mean Φ40 mm 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='40 10 Count RMS 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='4 10 2 20 10 30 HW 102 30 20 10 0 10 20 30 10 103 40 1 TIS(ppm, 1°≤≤750) X (mm) 9 Given the lack of commercially available options, scaling to AlGaAs coating diameters beyond 20 cm will entail the growth of custom GaAs boules for waferization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Sticking with traditional wafer geometries, 30 cm diameter would be an obvious choice as a next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In terms of multilayer epitaxy, production MBE systems have demonstrated sufficiently good optical performance and uniformity [27,40] and would not require customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' A dedicated system will be necessary to produce high-performance prototype and deliverable optics capable of meeting the strict optical specifications of GW-detectors [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' At the bonding stage, commercial vendors have demonstrated the production of silicon-on-insulator wafers up to 45-cm diameter [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' However, such systems are typically limited to a total bonded thickness of a few mm, while GW-relevant optics exhibit high mass (~40 kg) and much greater thickness (~20 cm), entailing modified tooling and unique challenges in production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' If sufficiently large mirrors with the desired performance metrics can be produced, it will then be necessary to explore impacts on the overall system operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' For instance, given the narrow bandgap of the high index GaAs layers, traditional cavity arm locking with frequency-doubled 532 nm light is no longer an option [43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' thus, alternative locking schemes must be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Given the long-wavelength transparency of AlGaAs, a dichroic coating with sufficient reflectance at both 1064 nm as well as an auxiliary wavelength of 2128 nm may be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' The auxiliary 2128 nm beam would be generated from 1064 nm beam, which is phase-locked to the main laser, using a degenerate optical parametric oscillator (DOPO) instead of a conventional frequency doubling [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' If low-noise dichroic AlGaAs coatings can be produced, then a locking system can be realized with minimal risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' In principle such a coating design is possible, and based on recent optical loss measurements, at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 μm, absorption at the 1 ppm level or less is expected for wavelengths near 2 μm [28,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Possible options for installing AlGaAs-coated input and end test masses (ITMs and ETMs) as an upgrade to the existing 4 km Advanced LIGO interferometers will require minimum coating diameters of 21-22 cm to exceed current requirements on coating thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Thus, the use of 20 cm GaAs wafers for both test masses is not possible without radical modifications and custom boules are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' However, mixed mirror sizes could be employed to avoid this, with larger IBS-coated ITMs focusing a smaller spot onto AlGaAs coated ETMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Such a design is limited by the size of the beamsplitter but has the advantage of keeping the power-recycling and signal-recycling cavity designs mostly unchanged (though it would require reshaping the anti-reflective side of the ITM to form a lens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Similar “mixed mirror” solutions leveraging 20 cm diameter AlGaAs will be evaluated for noise, alignment stability, resonances of higher order modes in the arms, sideband resonances in the arms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' These designs have the potential to serve as technology demonstrators for next-generation instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 10 As with cutting-edge ultrastable laser efforts, third-generation GW detectors such as the Einstein Telescope (ET) are proposing to incorporate cryogenics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' To maintain compatibility with available growth substrates, the ET low frequency interferometer could potentially implement similar mixed mirror designs, including cooled ETMs with a 13 cm beam and 70 cm diameter IBS coatings and ITMs having a 4 cm spot size with 20 cm diameter AlGaAs coatings (with or without cryogenic cooling depending on the ultimate outcome of the cryogenic crystalline coating noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' With cryogenic cooling, this geometry could employ ultrapure float-zone silicon substrates for the ITMs, which are currently available up to 20 cm diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Open questions relevant to AlGaAs in future GW detectors may go beyond that which can be answered in table-top experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' These include: (a) an accurate wideband (frequency) measurement of coating noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' (b) successful production of larger than ‘lab-scale’ mirrors, spanning substrate procurement, polishing, and bonding, to integration with relevant suspension systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' and (c) investigations of large-area coating performance when integrated in a complex and sensitive system at high laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Several platforms will be available within the gravitational-wave community in the near-term for such efforts: i) 10 m prototype at the AEI in Hannover, Germany [45], operating at 1064 nm and room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' A key aim is to investigate and overcome the standard quantum limit [25] and the use of low-noise AlGaAs-coated, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='8 cm diameter mirrors have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' ii) Gingin prototype in Western Australia [46] will investigate high-power effects in silicon mirrors at a wavelength near 2 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This is a three-phase project: (1) 7 m Fabry Perot cavity at 5 W laser power and 3 mm beam diameter with interchangeable fused silica and silicon mirrors, (2) 72 m Fabry Perot cavity using silicon mirrors with 10 cm diameter AlGaAs coatings and ~1 cm beam diameter, (3) 23 kW laser power in the arm cavities at 123 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This system can be used to explore the impact of point absorbers, wide angle scattering, thermal distortions and birefringence of coated and uncoated silicon substrates, and possibly electro-optic and non-linear effects at high laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' iii) There are currently two cryogenic protoypes under development, the ET-pathfinder in Maastricht, Netherlands [47] and a cryogenic upgrade to the 10 m prototype in Glasgow, UK [48], with the aim of testing technologies for low temperature operation of future GW detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Parameters such as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='5 μm and 2 μm laser wavelengths at temperatures of ~120 K and ~15-20 K are planned, using Si substrates for the mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' These systems could be ideal platforms to test large-area AlGaAs coatings beyond ongoing efforts with cm-scale reference cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' 11 These platforms will be instrumental in confirming the viability of AlGaAs coatings in these unique astronomical instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' We have outlined the historical background in the initial development of, as well as the current status and potential paths forward for, AlGaAs-based crystalline coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' These unique coatings exhibit promising optomechanical properties for enhanced sensitivity in GW detection and thus demand further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' We end by clarifying that the cost and timeline to realize the large-diameter production capabilities above (custom base wafers, epitaxy, and bonding) is comparable to the development of other important subsystems such as seismic isolation [49] and quantum squeezing [50], which is an appropriate comparison in that coating thermal noise is the limiting noise in the most sensitive frequency band of second-generation GW detectors [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This can also be compared to potential budgetary savings realized by putting off or even eliminating the need to develop cryogenics for future detectors [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' A proposed timeline (available to LIGO, Virgo, and KAGRA members) has been developed that would allow GW-relevant AlGaAs coatings to be realized within a span of 5 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' This will allow for AlGaAs to be considered for upgrades to the Advanced LIGO detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This research was supported by the National Science Foundation through the following grant awards: American University (PHYS-2012017, PHYS-2011787), Caltech (PHY-1764464), HWS (PHY-1912699, PHY-2011688, PHY-2208079), M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' (PHY-1764464), Embry-Riddle Aeronautical University (PHY- 2110598), Syracuse University (PHY-2011723, PHY- 2207640).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Steinlechner acknowledges the support of ETpathfinder (Interreg Vlaanderen-Nederland), E-TEST (Interreg Euregio Meuse-Rhine), the Province of Limburg, and support by the NWO Talent Programme Vidi 2020, project number VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='Vidi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content='062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Kedar and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Ye acknowledge support from NSF PHY-1734006, NIST, and AFRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Legero, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Sterr and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' Yu acknowledge support by the Project 20FUN08 NEXTLASERS, which has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 Research and Innovation Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E0T4oBgHgl3EQf0gKp/content/2301.02687v1.pdf'} +page_content=' A portion of this work was performed in the UCSB Nanofabrication Facility, an open access 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Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom +2)School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, +United Kingdom +(Dated: 10 January 2023) +We demonstrate nonreciprocal critical current in 65 nm thick polycrystalline and epitaxial Nb thin films +patterned into tracks. The nonreciprocal behavior gives a supercurrent diode effect, where the current passed +in one direction is a supercurrent and the other direction is a normal state (resistive) current. We study +the variation of the diode effect with temperature, magnetic field, and the width of the Nb tracks from +2-10 µm. For both polycrystalline and epitaxial samples, we find that tracks of width 4 µm provides the +largest supercurrent diode efficiency of up to ≈ 30%, with the effect reducing or disappearing in the widest +tracks of 10 µm. It is anticipated that the supercurrent diode will become a ubiquitous component of the +superconducting computer. +The supercurrent diode is an analogue to rectification +in semiconductor pn junctions, where current is allowed +to flow only in one direction. In the supercurrent diode, +the critical current of the device (Ic) is nonreciprocal +(I+ +c +̸= I− +c ), leading to the situation where a dissipa- +tionless supercurrent can pass in one direction, but upon +reversing the current direction, the device becomes resis- +tive. +Our work is motivated by recent experimental ob- +servations of supercurrent diode effect in a number of +systems. +Following on from early work1 and reports +in superconductor-ferromagnet hybrid systems2–5, recent +observations of supercurrent diode effect include from the +broken inversion symmetry in noncentrosymmetric ma- +terial systems6–9, and in junction devices10–15. +While +the underlying physics of these systems are interesting, +from a practical point of view, it will be difficult to in- +tegrate the somewhat exotic materials systems into es- +tablished industrial processes which tend to be based on +niobium. +It is therefore notable that the supercurrent +diode effect can be generated in single layer superconduc- +tor devices16–18 as a result of the Meissner effect and the +non-perfect edges in experimentally realized devices16,19. +These experimental observations have led to rapid devel- +opment of theory19–23. +The Meissner effect is attributed to the supercurrent +diode effect in the present work. Due to non-perfect de- +vice fabrication, our Nb tracks are unlikely to have iden- +tical edges. Such imperfections in the edges of the de- +vice can lead to the unequal generation, penetration and +expulsion of vortices on the two edges of the devices, re- +sulting in rectification16,19. +We observe a supercurrent diode effect in thin film Nb +patterned into tracks of varying width. We study two +samples, both of thickness 65 nm, one with polycrys- +talline and the other with epitaxial Nb. In order to char- +acterize the effect, it is helpful to introduce an efficiency +parameter η = I+ +c −I− +c +I+ +c +I− +c . We find a supercurrent diode ef- +a)Electronic mail: g.burnell@leeds.ac.uk +fect, where η depends upon the strength of applied out- +of-plane applied magnetic field, temperature, and track +width. +Nb films are deposited by dc magnetron sputtering +in the Royce Deposition System24. +The magnetrons +are mounted below, and confocal to, the substrate with +source-substrate distances of 134 mm. The base pressure +of the vacuum chamber is 3×10−9 mBar with the sub- +strate at room temperature and 1×10−8 mBar with the +substrate at 1000◦C. Nb is grown at a rate of 0.06 nm/s +at an Ar (6N purity) gas pressure of 3.6×10−3 mBar to +a nominal thickness of 65 nm. The growth rate and film +thicknesses are checked by x-ray reflectivity. The first Nb +sample was deposited at room temperature on Si/SiOx +substrate and the second Nb sample was deposited at ele- +vated temperature of 1000◦C onto a single crystal a-plane +Al2O3 substrate to promote epitaxial growth. +Samples are fabricated into tracks using a direct laser +writer to define resist masks in S1813 photoresist, and +reactive ion etching at 130 W in a 1:2 Ar:SF6 plasma +to etch the Nb films. After fabrication, devices are mea- +sured in a continuous flow 4He cryostat with 3 T horizon- +tal superconducting Helmholtz coils. Traditional 4-point- +probe transport geometry is used to measure the current- +voltage characteristic of the tracks with combined Keith- +ley 6221-2182A current source and nano-voltmeter in +pulse mode with 1 ms pulses and a 1% duty cycle to avoid +a reduced retracking Ic due to heating. The schematic +of the fabricated devices and measurement geometry is +shown in Figure 1 (a). +The sample grown at room temperature has a super- +conducting Tc of 8.75 K and a residual-resistivity ratio +(RRR) of 2.8, giving an estimate for the mean free path +(ℓ) of 6 nm – indicating a polycrystalline microsctructure. +The second sample has a higher Tc of 9.05 K and a RRR +of 30, giving an estimate for ℓ of 96 nm, consistent with an +epitaxial microstructure. The increased ℓ is expected for +the epitaxial Nb due to the decrease in crystallographic +defects such as grain boundaries. The properties of our +polycrystalline Nb thin films are reported elsewhere25. +Figure 1 shows the supercurrent diode effect in the +polycrystalline sample with 4 µm wide track. Figure 1 +arXiv:2301.02706v1 [cond-mat.supr-con] 6 Jan 2023 + +2 +-60 +-40 +-20 +0 +20 +40 +60 +0 +10 +20 +30 +|Ic| (mA) +μ0H (mT) +-5 +0 +5 +20 +25 +-60 +-40 +-20 +0 +20 +40 +60 +-30 +-20 +-10 +0 +10 +20 +30 +η (%) +μ0H (mT) +μ0H +I+ +I- +V+ +V- +w +-0.6 -0.4 -0.2 +0.0 +0.2 +0.4 +0.6 +-30 +-20 +-10 +0 +10 +20 +30 + -3 mT + +3 mT +Current (mA) +Voltage (V) +(a) +(b) +(c) +(d) + Ic ++ + Ic +- + Ic ++ + Ic +- +FIG. 1. +Supercurrent diode effect in polycrystalline 65 nm thick Nb patterned into a 4 µm wide track, measured at 5 K. +(a) Schematic cross section of the Nb track device showing the applied field and measurement current direction (not to scale). +(b) Current-voltage characteristic of the device measured at ±3 mT applied out-of-plane field. (c) Extracted critical currents +(I+ +c and I− +c ) with out-of-plane applied field, insert shows low field region with fitting to the model described in the text. The +uncertainty in determining Ic is the current step size and is smaller than the data points. (d) Diode efficiency η corresponding +to the dataset in (c). +(b) shows the current-voltage (I − V ) characteristic of +our track at applied fields where the diode effect is found +to be maximum. I+ +c and I− +c are extracted from the I −V +when the voltage reaches a small threshold value. Large +nonreciprocal I+ +c and I− +c can be seen in the I − V char- +acteristic. When the field polarity is reversed, the nonre- +ciprocal I+ +c and I− +c are also reversed. In the normal state +of the device, the I − V curve shows a slight non-linear +dependence, which we attribute to Joule heating as a re- +sult of the large current densities. Our measurements at +lower current densities (e.g. in wider tracks or at warmer +temperatures) show I − V curves following the expected +linear metallic behavior. +Figure 1 (c) shows the out-of-plane applied field de- +pendence of I+ +c +and I− +c . +The presented field sweep is +acquired by sweeping from negative to positive field. A +similar curve with the same features is obtained by sweep- +ing from positive to negative field. The sign of the I+ +c /I− +c +maximum with positive/negative field does not change +with temperature or field sweep direction and appears +to favor having the I+ +c (I− +c ) maximum in positive (neg- +ative) field. Across the 11 samples in our study showing +the diode effect, when keeping the device mounting and +wiring geometry the same, I+ +c (I− +c ) maximum appears in +positive (negative) field with a ratio of 8:3. The I+ +c /I− +c +maximum with positive/negative field can be reversed +by swapping the current wiring direction (Figure 1 (a)). +This apparent favoritism may be specific to our fabri- +cation processing, but may indicate that devices can be +fabricated with a deterministic bias. Figure 1 (d) shows +the extracted diode efficiency η, where peak efficiency is +achieved around the fields corresponding to I+ +c /I− +c max- +imum. +Our interpretation of the origin of the supercurrent +diode effect relies upon our samples being in the limit +where the critical current is determined by the vorticies +in the system, as opposed to being the depairing cur- +rent. From the Ic(B) dependence, it is possible to ex- +tract the maximum super-heating field of the Meissner +state16, Bs. +The inset to Figure 1 (c) shows the low +field Ic with linear fits (dashed lines). Bs corresponds +to the interpolated intercept of the fits to the low field +data, when the B offset is taken into account. From the +four linear fits shown in Figure 1 (c) inset, we obtain +Bs = 10 ± 1 mT. The expression16, Bs = φ0/( +√ +3πξw), +provides an order of magnitude estimate for Bs, where +φ0 is the flux quantum and ξ is the Ginzburg–Landau +coherence length (ξ = 11.6 nm for polycrystalline thin + +3 +-60 +-40 +-20 +0 +20 +40 +60 +10 +20 +30 +40 +50 + Ic ++ + Ic +- +|Ic| (mA) +μ0H (mT) +-60 +-40 +-20 +0 +20 +40 +60 +-20 +-10 +0 +10 +20 +η (%) +μ0H (mT) +(a) +(b) +FIG. 2. +Supercurrent diode effect in epitaxial 65 nm thick +Nb patterned into a 4 µm wide track, measured at 5 K. (a) +Extracted critical currents (I+ +c and I− +c ) with out-of-plane ap- +plied field. The uncertainty in determining Ic is the current +step size and is smaller than the data points. (d) Diode effi- +ciency η corresponding to the dataset in (a). +film Nb25). For w = 4 µm, Bs = 8.2 mT, in approx- +imate agreement with our experimental findings. This +indicates strongly that in the region where the diode ef- +fect is observed the critical current is determined by the +vorticies. +Figure 2 shows the supercurrent diode effect in the epi- +taxial 65 nm thick Nb sample patterned into 4 µm wide +track at 5 K. The epitaxial device shares many of the +same features as the polycrystalline device reported in +Figure 1. From the comparison between the two sam- +ples of the same width and measurement temperature, +the tracks patterned from the epitaxial sample has an +Ic about twice that of the polycrystalline track, shown +in Figure 2 (a). This is likely due to the difference in +Tc between the samples. As shown in Figure 2 (b), the +measured maximum η in this condition is smaller than +the polycrystalline sample. +We next study the track width dependence of the su- +percurrent diode effect in the polycrystalline and epitax- +ial Nb samples. For polycrystalline Nb, the London pene- +tration depth λL = 96 nm25, providing an estimate of the +Pearl penetration depth λP = 2λ2 +L/t ≈ 300 nm. For com- +parable epitaxial Nb, λL tends towards bulk26 providing +an estimate λP ≈ 50 nm. Other works have considered +tracks in the limit w/λP = 1/2517 and w/λP = 216, how- +ever here we explore a new limit where the width of the +tracks are far greater than the Pearl penetration depth, +w >> λP , covering between 10 < w/λP < 200. In this +new limit, we observe a significant superconducting diode +effect and report a width dependence in our samples. +Figure 3 shows the full track width and temperature +dependence of the supercurrent diode effect in the epitax- +ial and polycrystalline Nb samples. Considering first the +epitaxial sample, Figure 3 (a), two samples of at w = 7 +and 10 µm did not show finite η. In these two tracks, +Ic at low temperatures exceeded the maximum output of +our current source (100 mA), limiting the range of tem- +peratures we could measure. For the narrower samples +which showed η, the temperature dependence show sim- +ilar trends for all samples, with the largest η at 1.8 K, +and η decreasing with warming. +Figure 3 (b) and (c) presents the track width depen- +dence of η at fixed temperature. At 1.8 K, a clear peak +in η is found for the 4 µm track, with η decreasing lin- +early for narrower or wider tracks. At 5 K, the peak in +η is broader, with w = 4 and 5 µm showing similar η. +Again, for narrower or wider tracks η decreases linearly +with width. Linear fits to the decay of η for tracks of +4, 5, 6 and 7 µm are presented as dashed lines. +In the polycrystalline Nb samples, Figure 3 (d), tracks +in the width regime 3 ≤ w ≤ 5 µm follow a similar tem- +perature trend where η is largest for temperatures of 4 +or 5 K, and decreases from the maximum value as the +temperature is cooled or warmed. Tracks with w ≥ 7 µm +show the largest η at the lowest temperature, with η de- +creasing as the temperature is warmed. Considering the +track width dependence of η at fixed temperature, Figure +3 (e) and (f), η shows the largest value for tracks of w = 3 +or 4 µm. Upon increasing w, η decreases linearly between +7 ≤ w ≤ 10 µm at 1.8 K and between 5 ≤ w ≤ 10 µm at +1.8 K. The 5 µm track at 1.8 K is a outlier to this trend. +Linear fits to the decay of η for tracks of 7, 8 and 10 µm +are presented as dashed lines. Our data suggests that the +supercurrent diode effect depends upon the width of the +track with both an ideal track width and an upper limit +width. The supercurrent diode effect relies upon unequal +generation, penetration and expulsion of vortices due to +non-perfect edges arising during the lithography. When +the tracks are wide, vortex pinning likely plays a role and +the edge effects no longer determine the measured critical +current. From our data we can estimate the limit of track +width to observe the supercurrent diode effect. For the +polycrystalline film, extrapolating the decrease in η from +7-10 µm in Figures 3 (e) and (f), the presented linear fits +suggests that η = 0 will occur for w between about 12 +and 13 µm. In the epitaxial film, η = 0 is observed at 5 K +for w = 7µm. Extrapolating the data at 1.8 K suggests +that η = 0 for w ≈ 8 µm. In this scenario, the difference +between the polycrystalline and epitaxial samples could +be related to the differences in λP and increased disorder +in the polycrystalline sample and the different pinning +potentials of the two samples. + +4 +(d) +(e) +(f) +2 +4 +6 +8 +10 12 +0 +10 +20 +30 +η max (%) +Track Width (μm) +T = 1.8 K +2 +4 +6 +8 +10 12 +Track Width (μm) +T = 5 K +1 +2 +3 +4 +5 +6 +7 +8 +9 +0 +4 +8 +12 +16 +20 +24 +28 +32 +η max (%) +Temperature (K) + 3 μm + 4 μm + 5 μm + 7 μm + 8 μm + 10 μm +1 +2 +3 +4 +5 +6 +7 +8 +9 +0 +5 +10 +15 +20 +25 + 2 μm + 3 μm + 4 μm + 5 μm + 6 μm + 7 μm + 10 μm +η max (%) +Temperature (K) +(a) +(b) +(c) +2 +4 +6 +8 +0 +10 +20 +30 +η max (%) +Track Width (μm) +T = 1.8 K +2 +4 +6 +8 +Track Width (μm) +T = 5 K +Epitaxial Nb +Polycrystalline Nb +FIG. 3. +Supercurrent diode effect in epitaxial and polycrystalline 65 nm thick Nb patterned into tracks (a,d) The maximum +diode efficiency parameter, η, with temperature for a series of tracks with varying width for the epitaxial and polycrystalline +samples respectively. (b,c,e,f) Corresponding track width dependence of η at (b,e) 1.8 K and (c,f) 5 K. Solid lines in (a,d) +represent guides for the eye, while dashed lines in (b,c,e,f) show linear fits to the decay of η for the widest tracks. +In an attempt to further understand the origin of the +diode effect and the role of vorticies, we perform initial- +ization experiments on one of our tracks. Figure 4 shows +the epitaxial 65 nm thick Nb sample with 5 µm wide +track at 6 K. (a-f) shows the initial state of the device +after zero applied field cool and the conditions necessary +to initialize the supercurrent diode effect in the device. +At the zero field cooled condition in Figure 4 (a), the +device already shows a small diode effect of η = 6%, +presumably due to the small trapped flux in the super- +conducting magnet. +Figure 4 (a-f) present the results +of performing sequentially larger field sweeps. Starting +from zero applied field, we observe a cross over in I+ +c /I− +c +at approximately 1 mT, followed by a maximum in I+ +c at +about 4 mT. As in Figures 1 and 2, the maximum diode +effect, η, corresponds to the maximum in I+ +c . At larger +fields the effect reduces, and I+ +c = I− +c at about 20 mT. +In Figure 4, comparing the out and return field sweep +directions, there is a hysteretic behavior in Ic, and hence +η. Considering Figure 4 (e), the hysteresis is particularly +observable in I+ +c between 10 and 20 mT, but is present +over the whole field range where η ̸= 0. The hysteretic +behavior with field history suggests that the diode effect +is sensitive to the field history of the track. We expect +that the preceding field history establishes different vor- +tex states in the track, which therefore influences the +magnitude of η. The largest hysteresis between 10 and +20 mT is reproduced in subsequent larger field sweeps, +Figure 4 (f). +In conclusion, we report on supercurrent diode effect +in 65 nm thick polycrystalline and epitaxial Nb films pat- +terned into tracks. Our largest reported diode efficency is +η ≈ 30% in a 4 µm wide track at 5 K. We report that the +supercurrent diode effect can be observed in Nb tracks +over range of track widths, applied out-of-plane applied +fields and temperatures. Our results imply a track width +dependence, where for the widest tracks in this study, +we report that the diode effect reduces or disappears al- +together. Our results are consistent with the Meissner +effect edge vorticies mechanism determining the nonre- +ciprocal critical currents. +ACKNOWLEDGMENTS +We acknowledge support from the Henry Royce In- +stitute. The work was supported financially through the +following EPSRC grants: EP/V028138/1. We also would +like to thank N.O. Birge for helpful discussions. +AUTHOR DECLARATIONS +Conflict of Interest +The authors have no conflicts to disclose. +Author Contributions +N Satchell: Investigation (lead); methodology (lead); +conceptualization (equal); writing – original draft (lead); + +5 +0.1 +1 +10 +100 +10 +20 +30 +40 +50 +μ0H (mT) +10 +20 +30 +40 +50 +|Ic| (mA) +10 +20 +30 +40 +50 + Ic+ + Ic- +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +10 +20 +30 +40 +50 +(a) +(b) +(c) +(d) +(e) +(f) +FIG. 4. +Initialization of the supercurrent diode effect in +epitaxial 65 nm thick Nb patterned into 5 µm wide track +at 6 K. (a-e) I+ +c /I− +c with sequentially larger field sweeps on +a semi-log scale. Before each sweep the sample was briefly +warmed above Tc and cooled again in zero applied field. The +uncertainty in determining Ic is the current step size and is +smaller than the data points. +visualization (lead); formal analysis (lead); writing – +review and editing (equal); funding acquisition (sup- +porting). +PM Shepley: +Investigation (supporting); +methodology (supporting); writing – review and edit- +ing (equal). +MC Rosamond: +Investigation (sup- +porting); methodology (supporting); writing – review +and editing (equal). +G Burnell: Project administra- +tion (lead); funding acquisition (lead); conceptualization +(equal); methodology (supporting); investigation (sup- +porting); writing – review and editing (equal). +DATA AVAILABILITY +The datasets generated during the current study are +available in the University of Leeds repository, DOI:TBC. +REFERENCES +1P. S. Swartz and H. R. Hart, “Asymmetries of the Critical Surface +Current in Type-II Superconductors,” Phys. Rev. 156, 412–420 +(1967). +2N. Touitou, P. Bernstein, J. F. Hamet, C. Simon, L. M´echin, J. 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Prokscha, +and M. Horisberger, “Observation +of nonexponential magnetic penetration profiles in the meissner +state: A manifestation of nonlocal effects in superconductors,” +Phys. Rev. B 72, 024506 (2005). + diff --git a/StE0T4oBgHgl3EQf2AJK/content/tmp_files/load_file.txt b/StE0T4oBgHgl3EQf2AJK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98f09f4ee30cbbe045de40929068523275946634 --- /dev/null +++ b/StE0T4oBgHgl3EQf2AJK/content/tmp_files/load_file.txt @@ -0,0 +1,487 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf,len=486 +page_content='Supercurrent diode effect in thin film Nb tracks N Satchell,1 PM Shepley,1 MC Rosamond,2 and G Burnell1, a) 1)School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom 2)School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom (Dated: 10 January 2023) We demonstrate nonreciprocal critical current in 65 nm thick polycrystalline and epitaxial Nb thin films patterned into tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The nonreciprocal behavior gives a supercurrent diode effect, where the current passed in one direction is a supercurrent and the other direction is a normal state (resistive) current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' We study the variation of the diode effect with temperature, magnetic field, and the width of the Nb tracks from 2-10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' For both polycrystalline and epitaxial samples, we find that tracks of width 4 µm provides the largest supercurrent diode efficiency of up to ≈ 30%, with the effect reducing or disappearing in the widest tracks of 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' It is anticipated that the supercurrent diode will become a ubiquitous component of the superconducting computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The supercurrent diode is an analogue to rectification in semiconductor pn junctions, where current is allowed to flow only in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In the supercurrent diode, the critical current of the device (Ic) is nonreciprocal (I+ c ̸= I− c ), leading to the situation where a dissipa- tionless supercurrent can pass in one direction, but upon reversing the current direction, the device becomes resis- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Our work is motivated by recent experimental ob- servations of supercurrent diode effect in a number of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Following on from early work1 and reports in superconductor-ferromagnet hybrid systems2–5, recent observations of supercurrent diode effect include from the broken inversion symmetry in noncentrosymmetric ma- terial systems6–9, and in junction devices10–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' While the underlying physics of these systems are interesting, from a practical point of view, it will be difficult to in- tegrate the somewhat exotic materials systems into es- tablished industrial processes which tend to be based on niobium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' It is therefore notable that the supercurrent diode effect can be generated in single layer superconduc- tor devices16–18 as a result of the Meissner effect and the non-perfect edges in experimentally realized devices16,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' These experimental observations have led to rapid devel- opment of theory19–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The Meissner effect is attributed to the supercurrent diode effect in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Due to non-perfect de- vice fabrication, our Nb tracks are unlikely to have iden- tical edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Such imperfections in the edges of the de- vice can lead to the unequal generation, penetration and expulsion of vortices on the two edges of the devices, re- sulting in rectification16,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' We observe a supercurrent diode effect in thin film Nb patterned into tracks of varying width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' We study two samples, both of thickness 65 nm, one with polycrys- talline and the other with epitaxial Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In order to char- acterize the effect, it is helpful to introduce an efficiency parameter η = I+ c −I− c I+ c +I− c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' We find a supercurrent diode ef- a)Electronic mail: g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='burnell@leeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='uk fect, where η depends upon the strength of applied out- of-plane applied magnetic field, temperature, and track width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Nb films are deposited by dc magnetron sputtering in the Royce Deposition System24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The magnetrons are mounted below, and confocal to, the substrate with source-substrate distances of 134 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The base pressure of the vacuum chamber is 3×10−9 mBar with the sub- strate at room temperature and 1×10−8 mBar with the substrate at 1000◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Nb is grown at a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='06 nm/s at an Ar (6N purity) gas pressure of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='6×10−3 mBar to a nominal thickness of 65 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The growth rate and film thicknesses are checked by x-ray reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The first Nb sample was deposited at room temperature on Si/SiOx substrate and the second Nb sample was deposited at ele- vated temperature of 1000◦C onto a single crystal a-plane Al2O3 substrate to promote epitaxial growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Samples are fabricated into tracks using a direct laser writer to define resist masks in S1813 photoresist, and reactive ion etching at 130 W in a 1:2 Ar:SF6 plasma to etch the Nb films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' After fabrication, devices are mea- sured in a continuous flow 4He cryostat with 3 T horizon- tal superconducting Helmholtz coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Traditional 4-point- probe transport geometry is used to measure the current- voltage characteristic of the tracks with combined Keith- ley 6221-2182A current source and nano-voltmeter in pulse mode with 1 ms pulses and a 1% duty cycle to avoid a reduced retracking Ic due to heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The schematic of the fabricated devices and measurement geometry is shown in Figure 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The sample grown at room temperature has a super- conducting Tc of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='75 K and a residual-resistivity ratio (RRR) of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8, giving an estimate for the mean free path (ℓ) of 6 nm – indicating a polycrystalline microsctructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The second sample has a higher Tc of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='05 K and a RRR of 30, giving an estimate for ℓ of 96 nm, consistent with an epitaxial microstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The increased ℓ is expected for the epitaxial Nb due to the decrease in crystallographic defects such as grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The properties of our polycrystalline Nb thin films are reported elsewhere25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 1 shows the supercurrent diode effect in the polycrystalline sample with 4 µm wide track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='02706v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='supr-con] 6 Jan 2023 2 60 40 20 0 20 40 60 0 10 20 30 |Ic| (mA) μ0H (mT) 5 0 5 20 25 60 40 20 0 20 40 60 30 20 10 0 10 20 30 η (%) μ0H (mT) μ0H I+ I- V+ V- w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='6 30 20 10 0 10 20 30 3 mT +3 mT Current (mA) Voltage (V) (a) (b) (c) (d) Ic + Ic Ic + Ic FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Supercurrent diode effect in polycrystalline 65 nm thick Nb patterned into a 4 µm wide track, measured at 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (a) Schematic cross section of the Nb track device showing the applied field and measurement current direction (not to scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (b) Current-voltage characteristic of the device measured at ±3 mT applied out-of-plane field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (c) Extracted critical currents (I+ c and I− c ) with out-of-plane applied field, insert shows low field region with fitting to the model described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The uncertainty in determining Ic is the current step size and is smaller than the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (d) Diode efficiency η corresponding to the dataset in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (b) shows the current-voltage (I − V ) characteristic of our track at applied fields where the diode effect is found to be maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' I+ c and I− c are extracted from the I −V when the voltage reaches a small threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Large nonreciprocal I+ c and I− c can be seen in the I − V char- acteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' When the field polarity is reversed, the nonre- ciprocal I+ c and I− c are also reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In the normal state of the device, the I − V curve shows a slight non-linear dependence, which we attribute to Joule heating as a re- sult of the large current densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Our measurements at lower current densities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' in wider tracks or at warmer temperatures) show I − V curves following the expected linear metallic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 1 (c) shows the out-of-plane applied field de- pendence of I+ c and I− c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The presented field sweep is acquired by sweeping from negative to positive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' A similar curve with the same features is obtained by sweep- ing from positive to negative field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The sign of the I+ c /I− c maximum with positive/negative field does not change with temperature or field sweep direction and appears to favor having the I+ c (I− c ) maximum in positive (neg- ative) field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Across the 11 samples in our study showing the diode effect, when keeping the device mounting and wiring geometry the same, I+ c (I− c ) maximum appears in positive (negative) field with a ratio of 8:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The I+ c /I− c maximum with positive/negative field can be reversed by swapping the current wiring direction (Figure 1 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' This apparent favoritism may be specific to our fabri- cation processing, but may indicate that devices can be fabricated with a deterministic bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 1 (d) shows the extracted diode efficiency η, where peak efficiency is achieved around the fields corresponding to I+ c /I− c max- imum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Our interpretation of the origin of the supercurrent diode effect relies upon our samples being in the limit where the critical current is determined by the vorticies in the system, as opposed to being the depairing cur- rent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' From the Ic(B) dependence, it is possible to ex- tract the maximum super-heating field of the Meissner state16, Bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The inset to Figure 1 (c) shows the low field Ic with linear fits (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Bs corresponds to the interpolated intercept of the fits to the low field data, when the B offset is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' From the four linear fits shown in Figure 1 (c) inset, we obtain Bs = 10 ± 1 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The expression16, Bs = φ0/( √ 3πξw), provides an order of magnitude estimate for Bs, where φ0 is the flux quantum and ξ is the Ginzburg–Landau coherence length (ξ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='6 nm for polycrystalline thin 3 60 40 20 0 20 40 60 10 20 30 40 50 Ic + Ic |Ic| (mA) μ0H (mT) 60 40 20 0 20 40 60 20 10 0 10 20 η (%) μ0H (mT) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Supercurrent diode effect in epitaxial 65 nm thick Nb patterned into a 4 µm wide track, measured at 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (a) Extracted critical currents (I+ c and I− c ) with out-of-plane ap- plied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The uncertainty in determining Ic is the current step size and is smaller than the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (d) Diode effi- ciency η corresponding to the dataset in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' film Nb25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' For w = 4 µm, Bs = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='2 mT, in approx- imate agreement with our experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' This indicates strongly that in the region where the diode ef- fect is observed the critical current is determined by the vorticies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 2 shows the supercurrent diode effect in the epi- taxial 65 nm thick Nb sample patterned into 4 µm wide track at 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The epitaxial device shares many of the same features as the polycrystalline device reported in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' From the comparison between the two sam- ples of the same width and measurement temperature, the tracks patterned from the epitaxial sample has an Ic about twice that of the polycrystalline track, shown in Figure 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' This is likely due to the difference in Tc between the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' As shown in Figure 2 (b), the measured maximum η in this condition is smaller than the polycrystalline sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' We next study the track width dependence of the su- percurrent diode effect in the polycrystalline and epitax- ial Nb samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' For polycrystalline Nb, the London pene- tration depth λL = 96 nm25, providing an estimate of the Pearl penetration depth λP = 2λ2 L/t ≈ 300 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' For com- parable epitaxial Nb, λL tends towards bulk26 providing an estimate λP ≈ 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Other works have considered tracks in the limit w/λP = 1/2517 and w/λP = 216, how- ever here we explore a new limit where the width of the tracks are far greater than the Pearl penetration depth, w >> λP , covering between 10 < w/λP < 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In this new limit, we observe a significant superconducting diode effect and report a width dependence in our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 3 shows the full track width and temperature dependence of the supercurrent diode effect in the epitax- ial and polycrystalline Nb samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Considering first the epitaxial sample, Figure 3 (a), two samples of at w = 7 and 10 µm did not show finite η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In these two tracks, Ic at low temperatures exceeded the maximum output of our current source (100 mA), limiting the range of tem- peratures we could measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' For the narrower samples which showed η, the temperature dependence show sim- ilar trends for all samples, with the largest η at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K, and η decreasing with warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 3 (b) and (c) presents the track width depen- dence of η at fixed temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' At 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K, a clear peak in η is found for the 4 µm track, with η decreasing lin- early for narrower or wider tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' At 5 K, the peak in η is broader, with w = 4 and 5 µm showing similar η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Again, for narrower or wider tracks η decreases linearly with width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Linear fits to the decay of η for tracks of 4, 5, 6 and 7 µm are presented as dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In the polycrystalline Nb samples, Figure 3 (d), tracks in the width regime 3 ≤ w ≤ 5 µm follow a similar tem- perature trend where η is largest for temperatures of 4 or 5 K, and decreases from the maximum value as the temperature is cooled or warmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Tracks with w ≥ 7 µm show the largest η at the lowest temperature, with η de- creasing as the temperature is warmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Considering the track width dependence of η at fixed temperature, Figure 3 (e) and (f), η shows the largest value for tracks of w = 3 or 4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Upon increasing w, η decreases linearly between 7 ≤ w ≤ 10 µm at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K and between 5 ≤ w ≤ 10 µm at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The 5 µm track at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K is a outlier to this trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Linear fits to the decay of η for tracks of 7, 8 and 10 µm are presented as dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Our data suggests that the supercurrent diode effect depends upon the width of the track with both an ideal track width and an upper limit width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The supercurrent diode effect relies upon unequal generation, penetration and expulsion of vortices due to non-perfect edges arising during the lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' When the tracks are wide, vortex pinning likely plays a role and the edge effects no longer determine the measured critical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' From our data we can estimate the limit of track width to observe the supercurrent diode effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' For the polycrystalline film, extrapolating the decrease in η from 7-10 µm in Figures 3 (e) and (f), the presented linear fits suggests that η = 0 will occur for w between about 12 and 13 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In the epitaxial film, η = 0 is observed at 5 K for w = 7µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Extrapolating the data at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K suggests that η = 0 for w ≈ 8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In this scenario, the difference between the polycrystalline and epitaxial samples could be related to the differences in λP and increased disorder in the polycrystalline sample and the different pinning potentials of the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' 4 (d) (e) (f) 2 4 6 8 10 12 0 10 20 30 η max (%) Track Width (μm) T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K 2 4 6 8 10 12 Track Width (μm) T = 5 K 1 2 3 4 5 6 7 8 9 0 4 8 12 16 20 24 28 32 η max (%) Temperature (K) 3 μm 4 μm 5 μm 7 μm 8 μm 10 μm 1 2 3 4 5 6 7 8 9 0 5 10 15 20 25 2 μm 3 μm 4 μm 5 μm 6 μm 7 μm 10 μm η max (%) Temperature (K) (a) (b) (c) 2 4 6 8 0 10 20 30 η max (%) Track Width (μm) T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K 2 4 6 8 Track Width (μm) T = 5 K Epitaxial Nb Polycrystalline Nb FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Supercurrent diode effect in epitaxial and polycrystalline 65 nm thick Nb patterned into tracks (a,d) The maximum diode efficiency parameter, η, with temperature for a series of tracks with varying width for the epitaxial and polycrystalline samples respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (b,c,e,f) Corresponding track width dependence of η at (b,e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='8 K and (c,f) 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Solid lines in (a,d) represent guides for the eye, while dashed lines in (b,c,e,f) show linear fits to the decay of η for the widest tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In an attempt to further understand the origin of the diode effect and the role of vorticies, we perform initial- ization experiments on one of our tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 4 shows the epitaxial 65 nm thick Nb sample with 5 µm wide track at 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (a-f) shows the initial state of the device after zero applied field cool and the conditions necessary to initialize the supercurrent diode effect in the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' At the zero field cooled condition in Figure 4 (a), the device already shows a small diode effect of η = 6%, presumably due to the small trapped flux in the super- conducting magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Figure 4 (a-f) present the results of performing sequentially larger field sweeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Starting from zero applied field, we observe a cross over in I+ c /I− c at approximately 1 mT, followed by a maximum in I+ c at about 4 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' As in Figures 1 and 2, the maximum diode effect, η, corresponds to the maximum in I+ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' At larger fields the effect reduces, and I+ c = I− c at about 20 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In Figure 4, comparing the out and return field sweep directions, there is a hysteretic behavior in Ic, and hence η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Considering Figure 4 (e), the hysteresis is particularly observable in I+ c between 10 and 20 mT, but is present over the whole field range where η ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The hysteretic behavior with field history suggests that the diode effect is sensitive to the field history of the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' We expect that the preceding field history establishes different vor- tex states in the track, which therefore influences the magnitude of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The largest hysteresis between 10 and 20 mT is reproduced in subsequent larger field sweeps, Figure 4 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' In conclusion, we report on supercurrent diode effect in 65 nm thick polycrystalline and epitaxial Nb films pat- terned into tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Our largest reported diode efficency is η ≈ 30% in a 4 µm wide track at 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' We report that the supercurrent diode effect can be observed in Nb tracks over range of track widths, applied out-of-plane applied fields and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Our results imply a track width dependence, where for the widest tracks in this study, we report that the diode effect reduces or disappears al- together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Our results are consistent with the Meissner effect edge vorticies mechanism determining the nonre- ciprocal critical currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge support from the Henry Royce In- stitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The work was supported financially through the following EPSRC grants: EP/V028138/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' We also would like to thank N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Birge for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Author Contributions N Satchell: Investigation (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' methodology (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' conceptualization (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' writing – original draft (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content='1 1 10 100 10 20 30 40 50 μ0H (mT) 10 20 30 40 50 |Ic| (mA) 10 20 30 40 50 Ic+ Ic- 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 (a) (b) (c) (d) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Initialization of the supercurrent diode effect in epitaxial 65 nm thick Nb patterned into 5 µm wide track at 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' (a-e) I+ c /I− c with sequentially larger field sweeps on a semi-log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' Before each sweep the sample was briefly warmed above Tc and cooled again in zero applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' The uncertainty in determining Ic is the current step size and is smaller than the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' visualization (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' formal analysis (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' writing – review and editing (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' funding acquisition (sup- porting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' PM Shepley: Investigation (supporting);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' methodology (supporting);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' writing – review and edit- ing (equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' MC Rosamond: Investigation (sup- porting);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' methodology (supporting);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' writing – review and editing (equal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' G Burnell: Project administra- tion (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' funding acquisition (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' conceptualization (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' methodology (supporting);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE0T4oBgHgl3EQf2AJK/content/2301.02706v1.pdf'} +page_content=' investigation (sup- porting);' 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+increased conflict. Few studies have investigated how different +types of diversity impact Agile software teams. This study views +diversity through the lens of the categorization-elaboration model +(CEM). We investigated how diversity in gender, age, role, and +cultural background impacts team effectiveness and conflict, and +how these associations are moderated by psychological safety. +Our sample consisted of 1,118 participants from 161 teams +and was analyzed with Covariance-Based Structural Equation +Modeling (CB-SEM). We found a positive effect of age diversity +on team effectiveness and gender diversity on relational conflict. +Psychological safety contributed directly to effective teamwork +and less conflict but did not moderate the diversity-effectiveness +link. While our results are consistent with the CEM theory for +age and gender diversity, other types of diversity did not yield +similar results. We discuss several reasons for this, including +curvilinear effects, moderators such as task interdependence, or +the presence of a diversity mindset. With this paper, we argue +that a dichotomous nature of diversity is oversimplified. Indeed, +it is a complex relationship where context plays a pivotal role. +A deeper understanding of diversity through the lens of theories +such as the CEM may lead to more effective teamwork. +Index Terms—software teams, agile, diversity, psychological +safety, conflict +I. INTRODUCTION +Teams are increasingly crucial to organizations. This is +particularly relevant to organizations that use Agile software +methodologies. Agile represents a collaborative, iteration-based, +and human-oriented approach to product development [1]. It +originated in response to the perceived shortfalls of plan- +based approaches in the face of complex problems typical in +product development [2], [3]. Thus, “at its core, agile project +management is about managing the impact of complexity and +uncertainty on a project” [4, p. 281]. As a crucial aspect of +project management, scholars have attempted to identify the +factors and characteristics that influence the performance and +productivity of teams. One factor that has gained increased +attention in recent decades is team diversity [5], also in software +engineering specifically [6]. Team diversity is generally defined +as heterogeneity in member attributes, such as age, gender, +cultural background, tenure, role, or personality traits [7]. +While teams can be diverse on many attributes, most studies +focus on demographic diversity (e.g., age, gender, cultural +C. Verwijs is with The Liberators, The Netherlands. +D. Russo is with the Department of Computer Science, Aalborg University, +Denmark. Corresponding author. Email: daniel.russo@cs.aau.dk +Manuscript received Month 01, 2023; revised .... +background) or informational diversity (e.g., professional role, +education, experience). Many researchers have theorized that +diversity improves team performance [8], [9], [6]. However, +studies have provided mixed support. Investigations of how +diversity impacts teams [10], [11], [5], [12], [9], [6] generally +show that the effects are not clear-cut, vary by type of diversity, +and appear to be moderated by characteristics of the task, +the team, and its environment. However, diversity may also +negatively impact effectiveness through an increased conflict +between members [5], [10]. Several competing mechanisms +and integrated models have been proposed to explain these +conflicting results [13], [14], which are discussed in Section II. +Specifically +for +software +engineering +and +Agile +methodologies, Silveira & Prikladnicki [6] and Rodr´ıguez- +P´erez, Nadri & Nagappan [15] concluded from literature +reviews that our understanding of diversity in such teams +still needs to be improved. They found that most studies +have only investigated gender diversity [6] and argue that +a broader exploration of how diversity impacts software +engineering teams can be used to create better teams and +better results. Furthermore, studies have yet to explore how +diversity affects Agile methodologies and their effectiveness. +A more comprehensive examination is vital to understand +how to design more effective teams and achieve better results. +Henceforth, our research question (RQ) is: +RQ: How does diversity in Agile software teams impact +their effectiveness? +To answer our research question, we performed a quantitative +cross-sectional study with 1,118 team members representing +161 Agile software teams. Covariance Based Structural Equa- +tion Modeling (CB-SEM or SEM in short) was used to test how +four types of diversity (gender, age, cultural background, and +role) and one social moderator (psychological safety) interact +to impact team effectiveness and conflict in teams. Only age +diversity was positively associated with team effectiveness. +Concerning relational conflict, only gender diversity showed a +significant positive association. A replication package is also +openly available on Zenodo to support secondary studies. +The rest of the paper is structured as follows. In Section II, +we review the related works of team diversity and how it +impacts team outcomes. Subsequently, we clarify the research +gap this study intends to address and develop relevant hy- +potheses in Section II-D. Section III clarifies how we use +quantitative methods and a survey study to test our hypotheses. +arXiv:2301.12954v1 [cs.SE] 30 Jan 2023 + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +2 +The study results are reported in Section III-C, followed by a +comprehensive discussion of the results and their implications +in Section V. Finally, we conclude our paper outlining future +research opportunities in Section VI. +II. RELATED WORK +Scholars from several disciplines have shown mixed results +regarding how diversity impacts teams’ performance. Tshet- +shema & Chan conclude from a review of 35 studies that “a neg- +ative relationship between [demographic diversity] and team +performance is inferred as the most reported result.” [10, p. 9]. +However, they note that investigations of individual dimensions +of diversity often show a positive effect on team performance, +particularly gender and age. The complex relationship between +diversity and performance is also recognized by Patr´ıcio & +Franco [5]. They argue from a review of 80 studies that diversity +has a dual impact on performance. One is positive through +expanding perspectives, and the other is negative through +increased conflict. Bowers, Pharmer & Salas [11] performed +a meta-analysis of 13 empirical studies and found the effects +of team diversity on team performance to be dependent on +task complexity and difficulty instead. Their results suggest +that teams that perform tasks of low complexity may benefit +more from homogeneity, whereas teams that perform complex +tasks benefit from higher diversity. Another meta-analysis of +30 empirical studies by Horwitz & Horwitz [12] found no +significant effect of demographic diversity (age, gender, or +cultural background) on team performance but did find a +significant moderate effect of role diversity. +We now turn to investigations of individual dimensions of +diversity in teams commonly studied by scholars. +A. Diversity dimensions +For age diversity, Tshetshema & Chan [10] found a positive +effect on team performance in a review of empirical studies. +However, a meta-analysis of 74 empirical studies by Schneid +et al. [16] did not show a significant relationship, although +modest differences occurred as a result of moderators like task +complexity and team. Pesch, Bouncken & Kraus [17] attribute +the positive effect of age diversity primarily to differences +in tenure and work experience rather than age itself. They +also note that this diversity is likely to increase tension and +conflict in teams as members have to reconcile more diverse +perspectives on completing tasks. +Cultural diversity is defined as heterogeneity in shared be- +liefs, norms and values [18]. It is often operationalized through +surface-level ethnic or national diversity [10]. Tshetshema & +Chan [10] inventoried studies that investigated the link between +cultural diversity and team performance and inferred that a +positive relationship is the most reported result. However, the +relationship appears curvilinear: moderate cultural diversity is +beneficial, but too little or too much adversely affects team +performance [18]. +Scholars define gender diversity as heterogeneity in the +gender of team members. Most studies suggest a positive +relationship with team performance [10], [8]. Nevertheless, +too much diversity may lead to increased conflict, particularly +for complex tasks and high interdependence. Thus, Haas & +Hartmut [19] argue that gender diversity should be avoided in +such environments. +Role diversity is another dimension of diversity in teams +that is frequently studied. It represents the heterogeneity in the +functional disciplines and roles members bring to a team [20]. +Agile software methodologies emphasize the need for role +diversity in teams in order to solve complex problems [2], +[1]. Empirical studies have shown mixed results, with some +demonstrating positive effects and others negative [14]. Pelled, +Eisenhardt & Xin [20] found that role diversity increases +conflict due to the integration of diverse perspectives, which +positively influences task performance. Homberg & Bui [21] +found no significant effect of role diversity on the performance +of management teams in a meta-analysis of 53 empirical studies. +Instead, they attribute the mixed findings to publication bias. +Horwitz & Horwitz [12] did find a modest effect on the quality +of the work produced by teams in another meta-analysis, though +not on performance. +The empirical link between diversity and team performance +appears to be complicated. Several moderators have been found +to strengthen the positive impact or dampen the negative impact, +such as an inclusive team climate [22], task complexity and +difficulty [11], [12], psychological safety [23], management +support [24], or time [25]. +B. Diversity in Agile software teams +The importance of team diversity has also been recognized +for Agile software teams specifically [15], [2]. The assumption +is that diversity allows for a richer exploration of shared +problems due to the availability of more perspectives [9], +[6]. This is particularly relevant to the complex problem- +solving in Agile software teams, which require creativity and +the application of diverse skill sets [3]. Several studies have +investigated whether this assumption holds up in practice. Lee +& Xia [26] used a mixed-methods approach to investigate +how role diversity and team autonomy influence the ability of +Agile software teams to deliver on budget, on time, and on the +scope. They found a significant positive effect of diversity in +a survey study of 399 Agile software projects and follow-up +case studies. However, they found that role diversity improves +the quality of solutions emerging from problem-solving in +teams, but not speed. They also found evidence for the dual +impact of diversity, where diversity also increases conflict. +Melo et. al. [27] performed a multiple-case study of Agile +software teams in three large Brazilian software companies. +Their results suggest that teams are more productive when there +is diversity in the experience that members bring to the team. +Another study by Russo & Stol [8] surveyed 483 software +engineers to investigate how personality and gender influence +the productivity of software teams. Their results show that +men and women typically bring different positive and negative +traits to teams, and they argue that this explains some part +of why mixed-gender teams perform better. Rodr´ıguez-P´erez, +Nadri & Nagappan [15] conclude from a literature review that +gender differences between developers contribute significantly +to how they solve problems, debug issues, and work with + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +3 +others. The authors also note that gender diversity is most +frequently studied, but much less is known about how other +types of informational and demographic diversity affect Agile +software teams. A similar conclusion is reached by Silveira +& Prikladnicki [6] in a review of the literature on diversity +in Agile software teams. Thus, both groups of authors call +for more research to guide decision-making on how to design +better teams and generate better results. +C. Theories and moderators of the diversity-performance link +Two mechanisms have been proposed by which diversity +influences team performance [13]. The similarity-attraction +paradigm [7] derives from social psychology and social cate- +gorization to argue that similarity between members increases +mutual attraction, integration, and communication, which in +turn improves performance. Diversity of members, on the other +hand, results in more conflict and misunderstandings as people +categorize themselves into different subgroups. +Alternatively, cognitive resource diversity theory derives from +cognitive psychology. It treats teams as information processors +where individuals process information and then elaborate and +integrate it as a team [28]. In this conceptualization, diversity +allows teams to bring varied cognitive resources to bear when +information is processed individually and elaborated as a team, +which allows a richer exploration of shared challenges. +Thus, both mechanisms offer conflicting predictions about +how diversity will impact team performance. The former +expects relational conflict to increase and performance to +decrease, whereas the latter expects performance to increase. +However, the evidence mentioned above does not consistently +support one or the other. So the focus of academic inquiry has +shifted toward identifying potential moderators that allow both +mechanisms to be integrated [14], [29], [12], [13]. +One potential group of moderators concerns task character- +istics, like complexity and interdependence [13]. In this view, +homogeneity benefits low-complexity tasks with few inter- +dependencies, whereas heterogeneity benefits more complex +tasks with many inter-dependencies. This is primarily consistent +with findings from meta-analyses of the diversity-performance +relationship [11], [16]. However, other studies have found both +positive and negative effects of task interdependence on the +relationship between diversity and performance [14]. +Another potential moderator is psychological safety. Ed- +mondson [23, p. 9] defines it as “a shared belief held by +team members that the team is safe for interpersonal risk- +taking”. Several studies have already shown that psychological +safety contributes to more effective teamwork in software +teams [30], [31], [32], [33]. However, psychological safety is +also likely to moderate the relationship between diversity and +team performance. Diegmann & Rosenkranz [34] theorize that +psychological safety makes teams more resilient against the +disruptive effect of high diversity, such as increased conflict, +by providing a safe environment for members to elaborate +task information. Similarly, Roberge & Van Dick [35] expect +that psychological safety also interacts with the salience of a +collective identity. Diversity only contributes to higher team +performance when members feel safe and identify strongly +with their team. +To date, few studies have empirically investigated the +role of psychological safety as a moderator of the diversity- +performance association. Singh, Winkel & Selvarajan [36] +found that employee performance was higher among members +of diverse teams that also exhibited high psychological safety. +However, this study was contained to one organization and +only considered racial diversity. Furthermore, Kirkman et al. +[37] found that Communities of Practice (CoP) performed +better when diversity was paired with high psychological safety. +Virtual teams also experience fewer drawbacks from diversity +when they can elaborate information in psychologically safe +environments [38]. +Van Knippenberg, De Dreu & Homan [14] have proposed +the categorization-elaboration model (CEM) to integrate the +double-edged nature of diversity in teams and potential mod- +erators. The CEM is the most comprehensive model of work +group diversity and its moderators at the time of writing and +has received broad empirical support [39], [40], [41], [29], [42]. +It distinguishes between moderators related to the task, like +difficulty, complexity, and efficacy and moderators related to the +team and the social processes in it, like trust and commitment. +Both groups of moderators influence the ability of teams to +leverage the informational advantage offered through diversity, +though in different ways. In the case of task moderators, +complex and challenging tasks are more likely to elicit extensive +information processing in members [12], [43], which is +consistent with cognitive resource diversity theory. There is also +support for the moderating influence of task motivation through +process accountability [44]. Another potential task moderator +is task interdependence, which is generally defined as the +degree to which the completion of tasks requires collaboration +by team members [45]. Teams with low interdependence see +less interaction and thus experience fewer opportunities to +leverage the benefit of diverse cognitive resources. However, +empirical studies have found positive and negative effects of +task interdependence on the relationship between demographic +and role diversity and team performance [29]. This suggests +that the effect is either not linear or subject to other moderators. +At the same time, the CEM also proposes a mechanism +by which diversity can harm teamwork. As members grow +less similar and bring different perspectives to teamwork, this +diminishes performance when the social context of a team +encourages social categorization into subgroups and elicits +negative inter-group biases and identity threat [29], [7]. This +loss of social integration creates more potential for relational +conflict and negatively impacts the ability of teams to elaborate +information effectively and reduces their performance. However, +social moderators like trust and psychological safety allow +team members to integrate more effectively to bring diverse +perspectives and information-processing together and elaborate +on them, which is consistent with the similarity-attraction +paradigm. +A strength of this integrated approach is that it may explain +the conflicting results found in the literature. The different +mechanisms behind both groups of moderators independently +strengthen or diminish the ability of teams to leverage diversity +and can work in concert or in opposition. Thus, the CEM +broadens the discourse around team diversity from a one- + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +4 +dimensional approach where it is either a risk or an asset to +one where it can be both simultaneously. Finally, the CEM +has clear, practical implications for diversity management that +aim to reduce in-group bias, strengthen social moderators, and +match diversity with the nature of the task [14]. +D. Research Gap & Hypotheses +This study aims to address two related research gaps. +The first is that we want to answer the call by Silveira & +Prikladnicki [6] and Rodr´ıguez-P´erez, Nadri & Nagappan [15] +for more investigations into how diversity affects Agile software +teams, and not limited to only gender diversity. A more +comprehensive examination is vital to understand how to design +more effective teams and achieve better results. The second +research gap is that we want to investigate diversity in Agile +software teams through the lens of the CEM theory and its +opposing mechanisms. +To answer our research question, we will now develop seven +hypotheses we aim to test in this study. Our first hypothesis +is that diversity contributes to the effectiveness of Agile +software teams. Because such teams collaborate on complex +and interdependent tasks [3], [4], [2], they should benefit from +the expanded cognitive resources allowed by heterogeneity in +gender, age, cultural background, and role. This reflects one +mechanism by which diversity influences team effectiveness +and is in accordance with both cognitive resource diversity +theory and the CEM that integrates it. +Hypothesis 1 (H1). Agile software teams are more effective +when they are more diverse in gender (H1a), age (H1b), +cultural background (H1c), and role diversity (H1d). +Our second hypothesis concerns the second and opposing +mechanism of diversity. That is, we expect that increased +diversity also results in more relational conflict in teams. +This hypothesis reflects a core consequence of the similarity- +attraction paradigm and the CEM that integrates it. +Hypothesis 2 (H2). Agile software teams experience more +relational conflict when they are more diverse in gender (H2a), +age (H2b), cultural background (H2c), and role diversity (H2d). +Furthermore, we hypothesize that the increased relational +conflict, in turn, negatively impacts the effectiveness of +teams. This is consistent with the outcome expected by the +similarity-attraction paradigm and the CEM that integrates it. +Hypothesis +3 +(H3). +Relational +conflict +reduces +the +effectiveness of Agile software teams. +Following existing literature [30], [31], [23], [46], we +expect that psychological safety is a critical factor in enabling +team effectiveness through four different mechanisms. The +first involves a direct effect where psychological safety makes +teams more effective by creating more opportunities to openly +elaborate information, reconcile conflicting viewpoints, and +find creative solutions [30], [31]. +Hypothesis 4 (H4). Psychological safety increases the +effectiveness of Agile software teams. +In the second process, psychological safety decreases +relational conflict in teams by providing more opportunities to +air grievances and discuss the tension between members. +Hypothesis 5 (H5). Psychological safety reduces the amount +of relational conflict in Agile software teams. +Concerning diversity, we expect that psychological safety is +a social moderator of the association between diversity and +team effectiveness. Consistent with the CEM and Diegmann +& Rosenkranz [34], we anticipate that psychological safety is +a social moderator that creates an environment where diverse +teams can more effectively elaborate task-related information +and and experience less relational conflict than less diverse +teams. +Hypothesis 6 (H6). The relationship between diversity in +gender (H6a), age (H6b), cultural background (H6c), and role +(H6d) on the one hand and team effectiveness on the other is +moderated by psychological safety. +Hypothesis 7 (H7). The relationship between diversity in +gender (H7a), age (H7b), cultural background (H7c) and role +(H7d) on the one hand and relational conflict on the other is +moderated by psychological safety. +Our hypotheses are visualized in Figure 1. +III. RESEARCH DESIGN +We conducted a sample study with a sample of Agile +software teams to answer our research question. We used +Covariance-Based Structural Equation Modeling (CB-SEM) +to test our hypotheses. This section discusses the sample +(Sec. III-A), measurement instruments (Sec. III-B), and method +of analysis (Sec. III-C). +A. Participants +We performed our data collection process through a cus- +tomized online survey1 between September 2021 and January +2022. In total, 1.827 members from 733 distinct Agile software +teams completed the survey in that period. Because the survey +is public and accessible to anyone, we cannot properly calculate +a response rate. Scholars have emphasized that public surveys +are more susceptible to careless response [47]. So we applied +several strategies outlined in literature [47] to reduce potential +biases. First, we emphasized the anonymous nature of our +data collection. Second, we encouraged honest answers by +providing teams with a detailed team-level profile and relevant +feedback for their team upon completion. Third, we removed +118 participants with a completion time below the 5% percentile +(6.87 minutes) or answered very few questions (< 20). Finally, +1The GDPR-compliant survey has been designed so that teams can self- +assess their Agile development process. It is available at the following URL: +www.scrumteamsurvey.org. + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +5 +Fig. 1. Theoretical model and hypotheses. Sub-hypotheses are grouped, and control variables are omitted to retain visual clarity +we retained only those teams (161) with at least 4 participating +members to ensure a meaningful diversity measurement. The +composition of our sample is shown in table I. +Several variables in our model were measured at the individ- +ual level and aggregated to a higher (team) level in our analyses. +Such aggregation is only reasonable when sufficient variance +exists at the group level, not just between individuals. So we +calculated the Intraclass Correlation (ICC) [48] to determine +the proportion of variance at the higher level compared to the +total variance. The ICC ranged between 35% and 45% for our +independent variables, which exceeded the required threshold +of 10% suggested by Hair et al. Since no data was missing, +we did not deploy strategies to deal with missing data. +Finally, we performed a posthoc power analysis using +G*Power [49], version 3.1.9. We determined that the sample +size allows us to correctly capture medium effects (f = .15) +with a statistical power of 96% (1 − β = .96). In other words, +the probability of correctly rejecting the null hypothesis is 96% +given our sample. So we are confident that our sample is big +enough to provide a reliable outcome. +B. Measurements +Age, gender, role, and cultural diversity: To assess the +impact of diversity on team effectiveness, we identified three +dimensions of demographic diversity that are commonly studied +(age, cultural background, and gender) and one informational +dimension (role). The questions and the available categories are +shown in Table 2 in the Appendix. Participants were asked to +pick the most appropriate category for age, cultural background, +and functional role. For cultural background, participants were +asked to pick the region where they had lived the longest +instead of their country or race. We assume that the region +where one has lived the longest most substantially shapes the +mental models and perspectives one brings to a team. Gender +diversity was operationalized differently due to concerns that +asking participants to identify their gender would be in violation +of the European General Data Protection Regulation (GDPR). +Instead, the initiating participant of each team was asked to +indicate the gender distribution of men and women at the team +level. We recognize there are more genders. However, we had +to take this shortcut to obtain a reliable statistical analysis. +A team-level indicator for diversity was then created by +calculating a Gini-Simpson coefficient with the individual- +level responses for age, role, and cultural diversity. A Gini- +Simpson coefficient is a statistical indicator of the diversity +of the members in a sample, ranging between 0 (no diversity) +and 1 (maximum diversity) [50]. +Team Effectiveness was operationalized similarly to Verwijs +& Russo [30]. Team effectiveness is often defined as “the degree +to which a team meets the expectations of the quality of the +outcome” [51]. In this sense, stakeholder satisfaction is the +evaluation of team outcomes from the external perspective +of stakeholders (e.g., clients, customers, and users), whereas +team morale is the evaluation of team outcomes from the +internal perspective of team members. This is conceptually +similar to how team effectiveness is defined in the “Team +Diagnostic Survey (TDS)” [52]. For team morale, we used 3 +items from the “Utrecht Work Engagement Scale” (UWES) +scale [53] that were modified for use in teams by Van Boxmeer +et al. [54]. For stakeholder satisfaction, we used a 4-item scale +developed by the authors for another study [30]. Both measures +are self-reported. Reliability analysis showed that Team Morale +(α = .910) and Stakeholder Satisfaction (α = 0.832) were +consistently measured across participants. +Such conflicts represent interpersonal incompatibilities be- +tween team members that “typically includes tension, animosity, +and annoyance among members within a group” [55, p. 258]. +Relational conflict was operationalized by adapting three +items from a scale developed by Jehn et al. [55] to measure +relationship conflict. The items were adapted for use in teams by +the authors. The reliability of measurements across participants +was high (α = .892). + +H5(-) +Psychological Safety +Relational Conflict +H3 ( +[a,b, +H2a,b,c,d (+) +H6a,b.c,d ( +Team Diversity +H4 (+) +Gender Diversity +Team Effectiveness +Team Morale +H1a,b,c,d (+) +Age Diversity +Stakeholder Satisfaction +Cultural Diversity +Role DiversityIEEE TRANSACTIONS ON SOFTWARE ENGINEERING +6 +TABLE I +COMPOSITION OF THE SAMPLE +Variable +Category +N (%) +Respondents +1,118 +Teams +161 +Respondents per team +4-6 respondents +82 (50.9%) +7-9 respondents +64 (39.8%) +10+ respondents +15 (9.3%) +Product Type +Product for internal users +89 (55.3%) +Product for external users +72 (44.7%) +Scrum Team Size +1-4 members +1 (0.6%) +5-10 members +129 (80.1%) +11-16 members +26 (16.1%) +>16 members +5 (3.1%) +Scrum Team Experience +Low +5 (3.1%) +Moderate +75 (46.6%) +High +81 (50.3%) +Organization Sector +Technology +40 (24.8%) +Financial +29 (18%) +Healthcare +18 (11.2%) +Other +74 (46%) +Organization Size +1-50 employees +10 (6.2%) +51-500 employees +44 (27.3%) +501-5.000 employees +50 (31.1%) +>5.000 employees +55 (34.2%) +Unknown +2 (1.2%) +Role Diversity +Developer +534 (47.8%) +Scrum Master +127 (11.4%) +Product Owner +104 (9.3%) +Tester +93 (8.3%) +Analyst +73 (6.5%) +Visual/UX Designer +38 (3.4%) +Infrastructure +14 (1.3%) +Marketeer or sales +2 (0.2%) +Other +81 (7.2%) +Unknown +52 (4.7%) +Age Diversity +18-25 years +79 (7.1%) +26-35 years +494 (44.2%) +36-45 years +308 (27.5%) +46-55 years +128 (11.4%) +56-65 years +39 (3.5%) +66+ years +3 (0.3%) +Unknown +52 (4.7%) +Cultural Diversity +Western Europe +581 (52%) +Eastern Europe +104 (9.3%) +North America +103 (9.2%) +Central & South America +48 (4.3%) +Middle East +30 (2.7%) +South-East Asia +28 (2.5%) +South Asia +27 (2.4%) +East Asia +20 (1.8%) +Oceania +7 (0.6%) +Africa +2 (0.2%) +Other +51 (4.6%) +Unknown +11 (1%) +Gender Diversity +100% men or women +174 (15.6%) +80% men and 20% women +766 (68.5%) +50%-50% men and women +155 (13.9%) +20% men and 80% women +16 (1.4%) +Psychological Safety was operationalized by adapting three +items from the ’Inquiry & Dialogue’ scale that was developed +by Marsick & Watkins [56] as part of the Dimensions of +Organizational Learning Questionnaire (DLOQ). The items +were adapted for use in teams by the authors. The reliability +of measurements across participants was high (α = .791). +Control Variables: We included two items from the social +responsibility scale (SDRS5) [57] to control for socially +desirable answers and to control for common method bias [58]. +C. Analysis +We employed Structural Equation Modeling (SEM) with the +AMOS software package [59] to analyze the data. A strength +of SEM is that it is an inherently confirmatory approach that +combines multiple linear regressions and confirmatory factor +analysis (CFA) with Maximum Likelihood estimation (ML) to +produce more consistent and less biased estimates than those +derived through Ordinary Least Squares (OLS) that is typically +used in multiple regression and ANOVA [48]. Furthermore, +SEM allows researchers to simultaneously test both the struc- +tural part of a theory - the relationships between independent +and dependent variables - and the measurement model - the +inclusion of multiple indicators to measure latent factors [48], +[60], [61]. This is particularly useful for psychometric scales +that use multiple questions to operationalize an underlying +construct, as we do in this study. +In SEM, the statistical model is evaluated through several +“Goodness of Fit” indices and the statistical significance and +effect size of individual paths. The aim is to arrive at a model as +parsimonious as possible while providing a good fit (and thus +explanatory power). We discuss the fit indices in section III-D +Next, we tested our data for the necessary statistical as- +sumptions required for Structural Equation Modeling. First, we +assessed normality by comparing our independent, dependent, +and control variables against recommended thresholds for +kurtosis (< 3) and skew (< 2) [62] in literature. This was +satisfactory for all variables except cultural diversity, which +distribution was strongly leptokurtic. This means that only a +few teams showed some heterogeneity in cultural diversity, +whereas most were completely homogeneous. Although statis- +tical transformations can re-normalize such distributions, this +also inevitably complicates their interpretation, especially the +comparison with other effects in a model [48]. +Our measure for gender diversity was not continuous but +ordinal (no diversity, some diversity, or high diversity). We +treated this variable as continuous in our analyses because +such a model is more parsimonious than one that treats it +as categorical [63]. It also simplifies the interpretation and +retains more information than a model where the ordinal +variable is treated as categorical. However, this requires +that the relationship between the dependent variable and the +ordinal independent variable is linear and that each step is +approximately evenly spaced [63]. The relationship was linear, +but the steps were relatively not evenly spaced (respectively +.33 and .18 for both steps). However, the modest violation did +not warrant using a less parsimonious model with categorical +dummy variables for gender diversity instead of a single +variable. +We assessed homoscedasticity by inspecting the scatter +plots for all pairs of independent and dependent variables for +inconsistent patterns but found none. Finally, multicollinearity +was assessed by entering all independent variables one by one +into a linear regression [64]. The Variance Inflation Factor +(VIF) remained below the critical threshold of 10 [48] for all +measures. However, we did observe modest multicollinearity +of psychological safety with our indicators of diversity (age, +cultural background, role, and gender) and our control variable + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +7 +for the experience of teams with Scrum, with a VIF ranging +between 6.38 and 7.77 for these variables. +Using a single method - like a questionnaire - introduces +the potential for a systematic response bias where the method +itself influences answers [65]. To control for such common +method bias, the recommended approach in current literature +is using a marker variable that is theoretically unrelated to +other factors in the model [58]. We included two items from +the social responsibility scale (SDRS5) [57] and found a small +but significant unevenly distributed response bias. Following +recommendations in the literature, we retained the marker +variable “social desirability” in our causal model to control for +common method bias [58]. +We created a full latent variable model containing both +the measurement and structural models. The measurement +model defines relationships between indicator variables (survey +items) and underlying first-order latent factors and effectively +acts as a CFA-model [66]. The structural model defines +the hypothesized relations between latent variables and is a +regression model. This approach makes the results less prone +to convergence issues because of low indicator reliability and +offers more degrees of freedom to the analysis compared to a +non-latent model [67]. We began by assessing the measurement +model following the approach outlined in literature [60], [61], +[48]. Psychological safety, relational conflict, team morale, +stakeholder satisfaction, and psychological safety were entered +as first-order latent factors, with their respective survey items +as indicator variables. Once the measurement model exhibited +a good fit (see section III-D), we added the structural part of +the model. +In the structural part of the model, we created a second- +order latent factor to reflect the composite nature of “team +effectiveness”. The first-order latent factors for team morale and +stakeholder satisfaction were modeled as indicators, similar +to Verwijs & Russo [30]. We calculated interaction terms +by multiplying each team’s standardized factor score for +psychological safety with their standardized scores for each +diversity indicator (age, gender, role, cultural background) [68]. +The diversity indicators and the interaction terms were entered +into the model as exogenous variables. The exogenous variables +for psychological safety, the diversity indicators, and their +interaction terms were allowed to co-vary. No covariances +were allowed between endogenous variables as our model +predicted specific paths between them. +D. Model fit evaluations +We assessed reliability, convergent, and discriminant validity +for the resulting measurement model before testing for the +model fit. The individual steps involved in the model-fitting +process are in Table 1 in the Appendix. Discriminant validity +was assessed by analyzing the heterotrait-monotrait ratio of +correlations (HTMT) with a third-party plugin in AMOS [69] +and following the approach outlined in literature [48], [70]. +This ratio between trait correlations and within trait correlations +should remain below R = .90 to indicate good discriminant +validity from other constructs in different settings. This was +the case for all measures. We assessed convergent validity by +inspecting composite reliability (CR) and average extracted +variance (AVE). The AVE remained above the rule of thumb of +> .50% [48] for all pairs of factors, ranging between .621 and +.890. The CR was equal to or above the threshold of .70 [48] +for all scales. +We then proceeded with the fitting procedure. We investi- +gated local fit by inspecting the residual covariance matrix. +A standardized residual covariance is considered large when +it exceeds 2.58 [60]. This indicates that an item does not +sufficiently measure (only) its intended factor. One item from +Stakeholder Satisfaction (SH3) showed poor local fit, and we +removed it. +The overall goodness of fit was evaluated with indices +recommended by recent literature [66], [60], [48]; the Compar- +ative Fit Index (CFI), the Root Mean Error of Approximation +(RMSEA), the Standardized Root Mean Residual (SRMR) +and the Tucker Lewis Index (TLI). A commonly used index +that we reported but did not test for was χ2 (CMIN) and its +corollary CMIN/df. These indices are highly susceptible to +type I errors in larger samples (N > 400, [48]). So instead, +we used the Comparative Fit Index (CFI) [71], which offers +a similar test but with consideration of the sample size and +its reliable properties have made it the most commonly used +index today [48]. A cut-off value of .95 or higher is generally +considered to indicate good fit [60], [48], [72]. The Root Mean +Error of Approximation (RMSEA) by Steiger & Lind [73] also +provides an index that considers sample size but adds to this a +parsimony adjustment that leads it to favor the simplest model +out of potential models with the same explanatory power [66]. +A value below .05 is generally considered to indicate a good +fit [60], [48]; additionally, we follow the advice to report the +confidence interval in addition to only the absolute value [74]. +The Standardized Root Mean Residual (SRMR) calculates a +standardized mean of all the differences (residuals) between +each observed covariance and the hypothesized covariance +between variables [48]. A value below .08 is indicative of +a good fit. We also inspected local fit by looking at the +standardized residuals between pairs of variables, with values +beyond 2.58 as a cut-off value for poor local fit [60]. Finally, we +report and test the Tucker-Lewis Index (TLI). This is another +incremental fit index, like the CFI, that compares the relative +improvement of the hypothesized model from a model where +all variables are uncorrelated. Hair et al. +[48] considers a +value of .97 or above sufficient to conclude a good model +fit. In addition to overall model fit, we also evaluated our +model on the percentage of variance that is explained in team +effectiveness by all other variables in the model. +The measurement model fitted our data well (Chi2(79) = +127.650; TLI = .973; CFI = .980; RMSEA = .062; +SRMR = .0516). A Confirmatory Factor Analysis (CFA) +is reported in the Appendix (Table 3) that shows that all items +loaded primarily on their intended factors, except for PS2. This +item also loaded negatively on the factor for Relational Conflict. +The cumulative Eigenvalues of 5 factors explain 78% of the +total observed variance, which is well beyond the recommended +threshold of 60% [48]. +We then tested the path model for the effects we pre- +dicted from our theory. Our hypothesized theoretical model + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +8 +TABLE II +SCALES USED IN THE SURVEY STUDY, ALONG WITH ATTRIBUTION, NUMBER OF ITEMS, AND RELIABILITY (CRONBACH’S ALPHA) BASED ON +RESPONDENT-LEVEL RESPONSE DATA (N = 1, 118) +Construct variable +Items adapted from +# Items +Alpha +Psychological Safety +Adapted from ’Inquiry & Dialogue’ scale in DLOQ [56] +5 +.791 +Relational Conflict +Adapted from Jehn [55] +3 +.892 +Stakeholder Satisfaction +Created by authors from our case studies +4 +.874 +Team Morale +Adapted from Van Boxmeer et. al. [54] and Schaufeli [53] +3 +.910 +Social Desirability +Highest-loading items from SDRS-5 scale [57] +2 +.672 +fits the data well on each fit indices, as described in Ta- +ble IV: Chi2(129) = 156.282; TLI = .981; CFI = .988; +RMSEA = .036; SRMR = .051. The predictors in our +model explain respectively 40.7% of the variance in the latent +factor representing team effectiveness. For studies in the social +sciences, values above 26% are considered large [75]. +IV. RESULTS +We now turn to the results and hypothesis testing. The +means, standard deviations, and Pearson correlations of all +variables are reported in Table III-D. Significant effects are also +visualized in Figure 2. Following recommendations in statistical +literature [60], [66], we used a bootstrapping procedure with +2,000 samples and 95% bias-corrected confidence intervals +to more accurately estimate parameters and their p-values for +direct effects, factor loadings, and the hypothesized indirect +effects. This resulted in a standardized, bias-corrected estimate +(β) for each path, along with a p-value to test whether the null +hypothesis can be rejected. The parameter estimates relevant +to our hypotheses are reported in Table V. +Our results allowed us to reject the null hypotheses for 4 +out of 19 (sub)hypotheses. The primary hypothesis of this +study is that diversity makes Agile software teams more +effective because it broadens the cognitive resources available +for information processing (H1a-d). This is partially true for +our results, as only age diversity significantly contributes +to team effectiveness (H1b, β = .213, p < .05). Agile +software teams seem slightly more effective when there is +greater age heterogeneity. Nevertheless, heterogeneity in gender, +cultural background, or role does not appear relevant to team +effectiveness (H1a,c,d). +We also hypothesized that relational conflict in teams would +increase as heterogeneity increases and members become +less similar. This is also partially true, as only gender di- +versity significantly contributes to relational conflict (H2a, +β = .161, p < .01). Thus, there appears to be more conflict +as teams grow more heterogeneous in gender. There is no +discernible effect of diversity in age, cultural background, or +functional role on conflict. (H2b,c,d). +Contrary to our expectations, we did not find a significant +effect between relational conflict and team effectiveness (H3). +Teams that experience more relational conflict do not seem +to be more or less effective than teams that experience less +conflict. However, the results show a strong positive effect of +psychological safety on team effectiveness (H4, β = .660, p < +.01). Teams that experience more psychological safety are more +effective in that they have reported more satisfied stakeholders +and higher team morale. Psychological safety also strongly +decreases the amount of relational conflict reported by teams +(H5, β = −.636, p < .01). +Finally, we hypothesized that psychological safety moderates +the strength by which diversity contributes to team effectiveness +and relational conflict. However, none of the interactions were +significant (H6a-d; H7a-d). +V. DISCUSSION +This study investigated how diversity in age, role, cultural +background, and gender influences the effectiveness of Agile +software teams. 1.118 respondents from 161 Agile software +teams participated in our study. Overall, our results provide +mixed support for both the benefits and the risks of member +heterogeneity in teams. +According +to +the +categorization-elaboration +model +(CEM)[14] and cognitive resource diversity theory, we +hypothesized that Agile software teams benefit from diversity +as it expands the cognitive resources available for information +processing. However, only age diversity directly improves +team effectiveness directly. This finding is consistent with +the conclusions from a recent review of the literature by +Tshetshema & Chan [10], and a meta-analysis of 74 studies by +Schneid et al. [16], particularly for complex tasks. However, +another meta-analysis of 35 studies by Horwitz & Horwitz [12] +found no positive impact of demographic diversity (age, +gender, race). So our results are more nuanced than the +overall positive effect of team diversity that is reported by +Lee & Xia [26] for Agile software teams. We also did not +find a positive effect of gender diversity or cultural diversity, +whereas others did [10], [8]. All in all, the association +between demographic diversity and team effectiveness is more +complicated than the direct, positive effects we hypothesized. +In addition to demographic diversity, we also investigated +how role diversity improves team effectiveness. Agile software +methodologies emphasize this type of diversity as an important +characteristic of autonomous teams [78], [2]. In line with +cognitive resource diversity theory, role diversity allows teams +to leverage more perspectives and broader informational +resources to resolve complex problems [14], [13]. When +members bring more functional roles to their work together +(e.g., analyst, tester, developer, designer), their shared mental +models will be richer than when all members hold the same +role (e.g., developer) [11], [79]. However, we did not find +evidence for this. Teams with high role diversity were not + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +9 +TABLE III +MEANS, STANDARD DEVIATIONS, SKEWNESS, KURTOSIS AND CORRELATIONS (PEARSON) FOR CONTINUOUS VARIABLES. CORRELATIONS MARKED WITH +* ARE SIGNIFICANT AT p < 0.01 +. +Variable +Mean +SD +Skewness +Kurtosis +1 +2 +3 +4 +5 +6 +7 +8 +1 +Gender Diversity +1.97 +.55 +-.02 +.33 +1.00 +2 +Age Diversity +.71 +.20 +-.78 +.75 +.26* +1.00 +3 +Cultural Diversity +.05 +.16 +2.97 +7.98 +.03 +.00 +1.00 +4 +Role Diversity +.61 +.22 +-1.06 +1.01 +-.07 +.02 +-.06 +1.00 +5 +Psychological Safety +5.54 +.66 +-.80 +1.29 +.10 +-.06 +.04 +-.03 +1.00 +6 +Team Effectiveness +5.36 +.71 +-.38 +-.50 +.17 +.06 +.08 +.04 +.72* +1.00 +7 +Relational Conflict +2.45 +.98 +1.16 +1.56 +.06 +.00 +-.01 +-.07 +-.71* +-.50* +1.00 +Control Variables +8 +Social Desirability +5.72 +.48 +-.17 +-.03 +.16 +.11 +.02 +-.02 +.59* +.56* +-.40* +1.00 +TABLE IV +MODEL FIT INDICES +Model fit index +Value +Interpretation +Chi-Square ( χ2) +156.282 +n/a +Degrees of freedom (df) +129 +n/a +CMIN/df +1.211 +A value below 5 indicates an acceptable model fit [76], below 3 a +good fit [77] +Root Mean Square Error of Approximation (RMSEA) +.036 +Values ≤ .05 indicates good model fit [60] +RMSEA 90% Confidence Interval +.000-.055 +p of Close Fit (PCLOSE) +.873 +Probability that RMSEA ≤ 0.05, where higher is better +Comparative Fit Index (CFI) +.988 +Values ≥ .97 indicates good model fit [48] +Tucker Lewis Index (TLI) +.981 +Values ≥ .97 indicates good model fit [48] +Standardized Root Mean Square Residual (SRMR) +.051 +Values ≤ .08 indicates good model fit [48] +Variance explained by predictors (R2) of Team Effectiveness +40.7% +Values ≥ 26% indicates large effect [75] +Fig. 2. Standardized path coefficients for the model (∗∗ : p < .01, ∗ : p < 0.05). The dotted lines represent non-significant results. Indicator items and +non-significant paths for sub-hypotheses are omitted to improve readability. A detailed overview of the individual hypotheses is reported in Table V. + +H5(-)-.636** +Psychological Safety +Relational Conflict +H3 +1 +Team Diversity +Gender Diversity +Team Effectiveness +Team Morale +H1b: age (+).213* +Age Diversity +Stakeholder Satisfaction +Cultural Diversity +Role DiversityIEEE TRANSACTIONS ON SOFTWARE ENGINEERING +10 +TABLE V +PARAMETER ESTIMATES, CONFIDENCE INTERVALS, STANDARD ERRORS, STANDARDIZED COEFFICIENTS FOR DIRECT EFFECTS, INTERACTION TERMS +AND INDIRECT EFFECTS FOR HYPOTHESES (STATISTICALLY SIGNIFICANT HYPOTHESES AT p < 0.05 ARE SET IN BOLDFACE), AND FACTOR LOADINGS +Parameter +Unstandardized +95% CI +SE +p +Standardized +Direct Effects +H1a: Gender Diversity → Team Effectiveness +.037 +(-.075, .123) +-.075 +.622 +.056 +H1b: Age Diversity → Team Effectiveness +.391 +(.077, .800) +.077 +.041 +.213 +H1c: Cultural Diversity → Team Effectiveness +.024 +(-.382, .542) +-.382 +.872 +.010 +H1d: Role Diversity → Team Effectiveness +.022 +(-.315, .290) +-.315 +.956 +.013 +H2a: Gender Diversity → Relational Conflict +.241 +(.108, .417) +.108 +.008 +.161 +H2b: Age Diversity → Relational Conflict +-.490 +(-1.178, .026) +-1.178 +.118 +-.117 +H2c: Cultural Diversity → Relational Conflict +.032 +(-.824, 1.022) +-.824 +.855 +.006 +H2d: Role Diversity → Relational Conflict +-.332 +(-.834, .199) +-.834 +.306 +-.087 +H3: Relational Conflict → Team Effectiveness +.035 +(-.091, .181) +-.091 +.747 +.081 +H4: Psychological Safety → Team Effectiveness +.574 +(.300, .927) +.300 +.004 +.660 +H5: Psychological Safety → Relational Conflict +-1.262 +(-1.888, -.727) +-1.888 +.001 +-.636 +Interactions +H6a: Gender Diversity * Psychological Safety → Team Effectiveness +-.018 +(-.093, .037) +-.093 +.550 +-.052 +H6b: Age Diversity * Psychological Safety → Team Effectiveness +.007 +(-.069, .086) +-.069 +.885 +.020 +H6c: Cultural Diversity * Psychological Safety → Team Effectiveness +-.042 +(-.159, .061) +-.159 +.388 +-.081 +H6d: Role Diversity * Psychological Safety → Team Effectiveness +.026 +(-.070, .082) +-.070 +.550 +.076 +H7a: Gender Diversity * Psychological Safety → Relational Conflict +-.057 +(-.138, .054) +-.138 +.416 +-.072 +H7b: Age Diversity * Psychological Safety → Relational Conflict +.014 +(-.095, .129) +-.095 +.812 +.017 +H7c: Cultural Diversity * Psychological Safety → Relational Conflict +.133 +(-.047, .377) +-.047 +.206 +.112 +H7d: Role Diversity * Psychological Safety → Relational Conflict +-.044 +(-.162, .046) +-.162 +.406 +-.057 +Factor loadings from first to second-order factors +Team Effectiveness → Stakeholder Happiness +.752 +(.144, .614) +.114 +.003 +.389 +Team Effectiveness → Team Morale +1.000 +(.618, 1.800) +.873 +more or less effective than teams with lower role diversity. +This is partially consistent with extant literature. Homberg & +Bui [21] found no evidence for a link between role diversity +and team performance in a meta-analysis of other empirical +studies. Horwitz & Horwitz [12] also did not find an effect +on performance, although they did find one on the quality of +work done by teams. +Diversity in teams is often considered a double-edged +sword in the literature on diversity [13]. The CEM proposes +that diversity can also harm team effectiveness through the +similarity-attraction paradigm [7]. As members grow less +similar and bring different perspectives to teamwork, there +is more potential for tension and conflict. This decreases the +ability of teams to elaborate information effectively and reduces +their performance. Concerning the first assertion, our results +show that gender diversity does increase relational conflict but +not other kinds of diversity. This finding is consistent with some +studies [19], but not others [8], [10]. Regarding the second +assertion, we failed to find any impact of relational conflict +on team effectiveness. So while it appears true that gender +diversity increases relational conflict in teams to some extent, +we cannot conclude that this also harms team effectiveness +(i.e., the double-edged sword). +The CEM attempts to reconcile the conflicting results by +drawing attention to social- and task-related moderators that +shape how diversity impacts team performance. We investigated +one social moderator frequently associated with diversity, +relational conflict, and team effectiveness: psychological safety. +We hypothesized that a psychologically safe environment +would make it easier for diverse teams to elaborate on task +information effectively. Although psychological safety reduced +relational conflict and improved team effectiveness, we could +not reject the null hypotheses for psychological safety as +a moderator of the diversity-effectiveness link. In summary, +our results show some benefits of diversity (age) on team +effectiveness and some risks of diversity through relational +conflict (gender). Psychological safety also reduces relational +conflict and increases team effectiveness, but we found no +evidence for a moderating role in the diversity-effectiveness +link or the diversity-conflict link. +A. Alternative explanations +The mixed evidence suggests that there are factors at work +that moderate or mediate the effects of diversity on effectiveness +and conflict. Diversity alone does not make teams more +effective because it broadens cognitive resources, just as it +does not inherently and consistently create conflict because +members are less similar. +This study investigated psychological safety as one potential +social moderator of the diversity-effectiveness link. Our mixed +results suggest that other moderators are at play. One example +of this is task interdependence. A core element of Agile +software methodologies is that teams work together on complex +tasks [80], [2], [1]. Collective elaboration of task-related +information and the pooling of skills to accomplish tasks is +also a common thread in the definition of teamwork [51], +[81]. Without task interdependence, the two mechanisms of + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +11 +diversity diminish. Because there is less collective elaboration, +the benefits of the broadened cognitive resources that are offered +by diversity diminish. Furthermore, a major source of conflict +between members is removed because they spend much less +time together processing information. Members may have more +skin in the game when they feel they depend on others in their +team to be successful. Paradoxically, this may surface as a +higher degree of relational conflict than teams with very low +interdependence. In this sense, psychological safety is likely +only relevant as a moderator of the diversity-effectiveness link +in teams with high task interdependence but not low task +interdependence. Future studies can investigate if the effects of +diversity and psychological safety are more pronounced when +controlling for task interdependence. +Another explanation may be that the effect of diversity on +team effectiveness is not linear. Several authors [82], [83] have +argued for curvilinear models where diversity contributes to +performance only when it is moderated (inverted U) or when +it is either low or high (upright U). Which model applies +varies by diversity type. For example, Dahlin, Weingart & +Hinds [43] found that educational diversity contributed to +team performance when it was either low or high (inverted +U) but found the opposite for national diversity (upright U). +Richard et al. [84] found that management teams with moderate +gender diversity performed better than teams with low or high +diversity, but only in high-risk settings (upright U). However, +diversity in terms of age, gender, or function may contribute +to learning behavior in teams more strongly when diversity +is low or high but not moderate (inverted U) [85]. So while +there is some support for the curvilinear effects of diversity, +the relationship is complex. To further complicate matters, +the shape of the relationship may also be moderated by the +expectations that teams themselves have of the benefits of +diversity [29]. We performed a posthoc test to assess whether +a curvilinear relationship between dimensions of diversity and +team effectiveness better fitted the data. This was not the case. A +quadratic regression model was not significant for the following +diversity dimensions: age (R2 = .004, F(2, 158) = .321, p = +.726), gender (R2 = .021, F(2, 158) = 1.695, p = .187), +culture (R2 = .008, F(2, 158) = .664, p = .516), and role +(R2 = .000, F(2, 158) = .025, p = .975). Thus, the possibility +of a curvilinear relationship rather than a linear one does not +appear to explain the lack of results in this study. +We often assume that diversity in age, gender, function, +and cultural background inherently leads to a different un- +derstanding of the task and potential solutions. This is both +the strength and the weakness of diverse teams. In the day- +to-day practice of teams, such differences in understanding +may also lead to conflict if members need to adequately +express their view and integrate it with other members into a +synthesized solution. In addition to those mentioned above, task- +related and social moderators, it is reasonable to expect that +communication and conflict navigation skills are also highly +relevant, as well as the presence of an environment where +such different understandings can be elaborated effectively. +Few studies have investigated such moderators, particularly +for Agile software teams [6]. Furthermore, this ties into team +members’ beliefs about diversity, how to deal with it and +whether or not it benefits teamwork. Van Knippenberg et al. +[42], [29] call this a “Diversity Mind-Set”. Several studies +have shown that teams and organizations can better leverage +diversity when they recognize it as a strength and have learned +how to appreciate and deal with the resulting informational +diversity [86], [84], [29]. +For practitioners, it is important to notice that our results +are broadly consistent with existing research, showing that team +diversity is not unequivocally beneficial or harmful. Although +we found a positive effect of age diversity, the effects of +other types of diversity appear to be more conditional on +moderating factors. Several factors have been proposed to date, +like the autonomy that teams have [26], task difficulty [11], +psychological safety [23], team climate [22] and the beliefs that +teams have about diversity [29]. This suggests that context is +just as important as diversity alone. +B. Limitations +In the following section, we will discuss the threats to the +validity of our sample study. +Internal validity Internal validity refers to the confidence +with which changes in the dependent variables can be attributed +to the independent variables and not other uncontrolled fac- +tors [87]. We employed several strategies to maximize internal +validity. First, we recognize that online questionnaires are prone +to bias and self-selection as a result of their voluntary (non- +probabilistic) nature. We counteracted this by embedding our +questions in a tool that is regularly used by Agile software +teams to self-diagnose their process and identify improvements. +Team members were invited by people in their organization +to participate. Second, we thoroughly cleaned the dataset of +careless responses to prevent them from influencing the results. +Third, we did not inform the participants of our specific research +questions to prevent them from answering in a socially desirable +manner. We also controlled for social desirability in participants’ +responses, as well as common method bias introduced when a +single method is used to collect data. +Despite our safeguards, there may still be confounding +variables that we were unable to control for. This is particularly +relevant to the operationalization of team effectiveness, which is +based on self-reported scores on team morale and the perceived +satisfaction of stakeholders. Mathieu et al. [88] recognize that +such affect-based measures may suffer from a “halo effect”. +Future studies could ask stakeholders to rate their satisfaction +with team outcomes directly. This does not entirely rule out +a halo effect but is conceptually closer to what matters to +organizations. Future studies could also find more objective +measures for team effectiveness. +Construct validity Construct validity refers to the degree +to which the measures used in a study measure their intended +constructs [87]. We adapted items from established scales to +measure psychological safety [89], team effectiveness [30], +relational conflict [55] and social desirability [57]. A con- +firmatory factor analysis (CFA) showed that all items were +loaded primarily on their intended scales (see Table 3 in the +Appendix). A heterotrait-monotrait (HTMT) analysis confirmed +discriminant validity for all measures. The reliability for all + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +12 +TABLE VI +SUMMARY OF KEY FINDINGS & IMPLICATIONS +Findings +Implications +Diversity & team effec- +tiveness +Based on existing theory, we developed a Structural Equation +Model for how diversity and psychological safety interact to +impact team effectiveness and relational conflict. The model +fitted the data well (Chi2(129) = 156.282; TLI = .981; +CFI = .988; RMSEA = .036; SRMR = .051). +Age diversity showed a positive association with team +effectiveness (β = .213, p < .05), but not diversity in +gender, role, or cultural background. +Teams with members of different age groups will likely +benefit from the broader range of tenure and work/life +experience. The benefits of other types of diversity appear +more conditional on moderating factors. Organizations can +assess the extent to which teams are diverse. However, +psychological safety, communication skills, and a diversity +mindset seem important moderators that organizations need +to provide and encourage teams to leverage it. +Diversity +& +relational +conflict +Gender diversity was positively associated with relational +conflict in Agile software teams (β = .161, p < .01). +However, diversity in role, age, or cultural background did +not. In turn, relational conflict did not significantly affect +team effectiveness. +When teams grow more diverse, members’ different perspec- +tives may lead to more conflict and friction. This appears +particularly relevant to gender diversity. Such negative +consequences of diversity may be counteracted when teams +learn to see their diversity as a strength and recognize +that different perspectives can be reconciled through open +dialogue and elaboration. +Psychological +safety +& +team effectiveness +Psychological safety was positively associated with team +effectiveness (β = .660, p < .01) and negatively associated +with relational conflict (β = −.636, p < .01) +Teams that operate in environments where members can +openly and safely elaborate information are more effective +than other teams, regardless of their diversity. They also ex- +perience much less relational conflict. Organizations do well +to develop the skills, support structures, and management +styles that foster psychological safety in and around teams +Psychological safety as a +moderator +Psychological safety did not significantly moderate the +association between diversity and team effectiveness, nor +between diversity and relational conflict. +Psychological safety is paramount, but it does not appear to +strengthen the cognitive benefits of team diversity, nor does +not it appear to buffer against negative consequences. +measures exceeded the cutoff recommended in the literature +(CR >= .70 [48]), except social desirability. Thus, we are +confident that we reliably measured the intended constructs. +A limitation of our measure for team effectiveness is that +it only addressed (self-reported) stakeholder satisfaction and +team morale. Although both are reasonable and relevant aspects +of team effectiveness and are commonly used in team re- +search [51], effectiveness is also a more-faceted construct [90]. +Finally, we could not directly ask participants for their +gender due to privacy concerns. So it was not possible to +calculate a Gini index as we did for the diversity measures. The +resulting measure was ordinal instead of continuous, limiting +our analysis’s resolution for this variable. Future studies do +well to use a more continuous measure of gender distribution. +Conclusion validity Conclusion validity assesses the extent +to which the conclusions about the relationships between +variables are reasonable based on the results [91]. We used +Structural Equation Modeling to test the entire model simulta- +neously [66], [60]. The resulting model fits the data well on all +fit indices recommended by statistical literature and explains +a substantial amount of variance in the dependent variables. +Our sample was also large enough to identify medium effects +(f = .15) with a statistical power of 96%. +We published team-level data and syntax files to Zenodo for +reproducibility. +External validity Finally, external validity concerns the +extent to which the results actually represent the broader +population [92]. First, we assess the ecological validity of +our results to be high. Our questionnaire was integrated into a +more general tool that Agile software teams use to improve +their processes. Participants were invited by people in their +organization, usually Scrum Masters. Thus, the data is more +likely to reflect realistic teams than a stand-alone questionnaire +or an experimental design. +We do not know how well our sample reflects the total +population. However, our sample composition (Table I) shows +that a wide range of teams participated in the questionnaire, +with different levels of experience from different parts of +the world and different types of organizations. We also +observed a broad range of scores on the various measures. This +provides confidence that a wide range of teams participated. +Furthermore, our sample size and the aggregation of individual- +level responses to team-level aggregates reduce variability due +to non-systematic individual bias. +VI. CONCLUSION +A common thread in Agile software methodologies is their +emphasis on teams as the primary units where complex work +is performed. So it is not surprising that much research has +focused on what makes such teams more effective (i.e. [30], +[31], [93], [27], [94]). Although diversity is increasingly inves- +tigated in the broader literature on teams, scholarly knowledge +on how it impacts Agile software teams is still limited [6], +[15]. Such understanding can better equip organizations and +teams to leverage diversity more effectively or learn when +and how diversity is beneficial. Because what seems to be +clear about diversity is that while it brings more extensive +cognitive resources to teams, it can also bring more conflict as +members become less similar [13]. Several models have been +proposed to explain this “double-edged sword” of diversity, +with the categorization-elaboration model (CEM) [14] as the +most comprehensive one. + +IEEE TRANSACTIONS ON SOFTWARE ENGINEERING +13 +In this study, we explored how diversity impacts the +effectiveness of Agile software teams through the lens of the +CEM theory. Our sample consisted of 1,118 team members +representing 161 Agile software teams. Our results show that +age diversity contributes to more effective teamwork but not +diversity in gender, role, or cultural background. This may +reflect the value of having more varied levels of experience in +teams. Furthermore, the CEM also predicts a negative effect of +diversity through social categorization and identity threat, which +can surface through increased conflict. While our results support +this effect, we only found evidence for gender diversity. Finally, +the CEM predicts that task- and social moderators influence +the impact of diversity. One such moderator that is frequently +studied is psychological safety [23]. While our results show +that it contributes to more effective teamwork and less conflict +in teams, it did not moderate the link between diversity and +effectiveness or diversity and conflict. Thus, the presence of +psychological safety in a team does not in itself allow teams +to leverage their diversity better. Despite the strong focus on +role diversity and cross-functional teamwork in Agile software +methodologies [80], [2], we found no apparent effect on team +effectiveness. So while our results are broadly consistent with +the CEM for age and gender diversity, it is surprising that +heterogeneity in role or cultural background did not produce +similar effects. One moderator that may be particularly relevant +here is task interdependence. Teams vary broadly in the degree +to which members actually (need to) work together on tasks +and, thus, the opportunities that arise to leverage the broader +cognitive resources of diverse teams. +This study has several implications for future studies of +how diversity impacts the effectiveness of Agile software +teams. First, the role of task-related and social moderators +should be investigated more thoroughly. The categorization- +elaboration model [14] provides a valuable framework for +such research because it integrates the opposing mechanisms +of diversity proposed by cognitive resource diversity theory and +the similarity-attraction paradigm. From a practical viewpoint, +such research can also drive the development of training and +methods to help teams and organizations to leverage their +diversity on all sorts of dimensions, and not limited to gender, +age, cultural background, and functional role. Second, more +attention should be paid to the beliefs that teams have about +diversity and its effects. Such a “Diversity Mind-Set” [29] can +act as a powerful moderator by making teams aware of their +diversity and how it can expand their experience as a team. +Finally, future research should investigate broader definitions +of performance and effectiveness. In this study, we mainly +focused on stakeholder satisfaction and team morale. Since +effectiveness is a multi-faceted construct [90], we likely missed +aspects that are affected by diversity in teams, like speed, +quality, or innovativeness. +ACKNOWLEDGMENT +The authors would like to thank all participants in our study +for their efforts and time. We would also like to thank The +Liberators BV and its community of patrons for funding part +of this research. +VII. SUPPLEMENTARY MATERIALS +A replication package for the sample study is available at the +following DOI to support Open Science: https://www.doi.org/ +10.5281/zenodo.7537784 under a CC-BY-NC-SA 4.0 license. +The package includes the model definitions (AMOS), syntaxes +for SPSS, and a fully anonymized, cleaned, and aggregated +dataset of the analyzed teams. +VIII. RESPONSIBLE DISCLOSURE +Data has been collected and stored according to the policy +for research data management of Aalborg University, respecting +the total anonymity of informants. +Christiaan Verwijs has a financial interest in The Liberators +BV. +REFERENCES +[1] A. Manifesto, “Agile manifesto,” Haettu, vol. 14, p. 2012, 2001. +[2] C. Larman, Agile and iterative development: a manager’s guide. Pearson +Education India, 2004. +[3] J. Highsmith and A. 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Van Vliet, “When agile meets the enterprise,” +Information and software technology, vol. 55, no. 12, pp. 2154–2171, +2013. + diff --git a/UNFOT4oBgHgl3EQf5jRf/content/tmp_files/load_file.txt b/UNFOT4oBgHgl3EQf5jRf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0de6258d8fef078f4e8ef9a5fac601ed966b7c5 --- /dev/null +++ b/UNFOT4oBgHgl3EQf5jRf/content/tmp_files/load_file.txt @@ -0,0 +1,1597 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf,len=1596 +page_content='IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 1 The Double-Edged Sword of Diversity: How Diversity, Conflict, and Psychological Safety Impact Agile Software Teams Christiaan Verwijs and Daniel Russo Abstract—Team diversity can be seen as a double-edged sword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' It brings additional cognitive resources to teams at the risk of increased conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Few studies have investigated how different types of diversity impact Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This study views diversity through the lens of the categorization-elaboration model (CEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We investigated how diversity in gender, age, role, and cultural background impacts team effectiveness and conflict, and how these associations are moderated by psychological safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our sample consisted of 1,118 participants from 161 teams and was analyzed with Covariance-Based Structural Equation Modeling (CB-SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We found a positive effect of age diversity on team effectiveness and gender diversity on relational conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Psychological safety contributed directly to effective teamwork and less conflict but did not moderate the diversity-effectiveness link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' While our results are consistent with the CEM theory for age and gender diversity, other types of diversity did not yield similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We discuss several reasons for this, including curvilinear effects, moderators such as task interdependence, or the presence of a diversity mindset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' With this paper, we argue that a dichotomous nature of diversity is oversimplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Indeed, it is a complex relationship where context plays a pivotal role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A deeper understanding of diversity through the lens of theories such as the CEM may lead to more effective teamwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Index Terms—software teams, agile, diversity, psychological safety, conflict I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' INTRODUCTION Teams are increasingly crucial to organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is particularly relevant to organizations that use Agile software methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Agile represents a collaborative, iteration-based, and human-oriented approach to product development [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' It originated in response to the perceived shortfalls of plan- based approaches in the face of complex problems typical in product development [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, “at its core, agile project management is about managing the impact of complexity and uncertainty on a project” [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' 281].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' As a crucial aspect of project management, scholars have attempted to identify the factors and characteristics that influence the performance and productivity of teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' One factor that has gained increased attention in recent decades is team diversity [5], also in software engineering specifically [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Team diversity is generally defined as heterogeneity in member attributes, such as age, gender, cultural background, tenure, role, or personality traits [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' While teams can be diverse on many attributes, most studies focus on demographic diversity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=', age, gender, cultural C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Verwijs is with The Liberators, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Russo is with the Department of Computer Science, Aalborg University, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Email: daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='russo@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='aau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='dk Manuscript received Month 01, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' revised .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='. background) or informational diversity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=', professional role, education, experience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Many researchers have theorized that diversity improves team performance [8], [9], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, studies have provided mixed support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Investigations of how diversity impacts teams [10], [11], [5], [12], [9], [6] generally show that the effects are not clear-cut, vary by type of diversity, and appear to be moderated by characteristics of the task, the team, and its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, diversity may also negatively impact effectiveness through an increased conflict between members [5], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several competing mechanisms and integrated models have been proposed to explain these conflicting results [13], [14], which are discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Specifically for software engineering and Agile methodologies, Silveira & Prikladnicki [6] and Rodr´ıguez- P´erez, Nadri & Nagappan [15] concluded from literature reviews that our understanding of diversity in such teams still needs to be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' They found that most studies have only investigated gender diversity [6] and argue that a broader exploration of how diversity impacts software engineering teams can be used to create better teams and better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Furthermore, studies have yet to explore how diversity affects Agile methodologies and their effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A more comprehensive examination is vital to understand how to design more effective teams and achieve better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Henceforth, our research question (RQ) is: RQ: How does diversity in Agile software teams impact their effectiveness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' To answer our research question, we performed a quantitative cross-sectional study with 1,118 team members representing 161 Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Covariance Based Structural Equa- tion Modeling (CB-SEM or SEM in short) was used to test how four types of diversity (gender, age, cultural background, and role) and one social moderator (psychological safety) interact to impact team effectiveness and conflict in teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Only age diversity was positively associated with team effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Concerning relational conflict, only gender diversity showed a significant positive association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A replication package is also openly available on Zenodo to support secondary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In Section II, we review the related works of team diversity and how it impacts team outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Subsequently, we clarify the research gap this study intends to address and develop relevant hy- potheses in Section II-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Section III clarifies how we use quantitative methods and a survey study to test our hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='12954v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='SE] 30 Jan 2023 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 2 The study results are reported in Section III-C, followed by a comprehensive discussion of the results and their implications in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, we conclude our paper outlining future research opportunities in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' RELATED WORK Scholars from several disciplines have shown mixed results regarding how diversity impacts teams’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Tshet- shema & Chan conclude from a review of 35 studies that “a neg- ative relationship between [demographic diversity] and team performance is inferred as the most reported result.” [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, they note that investigations of individual dimensions of diversity often show a positive effect on team performance, particularly gender and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The complex relationship between diversity and performance is also recognized by Patr´ıcio & Franco [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' They argue from a review of 80 studies that diversity has a dual impact on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' One is positive through expanding perspectives, and the other is negative through increased conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Bowers, Pharmer & Salas [11] performed a meta-analysis of 13 empirical studies and found the effects of team diversity on team performance to be dependent on task complexity and difficulty instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Their results suggest that teams that perform tasks of low complexity may benefit more from homogeneity, whereas teams that perform complex tasks benefit from higher diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Another meta-analysis of 30 empirical studies by Horwitz & Horwitz [12] found no significant effect of demographic diversity (age, gender, or cultural background) on team performance but did find a significant moderate effect of role diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We now turn to investigations of individual dimensions of diversity in teams commonly studied by scholars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Diversity dimensions For age diversity, Tshetshema & Chan [10] found a positive effect on team performance in a review of empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, a meta-analysis of 74 empirical studies by Schneid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [16] did not show a significant relationship, although modest differences occurred as a result of moderators like task complexity and team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Pesch, Bouncken & Kraus [17] attribute the positive effect of age diversity primarily to differences in tenure and work experience rather than age itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' They also note that this diversity is likely to increase tension and conflict in teams as members have to reconcile more diverse perspectives on completing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Cultural diversity is defined as heterogeneity in shared be- liefs, norms and values [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' It is often operationalized through surface-level ethnic or national diversity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Tshetshema & Chan [10] inventoried studies that investigated the link between cultural diversity and team performance and inferred that a positive relationship is the most reported result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, the relationship appears curvilinear: moderate cultural diversity is beneficial, but too little or too much adversely affects team performance [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Scholars define gender diversity as heterogeneity in the gender of team members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Most studies suggest a positive relationship with team performance [10], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Nevertheless, too much diversity may lead to increased conflict, particularly for complex tasks and high interdependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, Haas & Hartmut [19] argue that gender diversity should be avoided in such environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Role diversity is another dimension of diversity in teams that is frequently studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' It represents the heterogeneity in the functional disciplines and roles members bring to a team [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Agile software methodologies emphasize the need for role diversity in teams in order to solve complex problems [2], [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Empirical studies have shown mixed results, with some demonstrating positive effects and others negative [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Pelled, Eisenhardt & Xin [20] found that role diversity increases conflict due to the integration of diverse perspectives, which positively influences task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Homberg & Bui [21] found no significant effect of role diversity on the performance of management teams in a meta-analysis of 53 empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Instead, they attribute the mixed findings to publication bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Horwitz & Horwitz [12] did find a modest effect on the quality of the work produced by teams in another meta-analysis, though not on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The empirical link between diversity and team performance appears to be complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several moderators have been found to strengthen the positive impact or dampen the negative impact, such as an inclusive team climate [22], task complexity and difficulty [11], [12], psychological safety [23], management support [24], or time [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Diversity in Agile software teams The importance of team diversity has also been recognized for Agile software teams specifically [15], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The assumption is that diversity allows for a richer exploration of shared problems due to the availability of more perspectives [9], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is particularly relevant to the complex problem- solving in Agile software teams, which require creativity and the application of diverse skill sets [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several studies have investigated whether this assumption holds up in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Lee & Xia [26] used a mixed-methods approach to investigate how role diversity and team autonomy influence the ability of Agile software teams to deliver on budget, on time, and on the scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' They found a significant positive effect of diversity in a survey study of 399 Agile software projects and follow-up case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, they found that role diversity improves the quality of solutions emerging from problem-solving in teams, but not speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' They also found evidence for the dual impact of diversity, where diversity also increases conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Melo et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [27] performed a multiple-case study of Agile software teams in three large Brazilian software companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Their results suggest that teams are more productive when there is diversity in the experience that members bring to the team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Another study by Russo & Stol [8] surveyed 483 software engineers to investigate how personality and gender influence the productivity of software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Their results show that men and women typically bring different positive and negative traits to teams, and they argue that this explains some part of why mixed-gender teams perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Rodr´ıguez-P´erez, Nadri & Nagappan [15] conclude from a literature review that gender differences between developers contribute significantly to how they solve problems, debug issues, and work with IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 3 others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The authors also note that gender diversity is most frequently studied, but much less is known about how other types of informational and demographic diversity affect Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A similar conclusion is reached by Silveira & Prikladnicki [6] in a review of the literature on diversity in Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, both groups of authors call for more research to guide decision-making on how to design better teams and generate better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Theories and moderators of the diversity-performance link Two mechanisms have been proposed by which diversity influences team performance [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The similarity-attraction paradigm [7] derives from social psychology and social cate- gorization to argue that similarity between members increases mutual attraction, integration, and communication, which in turn improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Diversity of members, on the other hand, results in more conflict and misunderstandings as people categorize themselves into different subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Alternatively, cognitive resource diversity theory derives from cognitive psychology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' It treats teams as information processors where individuals process information and then elaborate and integrate it as a team [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In this conceptualization, diversity allows teams to bring varied cognitive resources to bear when information is processed individually and elaborated as a team, which allows a richer exploration of shared challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, both mechanisms offer conflicting predictions about how diversity will impact team performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The former expects relational conflict to increase and performance to decrease, whereas the latter expects performance to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, the evidence mentioned above does not consistently support one or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So the focus of academic inquiry has shifted toward identifying potential moderators that allow both mechanisms to be integrated [14], [29], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' One potential group of moderators concerns task character- istics, like complexity and interdependence [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In this view, homogeneity benefits low-complexity tasks with few inter- dependencies, whereas heterogeneity benefits more complex tasks with many inter-dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is primarily consistent with findings from meta-analyses of the diversity-performance relationship [11], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, other studies have found both positive and negative effects of task interdependence on the relationship between diversity and performance [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Another potential moderator is psychological safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Ed- mondson [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' 9] defines it as “a shared belief held by team members that the team is safe for interpersonal risk- taking”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several studies have already shown that psychological safety contributes to more effective teamwork in software teams [30], [31], [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, psychological safety is also likely to moderate the relationship between diversity and team performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Diegmann & Rosenkranz [34] theorize that psychological safety makes teams more resilient against the disruptive effect of high diversity, such as increased conflict, by providing a safe environment for members to elaborate task information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Similarly, Roberge & Van Dick [35] expect that psychological safety also interacts with the salience of a collective identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Diversity only contributes to higher team performance when members feel safe and identify strongly with their team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' To date, few studies have empirically investigated the role of psychological safety as a moderator of the diversity- performance association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Singh, Winkel & Selvarajan [36] found that employee performance was higher among members of diverse teams that also exhibited high psychological safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, this study was contained to one organization and only considered racial diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Furthermore, Kirkman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [37] found that Communities of Practice (CoP) performed better when diversity was paired with high psychological safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Virtual teams also experience fewer drawbacks from diversity when they can elaborate information in psychologically safe environments [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Van Knippenberg, De Dreu & Homan [14] have proposed the categorization-elaboration model (CEM) to integrate the double-edged nature of diversity in teams and potential mod- erators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The CEM is the most comprehensive model of work group diversity and its moderators at the time of writing and has received broad empirical support [39], [40], [41], [29], [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' It distinguishes between moderators related to the task, like difficulty, complexity, and efficacy and moderators related to the team and the social processes in it, like trust and commitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Both groups of moderators influence the ability of teams to leverage the informational advantage offered through diversity, though in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In the case of task moderators, complex and challenging tasks are more likely to elicit extensive information processing in members [12], [43], which is consistent with cognitive resource diversity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' There is also support for the moderating influence of task motivation through process accountability [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Another potential task moderator is task interdependence, which is generally defined as the degree to which the completion of tasks requires collaboration by team members [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Teams with low interdependence see less interaction and thus experience fewer opportunities to leverage the benefit of diverse cognitive resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, empirical studies have found positive and negative effects of task interdependence on the relationship between demographic and role diversity and team performance [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This suggests that the effect is either not linear or subject to other moderators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' At the same time, the CEM also proposes a mechanism by which diversity can harm teamwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' As members grow less similar and bring different perspectives to teamwork, this diminishes performance when the social context of a team encourages social categorization into subgroups and elicits negative inter-group biases and identity threat [29], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This loss of social integration creates more potential for relational conflict and negatively impacts the ability of teams to elaborate information effectively and reduces their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, social moderators like trust and psychological safety allow team members to integrate more effectively to bring diverse perspectives and information-processing together and elaborate on them, which is consistent with the similarity-attraction paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A strength of this integrated approach is that it may explain the conflicting results found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The different mechanisms behind both groups of moderators independently strengthen or diminish the ability of teams to leverage diversity and can work in concert or in opposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, the CEM broadens the discourse around team diversity from a one- IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 4 dimensional approach where it is either a risk or an asset to one where it can be both simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, the CEM has clear, practical implications for diversity management that aim to reduce in-group bias, strengthen social moderators, and match diversity with the nature of the task [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Research Gap & Hypotheses This study aims to address two related research gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The first is that we want to answer the call by Silveira & Prikladnicki [6] and Rodr´ıguez-P´erez, Nadri & Nagappan [15] for more investigations into how diversity affects Agile software teams, and not limited to only gender diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A more comprehensive examination is vital to understand how to design more effective teams and achieve better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The second research gap is that we want to investigate diversity in Agile software teams through the lens of the CEM theory and its opposing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' To answer our research question, we will now develop seven hypotheses we aim to test in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our first hypothesis is that diversity contributes to the effectiveness of Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Because such teams collaborate on complex and interdependent tasks [3], [4], [2], they should benefit from the expanded cognitive resources allowed by heterogeneity in gender, age, cultural background, and role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This reflects one mechanism by which diversity influences team effectiveness and is in accordance with both cognitive resource diversity theory and the CEM that integrates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Hypothesis 1 (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Agile software teams are more effective when they are more diverse in gender (H1a), age (H1b), cultural background (H1c), and role diversity (H1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our second hypothesis concerns the second and opposing mechanism of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' That is, we expect that increased diversity also results in more relational conflict in teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This hypothesis reflects a core consequence of the similarity- attraction paradigm and the CEM that integrates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Hypothesis 2 (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Agile software teams experience more relational conflict when they are more diverse in gender (H2a), age (H2b), cultural background (H2c), and role diversity (H2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Furthermore, we hypothesize that the increased relational conflict, in turn, negatively impacts the effectiveness of teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is consistent with the outcome expected by the similarity-attraction paradigm and the CEM that integrates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Hypothesis 3 (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Relational conflict reduces the effectiveness of Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Following existing literature [30], [31], [23], [46], we expect that psychological safety is a critical factor in enabling team effectiveness through four different mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The first involves a direct effect where psychological safety makes teams more effective by creating more opportunities to openly elaborate information, reconcile conflicting viewpoints, and find creative solutions [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Hypothesis 4 (H4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Psychological safety increases the effectiveness of Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In the second process, psychological safety decreases relational conflict in teams by providing more opportunities to air grievances and discuss the tension between members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Hypothesis 5 (H5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Psychological safety reduces the amount of relational conflict in Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Concerning diversity, we expect that psychological safety is a social moderator of the association between diversity and team effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Consistent with the CEM and Diegmann & Rosenkranz [34], we anticipate that psychological safety is a social moderator that creates an environment where diverse teams can more effectively elaborate task-related information and and experience less relational conflict than less diverse teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Hypothesis 6 (H6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The relationship between diversity in gender (H6a), age (H6b), cultural background (H6c), and role (H6d) on the one hand and team effectiveness on the other is moderated by psychological safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Hypothesis 7 (H7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The relationship between diversity in gender (H7a), age (H7b), cultural background (H7c) and role (H7d) on the one hand and relational conflict on the other is moderated by psychological safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our hypotheses are visualized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' RESEARCH DESIGN We conducted a sample study with a sample of Agile software teams to answer our research question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We used Covariance-Based Structural Equation Modeling (CB-SEM) to test our hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This section discusses the sample (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' III-A), measurement instruments (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' III-B), and method of analysis (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Participants We performed our data collection process through a cus- tomized online survey1 between September 2021 and January 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In total, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='827 members from 733 distinct Agile software teams completed the survey in that period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Because the survey is public and accessible to anyone, we cannot properly calculate a response rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Scholars have emphasized that public surveys are more susceptible to careless response [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So we applied several strategies outlined in literature [47] to reduce potential biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' First, we emphasized the anonymous nature of our data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Second, we encouraged honest answers by providing teams with a detailed team-level profile and relevant feedback for their team upon completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Third, we removed 118 participants with a completion time below the 5% percentile (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='87 minutes) or answered very few questions (< 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, 1The GDPR-compliant survey has been designed so that teams can self- assess their Agile development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' It is available at the following URL: www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='scrumteamsurvey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Theoretical model and hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Sub-hypotheses are grouped, and control variables are omitted to retain visual clarity we retained only those teams (161) with at least 4 participating members to ensure a meaningful diversity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The composition of our sample is shown in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several variables in our model were measured at the individ- ual level and aggregated to a higher (team) level in our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Such aggregation is only reasonable when sufficient variance exists at the group level, not just between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So we calculated the Intraclass Correlation (ICC) [48] to determine the proportion of variance at the higher level compared to the total variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The ICC ranged between 35% and 45% for our independent variables, which exceeded the required threshold of 10% suggested by Hair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Since no data was missing, we did not deploy strategies to deal with missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, we performed a posthoc power analysis using G*Power [49], version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We determined that the sample size allows us to correctly capture medium effects (f = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='15) with a statistical power of 96% (1 − β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In other words, the probability of correctly rejecting the null hypothesis is 96% given our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So we are confident that our sample is big enough to provide a reliable outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Measurements Age, gender, role, and cultural diversity: To assess the impact of diversity on team effectiveness, we identified three dimensions of demographic diversity that are commonly studied (age, cultural background, and gender) and one informational dimension (role).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The questions and the available categories are shown in Table 2 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Participants were asked to pick the most appropriate category for age, cultural background, and functional role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' For cultural background, participants were asked to pick the region where they had lived the longest instead of their country or race.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We assume that the region where one has lived the longest most substantially shapes the mental models and perspectives one brings to a team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Gender diversity was operationalized differently due to concerns that asking participants to identify their gender would be in violation of the European General Data Protection Regulation (GDPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Instead, the initiating participant of each team was asked to indicate the gender distribution of men and women at the team level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We recognize there are more genders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, we had to take this shortcut to obtain a reliable statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A team-level indicator for diversity was then created by calculating a Gini-Simpson coefficient with the individual- level responses for age, role, and cultural diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A Gini- Simpson coefficient is a statistical indicator of the diversity of the members in a sample, ranging between 0 (no diversity) and 1 (maximum diversity) [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Team Effectiveness was operationalized similarly to Verwijs & Russo [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Team effectiveness is often defined as “the degree to which a team meets the expectations of the quality of the outcome” [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In this sense, stakeholder satisfaction is the evaluation of team outcomes from the external perspective of stakeholders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=', clients, customers, and users), whereas team morale is the evaluation of team outcomes from the internal perspective of team members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is conceptually similar to how team effectiveness is defined in the “Team Diagnostic Survey (TDS)” [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' For team morale, we used 3 items from the “Utrecht Work Engagement Scale” (UWES) scale [53] that were modified for use in teams by Van Boxmeer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' For stakeholder satisfaction, we used a 4-item scale developed by the authors for another study [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Both measures are self-reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Reliability analysis showed that Team Morale (α = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='910) and Stakeholder Satisfaction (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='832) were consistently measured across participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Such conflicts represent interpersonal incompatibilities be- tween team members that “typically includes tension, animosity, and annoyance among members within a group” [55, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' 258].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Relational conflict was operationalized by adapting three items from a scale developed by Jehn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [55] to measure relationship conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The items were adapted for use in teams by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The reliability of measurements across participants was high (α = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='892).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' H5(-) Psychological Safety Relational Conflict H3 ( [a,b, H2a,b,c,d (+) H6a,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='c,d ( Team Diversity H4 (+) Gender Diversity Team Effectiveness Team Morale H1a,b,c,d (+) Age Diversity Stakeholder Satisfaction Cultural Diversity Role DiversityIEEE TRANSACTIONS ON SOFTWARE ENGINEERING 6 TABLE I COMPOSITION OF THE SAMPLE Variable Category N (%) Respondents 1,118 Teams 161 Respondents per team 4-6 respondents 82 (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='9%) 7-9 respondents 64 (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='8%) 10+ respondents 15 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) Product Type Product for internal users 89 (55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) Product for external users 72 (44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='7%) Scrum Team Size 1-4 members 1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='6%) 5-10 members 129 (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='1%) 11-16 members 26 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='1%) >16 members 5 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='1%) Scrum Team Experience Low 5 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='1%) Moderate 75 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='6%) High 81 (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) Organization Sector Technology 40 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='8%) Financial 29 (18%) Healthcare 18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) Other 74 (46%) Organization Size 1-50 employees 10 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) 51-500 employees 44 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) 501-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='000 employees 50 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='1%) >5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='000 employees 55 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) Unknown 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) Role Diversity Developer 534 (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='8%) Scrum Master 127 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='4%) Product Owner 104 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) Tester 93 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) Analyst 73 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='5%) Visual/UX Designer 38 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='4%) Infrastructure 14 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) Marketeer or sales 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) Other 81 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) Unknown 52 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='7%) Age Diversity 18-25 years 79 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='1%) 26-35 years 494 (44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) 36-45 years 308 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='5%) 46-55 years 128 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='4%) 56-65 years 39 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='5%) 66+ years 3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) Unknown 52 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='7%) Cultural Diversity Western Europe 581 (52%) Eastern Europe 104 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) North America 103 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) Central & South America 48 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='3%) Middle East 30 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='7%) South-East Asia 28 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='5%) South Asia 27 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='4%) East Asia 20 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='8%) Oceania 7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='6%) Africa 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='2%) Other 51 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='6%) Unknown 11 (1%) Gender Diversity 100% men or women 174 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='6%) 80% men and 20% women 766 (68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='5%) 50%-50% men and women 155 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='9%) 20% men and 80% women 16 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='4%) Psychological Safety was operationalized by adapting three items from the ’Inquiry & Dialogue’ scale that was developed by Marsick & Watkins [56] as part of the Dimensions of Organizational Learning Questionnaire (DLOQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The items were adapted for use in teams by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The reliability of measurements across participants was high (α = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='791).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Control Variables: We included two items from the social responsibility scale (SDRS5) [57] to control for socially desirable answers and to control for common method bias [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Analysis We employed Structural Equation Modeling (SEM) with the AMOS software package [59] to analyze the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A strength of SEM is that it is an inherently confirmatory approach that combines multiple linear regressions and confirmatory factor analysis (CFA) with Maximum Likelihood estimation (ML) to produce more consistent and less biased estimates than those derived through Ordinary Least Squares (OLS) that is typically used in multiple regression and ANOVA [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Furthermore, SEM allows researchers to simultaneously test both the struc- tural part of a theory - the relationships between independent and dependent variables - and the measurement model - the inclusion of multiple indicators to measure latent factors [48], [60], [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is particularly useful for psychometric scales that use multiple questions to operationalize an underlying construct, as we do in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In SEM, the statistical model is evaluated through several “Goodness of Fit” indices and the statistical significance and effect size of individual paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The aim is to arrive at a model as parsimonious as possible while providing a good fit (and thus explanatory power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We discuss the fit indices in section III-D Next, we tested our data for the necessary statistical as- sumptions required for Structural Equation Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' First, we assessed normality by comparing our independent, dependent, and control variables against recommended thresholds for kurtosis (< 3) and skew (< 2) [62] in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This was satisfactory for all variables except cultural diversity, which distribution was strongly leptokurtic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This means that only a few teams showed some heterogeneity in cultural diversity, whereas most were completely homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Although statis- tical transformations can re-normalize such distributions, this also inevitably complicates their interpretation, especially the comparison with other effects in a model [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our measure for gender diversity was not continuous but ordinal (no diversity, some diversity, or high diversity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We treated this variable as continuous in our analyses because such a model is more parsimonious than one that treats it as categorical [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' It also simplifies the interpretation and retains more information than a model where the ordinal variable is treated as categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, this requires that the relationship between the dependent variable and the ordinal independent variable is linear and that each step is approximately evenly spaced [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The relationship was linear, but the steps were relatively not evenly spaced (respectively .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='33 and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='18 for both steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, the modest violation did not warrant using a less parsimonious model with categorical dummy variables for gender diversity instead of a single variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We assessed homoscedasticity by inspecting the scatter plots for all pairs of independent and dependent variables for inconsistent patterns but found none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, multicollinearity was assessed by entering all independent variables one by one into a linear regression [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The Variance Inflation Factor (VIF) remained below the critical threshold of 10 [48] for all measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, we did observe modest multicollinearity of psychological safety with our indicators of diversity (age, cultural background, role, and gender) and our control variable IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 7 for the experience of teams with Scrum, with a VIF ranging between 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='38 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='77 for these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Using a single method - like a questionnaire - introduces the potential for a systematic response bias where the method itself influences answers [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' To control for such common method bias, the recommended approach in current literature is using a marker variable that is theoretically unrelated to other factors in the model [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We included two items from the social responsibility scale (SDRS5) [57] and found a small but significant unevenly distributed response bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Following recommendations in the literature, we retained the marker variable “social desirability” in our causal model to control for common method bias [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We created a full latent variable model containing both the measurement and structural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The measurement model defines relationships between indicator variables (survey items) and underlying first-order latent factors and effectively acts as a CFA-model [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The structural model defines the hypothesized relations between latent variables and is a regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This approach makes the results less prone to convergence issues because of low indicator reliability and offers more degrees of freedom to the analysis compared to a non-latent model [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We began by assessing the measurement model following the approach outlined in literature [60], [61], [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Psychological safety, relational conflict, team morale, stakeholder satisfaction, and psychological safety were entered as first-order latent factors, with their respective survey items as indicator variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Once the measurement model exhibited a good fit (see section III-D), we added the structural part of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In the structural part of the model, we created a second- order latent factor to reflect the composite nature of “team effectiveness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The first-order latent factors for team morale and stakeholder satisfaction were modeled as indicators, similar to Verwijs & Russo [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We calculated interaction terms by multiplying each team’s standardized factor score for psychological safety with their standardized scores for each diversity indicator (age, gender, role, cultural background) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The diversity indicators and the interaction terms were entered into the model as exogenous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The exogenous variables for psychological safety, the diversity indicators, and their interaction terms were allowed to co-vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' No covariances were allowed between endogenous variables as our model predicted specific paths between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Model fit evaluations We assessed reliability, convergent, and discriminant validity for the resulting measurement model before testing for the model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The individual steps involved in the model-fitting process are in Table 1 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Discriminant validity was assessed by analyzing the heterotrait-monotrait ratio of correlations (HTMT) with a third-party plugin in AMOS [69] and following the approach outlined in literature [48], [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This ratio between trait correlations and within trait correlations should remain below R = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='90 to indicate good discriminant validity from other constructs in different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This was the case for all measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We assessed convergent validity by inspecting composite reliability (CR) and average extracted variance (AVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The AVE remained above the rule of thumb of > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='50% [48] for all pairs of factors, ranging between .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='621 and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The CR was equal to or above the threshold of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='70 [48] for all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We then proceeded with the fitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We investi- gated local fit by inspecting the residual covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A standardized residual covariance is considered large when it exceeds 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='58 [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This indicates that an item does not sufficiently measure (only) its intended factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' One item from Stakeholder Satisfaction (SH3) showed poor local fit, and we removed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The overall goodness of fit was evaluated with indices recommended by recent literature [66], [60], [48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' the Compar- ative Fit Index (CFI), the Root Mean Error of Approximation (RMSEA), the Standardized Root Mean Residual (SRMR) and the Tucker Lewis Index (TLI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A commonly used index that we reported but did not test for was χ2 (CMIN) and its corollary CMIN/df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' These indices are highly susceptible to type I errors in larger samples (N > 400, [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So instead, we used the Comparative Fit Index (CFI) [71], which offers a similar test but with consideration of the sample size and its reliable properties have made it the most commonly used index today [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A cut-off value of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='95 or higher is generally considered to indicate good fit [60], [48], [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The Root Mean Error of Approximation (RMSEA) by Steiger & Lind [73] also provides an index that considers sample size but adds to this a parsimony adjustment that leads it to favor the simplest model out of potential models with the same explanatory power [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A value below .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='05 is generally considered to indicate a good fit [60], [48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' additionally, we follow the advice to report the confidence interval in addition to only the absolute value [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The Standardized Root Mean Residual (SRMR) calculates a standardized mean of all the differences (residuals) between each observed covariance and the hypothesized covariance between variables [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A value below .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='08 is indicative of a good fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We also inspected local fit by looking at the standardized residuals between pairs of variables, with values beyond 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='58 as a cut-off value for poor local fit [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, we report and test the Tucker-Lewis Index (TLI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is another incremental fit index, like the CFI, that compares the relative improvement of the hypothesized model from a model where all variables are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Hair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [48] considers a value of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='97 or above sufficient to conclude a good model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In addition to overall model fit, we also evaluated our model on the percentage of variance that is explained in team effectiveness by all other variables in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The measurement model fitted our data well (Chi2(79) = 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='650;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' TLI = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' CFI = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' RMSEA = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='062;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' SRMR = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='0516).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A Confirmatory Factor Analysis (CFA) is reported in the Appendix (Table 3) that shows that all items loaded primarily on their intended factors, except for PS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This item also loaded negatively on the factor for Relational Conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The cumulative Eigenvalues of 5 factors explain 78% of the total observed variance, which is well beyond the recommended threshold of 60% [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We then tested the path model for the effects we pre- dicted from our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our hypothesized theoretical model IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 8 TABLE II SCALES USED IN THE SURVEY STUDY, ALONG WITH ATTRIBUTION, NUMBER OF ITEMS, AND RELIABILITY (CRONBACH’S ALPHA) BASED ON RESPONDENT-LEVEL RESPONSE DATA (N = 1, 118) Construct variable Items adapted from # Items Alpha Psychological Safety Adapted from ’Inquiry & Dialogue’ scale in DLOQ [56] 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='791 Relational Conflict Adapted from Jehn [55] 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='892 Stakeholder Satisfaction Created by authors from our case studies 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='874 Team Morale Adapted from Van Boxmeer et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [54] and Schaufeli [53] 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='910 Social Desirability Highest-loading items from SDRS-5 scale [57] 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='672 fits the data well on each fit indices, as described in Ta- ble IV: Chi2(129) = 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='282;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' TLI = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' CFI = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' RMSEA = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='036;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' SRMR = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The predictors in our model explain respectively 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='7% of the variance in the latent factor representing team effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' For studies in the social sciences, values above 26% are considered large [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' RESULTS We now turn to the results and hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The means, standard deviations, and Pearson correlations of all variables are reported in Table III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Significant effects are also visualized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Following recommendations in statistical literature [60], [66], we used a bootstrapping procedure with 2,000 samples and 95% bias-corrected confidence intervals to more accurately estimate parameters and their p-values for direct effects, factor loadings, and the hypothesized indirect effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This resulted in a standardized, bias-corrected estimate (β) for each path, along with a p-value to test whether the null hypothesis can be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The parameter estimates relevant to our hypotheses are reported in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our results allowed us to reject the null hypotheses for 4 out of 19 (sub)hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The primary hypothesis of this study is that diversity makes Agile software teams more effective because it broadens the cognitive resources available for information processing (H1a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is partially true for our results, as only age diversity significantly contributes to team effectiveness (H1b, β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='213, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Agile software teams seem slightly more effective when there is greater age heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Nevertheless, heterogeneity in gender, cultural background, or role does not appear relevant to team effectiveness (H1a,c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We also hypothesized that relational conflict in teams would increase as heterogeneity increases and members become less similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is also partially true, as only gender di- versity significantly contributes to relational conflict (H2a, β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='161, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, there appears to be more conflict as teams grow more heterogeneous in gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' There is no discernible effect of diversity in age, cultural background, or functional role on conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' (H2b,c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Contrary to our expectations, we did not find a significant effect between relational conflict and team effectiveness (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Teams that experience more relational conflict do not seem to be more or less effective than teams that experience less conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, the results show a strong positive effect of psychological safety on team effectiveness (H4, β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='660, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Teams that experience more psychological safety are more effective in that they have reported more satisfied stakeholders and higher team morale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Psychological safety also strongly decreases the amount of relational conflict reported by teams (H5, β = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='636, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, we hypothesized that psychological safety moderates the strength by which diversity contributes to team effectiveness and relational conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, none of the interactions were significant (H6a-d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' H7a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' DISCUSSION This study investigated how diversity in age, role, cultural background, and gender influences the effectiveness of Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='118 respondents from 161 Agile software teams participated in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Overall, our results provide mixed support for both the benefits and the risks of member heterogeneity in teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' According to the categorization-elaboration model (CEM)[14] and cognitive resource diversity theory, we hypothesized that Agile software teams benefit from diversity as it expands the cognitive resources available for information processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, only age diversity directly improves team effectiveness directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This finding is consistent with the conclusions from a recent review of the literature by Tshetshema & Chan [10], and a meta-analysis of 74 studies by Schneid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [16], particularly for complex tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, another meta-analysis of 35 studies by Horwitz & Horwitz [12] found no positive impact of demographic diversity (age, gender, race).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So our results are more nuanced than the overall positive effect of team diversity that is reported by Lee & Xia [26] for Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We also did not find a positive effect of gender diversity or cultural diversity, whereas others did [10], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' All in all, the association between demographic diversity and team effectiveness is more complicated than the direct, positive effects we hypothesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In addition to demographic diversity, we also investigated how role diversity improves team effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Agile software methodologies emphasize this type of diversity as an important characteristic of autonomous teams [78], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In line with cognitive resource diversity theory, role diversity allows teams to leverage more perspectives and broader informational resources to resolve complex problems [14], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' When members bring more functional roles to their work together (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=', analyst, tester, developer, designer), their shared mental models will be richer than when all members hold the same role (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=', developer) [11], [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, we did not find evidence for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Teams with high role diversity were not IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 9 TABLE III MEANS, STANDARD DEVIATIONS, SKEWNESS, KURTOSIS AND CORRELATIONS (PEARSON) FOR CONTINUOUS VARIABLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' CORRELATIONS MARKED WITH ARE SIGNIFICANT AT p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Variable Mean SD Skewness Kurtosis 1 2 3 4 5 6 7 8 1 Gender Diversity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='97 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='55 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 2 Age Diversity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='71 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='78 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='75 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='26* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 3 Cultural Diversity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='97 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='98 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='03 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 4 Role Diversity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='61 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 5 Psychological Safety 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='54 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='66 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='04 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 6 Team Effectiveness 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='36 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='71 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='38 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='17 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='08 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='04 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='72* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 7 Relational Conflict 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='45 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='56 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='71* .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='50* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 Control Variables 8 Social Desirability 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='72 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='48 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='17 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='03 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='59* .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='56* .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='40* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='00 TABLE IV MODEL FIT INDICES Model fit index Value Interpretation Chi-Square ( χ2) 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='282 n/a Degrees of freedom (df) 129 n/a CMIN/df 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='211 A value below 5 indicates an acceptable model fit [76], below 3 a good fit [77] Root Mean Square Error of Approximation (RMSEA) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='036 Values ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='05 indicates good model fit [60] RMSEA 90% Confidence Interval .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='000-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='055 p of Close Fit (PCLOSE) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='873 Probability that RMSEA ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='05, where higher is better Comparative Fit Index (CFI) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='988 Values ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='97 indicates good model fit [48] Tucker Lewis Index (TLI) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='981 Values ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='97 indicates good model fit [48] Standardized Root Mean Square Residual (SRMR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='051 Values ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='08 indicates good model fit [48] Variance explained by predictors (R2) of Team Effectiveness 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='7% Values ≥ 26% indicates large effect [75] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Standardized path coefficients for the model (∗∗ : p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01, ∗ : p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The dotted lines represent non-significant results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Indicator items and non-significant paths for sub-hypotheses are omitted to improve readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A detailed overview of the individual hypotheses is reported in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' H5(-)-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='636** Psychological Safety Relational Conflict H3 1 Team Diversity Gender Diversity Team Effectiveness Team Morale H1b: age (+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='213* Age Diversity Stakeholder Satisfaction Cultural Diversity Role DiversityIEEE TRANSACTIONS ON SOFTWARE ENGINEERING 10 TABLE V PARAMETER ESTIMATES, CONFIDENCE INTERVALS, STANDARD ERRORS, STANDARDIZED COEFFICIENTS FOR DIRECT EFFECTS, INTERACTION TERMS AND INDIRECT EFFECTS FOR HYPOTHESES (STATISTICALLY SIGNIFICANT HYPOTHESES AT p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='05 ARE SET IN BOLDFACE), AND FACTOR LOADINGS Parameter Unstandardized 95% CI SE p Standardized Direct Effects H1a: Gender Diversity → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='037 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='075, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='123) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='075 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='622 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='056 H1b: Age Diversity → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='391 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='077, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='800) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='077 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='041 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='213 H1c: Cultural Diversity → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='024 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='382, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='542) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='382 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='872 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='010 H1d: Role Diversity → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='022 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='315, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='290) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='315 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='956 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='013 H2a: Gender Diversity → Relational Conflict .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='241 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='108, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='417) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='108 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='008 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='161 H2b: Age Diversity → Relational Conflict .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='490 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='178, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='026) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='178 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='118 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='117 H2c: Cultural Diversity → Relational Conflict .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='032 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='824, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='824 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='855 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='006 H2d: Role Diversity → Relational Conflict .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='332 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='834, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='199) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='834 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='306 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='087 H3: Relational Conflict → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='035 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='091, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='181) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='091 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='747 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='081 H4: Psychological Safety → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='574 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='300, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='927) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='300 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='004 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='660 H5: Psychological Safety → Relational Conflict 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='262 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='888, -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='727) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='888 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='001 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='636 Interactions H6a: Gender Diversity * Psychological Safety → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='018 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='093, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='037) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='093 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='550 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='052 H6b: Age Diversity * Psychological Safety → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='007 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='069, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='086) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='069 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='885 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='020 H6c: Cultural Diversity * Psychological Safety → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='042 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='159, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='061) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='159 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='388 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='081 H6d: Role Diversity * Psychological Safety → Team Effectiveness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='026 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='070, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='082) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='070 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='550 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='076 H7a: Gender Diversity * Psychological Safety → Relational Conflict .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='057 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='138, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='054) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='138 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='416 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='072 H7b: Age Diversity * Psychological Safety → Relational Conflict .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='014 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='095, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='129) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='095 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='812 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='017 H7c: Cultural Diversity * Psychological Safety → Relational Conflict .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='133 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='047, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='377) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='047 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='206 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='112 H7d: Role Diversity * Psychological Safety → Relational Conflict .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='044 (-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='162, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='046) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='162 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='406 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='057 Factor loadings from first to second-order factors Team Effectiveness → Stakeholder Happiness .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='752 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='144, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='614) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='114 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='003 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='389 Team Effectiveness → Team Morale 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='000 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='618, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='800) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='873 more or less effective than teams with lower role diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is partially consistent with extant literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Homberg & Bui [21] found no evidence for a link between role diversity and team performance in a meta-analysis of other empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Horwitz & Horwitz [12] also did not find an effect on performance, although they did find one on the quality of work done by teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Diversity in teams is often considered a double-edged sword in the literature on diversity [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The CEM proposes that diversity can also harm team effectiveness through the similarity-attraction paradigm [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' As members grow less similar and bring different perspectives to teamwork, there is more potential for tension and conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This decreases the ability of teams to elaborate information effectively and reduces their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Concerning the first assertion, our results show that gender diversity does increase relational conflict but not other kinds of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This finding is consistent with some studies [19], but not others [8], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Regarding the second assertion, we failed to find any impact of relational conflict on team effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So while it appears true that gender diversity increases relational conflict in teams to some extent, we cannot conclude that this also harms team effectiveness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=', the double-edged sword).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The CEM attempts to reconcile the conflicting results by drawing attention to social- and task-related moderators that shape how diversity impacts team performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We investigated one social moderator frequently associated with diversity, relational conflict, and team effectiveness: psychological safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We hypothesized that a psychologically safe environment would make it easier for diverse teams to elaborate on task information effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Although psychological safety reduced relational conflict and improved team effectiveness, we could not reject the null hypotheses for psychological safety as a moderator of the diversity-effectiveness link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In summary, our results show some benefits of diversity (age) on team effectiveness and some risks of diversity through relational conflict (gender).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Psychological safety also reduces relational conflict and increases team effectiveness, but we found no evidence for a moderating role in the diversity-effectiveness link or the diversity-conflict link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Alternative explanations The mixed evidence suggests that there are factors at work that moderate or mediate the effects of diversity on effectiveness and conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Diversity alone does not make teams more effective because it broadens cognitive resources, just as it does not inherently and consistently create conflict because members are less similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This study investigated psychological safety as one potential social moderator of the diversity-effectiveness link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our mixed results suggest that other moderators are at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' One example of this is task interdependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A core element of Agile software methodologies is that teams work together on complex tasks [80], [2], [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Collective elaboration of task-related information and the pooling of skills to accomplish tasks is also a common thread in the definition of teamwork [51], [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Without task interdependence, the two mechanisms of IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 11 diversity diminish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Because there is less collective elaboration, the benefits of the broadened cognitive resources that are offered by diversity diminish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Furthermore, a major source of conflict between members is removed because they spend much less time together processing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Members may have more skin in the game when they feel they depend on others in their team to be successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Paradoxically, this may surface as a higher degree of relational conflict than teams with very low interdependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In this sense, psychological safety is likely only relevant as a moderator of the diversity-effectiveness link in teams with high task interdependence but not low task interdependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Future studies can investigate if the effects of diversity and psychological safety are more pronounced when controlling for task interdependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Another explanation may be that the effect of diversity on team effectiveness is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several authors [82], [83] have argued for curvilinear models where diversity contributes to performance only when it is moderated (inverted U) or when it is either low or high (upright U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Which model applies varies by diversity type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' For example, Dahlin, Weingart & Hinds [43] found that educational diversity contributed to team performance when it was either low or high (inverted U) but found the opposite for national diversity (upright U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Richard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [84] found that management teams with moderate gender diversity performed better than teams with low or high diversity, but only in high-risk settings (upright U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, diversity in terms of age, gender, or function may contribute to learning behavior in teams more strongly when diversity is low or high but not moderate (inverted U) [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So while there is some support for the curvilinear effects of diversity, the relationship is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' To further complicate matters, the shape of the relationship may also be moderated by the expectations that teams themselves have of the benefits of diversity [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We performed a posthoc test to assess whether a curvilinear relationship between dimensions of diversity and team effectiveness better fitted the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This was not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A quadratic regression model was not significant for the following diversity dimensions: age (R2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='004, F(2, 158) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='321, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='726), gender (R2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='021, F(2, 158) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='695, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='187), culture (R2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='008, F(2, 158) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='664, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='516), and role (R2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='000, F(2, 158) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='025, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, the possibility of a curvilinear relationship rather than a linear one does not appear to explain the lack of results in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We often assume that diversity in age, gender, function, and cultural background inherently leads to a different un- derstanding of the task and potential solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is both the strength and the weakness of diverse teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In the day- to-day practice of teams, such differences in understanding may also lead to conflict if members need to adequately express their view and integrate it with other members into a synthesized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In addition to those mentioned above, task- related and social moderators, it is reasonable to expect that communication and conflict navigation skills are also highly relevant, as well as the presence of an environment where such different understandings can be elaborated effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Few studies have investigated such moderators, particularly for Agile software teams [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Furthermore, this ties into team members’ beliefs about diversity, how to deal with it and whether or not it benefits teamwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Van Knippenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [42], [29] call this a “Diversity Mind-Set”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several studies have shown that teams and organizations can better leverage diversity when they recognize it as a strength and have learned how to appreciate and deal with the resulting informational diversity [86], [84], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' For practitioners, it is important to notice that our results are broadly consistent with existing research, showing that team diversity is not unequivocally beneficial or harmful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Although we found a positive effect of age diversity, the effects of other types of diversity appear to be more conditional on moderating factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several factors have been proposed to date, like the autonomy that teams have [26], task difficulty [11], psychological safety [23], team climate [22] and the beliefs that teams have about diversity [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This suggests that context is just as important as diversity alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Limitations In the following section, we will discuss the threats to the validity of our sample study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Internal validity Internal validity refers to the confidence with which changes in the dependent variables can be attributed to the independent variables and not other uncontrolled fac- tors [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We employed several strategies to maximize internal validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' First, we recognize that online questionnaires are prone to bias and self-selection as a result of their voluntary (non- probabilistic) nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We counteracted this by embedding our questions in a tool that is regularly used by Agile software teams to self-diagnose their process and identify improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Team members were invited by people in their organization to participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Second, we thoroughly cleaned the dataset of careless responses to prevent them from influencing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Third, we did not inform the participants of our specific research questions to prevent them from answering in a socially desirable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We also controlled for social desirability in participants’ responses, as well as common method bias introduced when a single method is used to collect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Despite our safeguards, there may still be confounding variables that we were unable to control for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This is particularly relevant to the operationalization of team effectiveness, which is based on self-reported scores on team morale and the perceived satisfaction of stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Mathieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [88] recognize that such affect-based measures may suffer from a “halo effect”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Future studies could ask stakeholders to rate their satisfaction with team outcomes directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This does not entirely rule out a halo effect but is conceptually closer to what matters to organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Future studies could also find more objective measures for team effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Construct validity Construct validity refers to the degree to which the measures used in a study measure their intended constructs [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We adapted items from established scales to measure psychological safety [89], team effectiveness [30], relational conflict [55] and social desirability [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A con- firmatory factor analysis (CFA) showed that all items were loaded primarily on their intended scales (see Table 3 in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A heterotrait-monotrait (HTMT) analysis confirmed discriminant validity for all measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The reliability for all IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 12 TABLE VI SUMMARY OF KEY FINDINGS & IMPLICATIONS Findings Implications Diversity & team effec- tiveness Based on existing theory, we developed a Structural Equation Model for how diversity and psychological safety interact to impact team effectiveness and relational conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The model fitted the data well (Chi2(129) = 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='282;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' TLI = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' CFI = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' RMSEA = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='036;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' SRMR = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='051).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Age diversity showed a positive association with team effectiveness (β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='213, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='05), but not diversity in gender, role, or cultural background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Teams with members of different age groups will likely benefit from the broader range of tenure and work/life experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The benefits of other types of diversity appear more conditional on moderating factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Organizations can assess the extent to which teams are diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, psychological safety, communication skills, and a diversity mindset seem important moderators that organizations need to provide and encourage teams to leverage it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Diversity & relational conflict Gender diversity was positively associated with relational conflict in Agile software teams (β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='161, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, diversity in role, age, or cultural background did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In turn, relational conflict did not significantly affect team effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' When teams grow more diverse, members’ different perspec- tives may lead to more conflict and friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This appears particularly relevant to gender diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Such negative consequences of diversity may be counteracted when teams learn to see their diversity as a strength and recognize that different perspectives can be reconciled through open dialogue and elaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Psychological safety & team effectiveness Psychological safety was positively associated with team effectiveness (β = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='660, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01) and negatively associated with relational conflict (β = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='636, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='01) Teams that operate in environments where members can openly and safely elaborate information are more effective than other teams, regardless of their diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' They also ex- perience much less relational conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Organizations do well to develop the skills, support structures, and management styles that foster psychological safety in and around teams Psychological safety as a moderator Psychological safety did not significantly moderate the association between diversity and team effectiveness, nor between diversity and relational conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Psychological safety is paramount, but it does not appear to strengthen the cognitive benefits of team diversity, nor does not it appear to buffer against negative consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' measures exceeded the cutoff recommended in the literature (CR >= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='70 [48]), except social desirability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, we are confident that we reliably measured the intended constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' A limitation of our measure for team effectiveness is that it only addressed (self-reported) stakeholder satisfaction and team morale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Although both are reasonable and relevant aspects of team effectiveness and are commonly used in team re- search [51], effectiveness is also a more-faceted construct [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, we could not directly ask participants for their gender due to privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So it was not possible to calculate a Gini index as we did for the diversity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The resulting measure was ordinal instead of continuous, limiting our analysis’s resolution for this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Future studies do well to use a more continuous measure of gender distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Conclusion validity Conclusion validity assesses the extent to which the conclusions about the relationships between variables are reasonable based on the results [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We used Structural Equation Modeling to test the entire model simulta- neously [66], [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The resulting model fits the data well on all fit indices recommended by statistical literature and explains a substantial amount of variance in the dependent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our sample was also large enough to identify medium effects (f = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='15) with a statistical power of 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We published team-level data and syntax files to Zenodo for reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' External validity Finally, external validity concerns the extent to which the results actually represent the broader population [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' First, we assess the ecological validity of our results to be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our questionnaire was integrated into a more general tool that Agile software teams use to improve their processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Participants were invited by people in their organization, usually Scrum Masters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, the data is more likely to reflect realistic teams than a stand-alone questionnaire or an experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We do not know how well our sample reflects the total population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' However, our sample composition (Table I) shows that a wide range of teams participated in the questionnaire, with different levels of experience from different parts of the world and different types of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We also observed a broad range of scores on the various measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This provides confidence that a wide range of teams participated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Furthermore, our sample size and the aggregation of individual- level responses to team-level aggregates reduce variability due to non-systematic individual bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' CONCLUSION A common thread in Agile software methodologies is their emphasis on teams as the primary units where complex work is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So it is not surprising that much research has focused on what makes such teams more effective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' [30], [31], [93], [27], [94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Although diversity is increasingly inves- tigated in the broader literature on teams, scholarly knowledge on how it impacts Agile software teams is still limited [6], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Such understanding can better equip organizations and teams to leverage diversity more effectively or learn when and how diversity is beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Because what seems to be clear about diversity is that while it brings more extensive cognitive resources to teams, it can also bring more conflict as members become less similar [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Several models have been proposed to explain this “double-edged sword” of diversity, with the categorization-elaboration model (CEM) [14] as the most comprehensive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' IEEE TRANSACTIONS ON SOFTWARE ENGINEERING 13 In this study, we explored how diversity impacts the effectiveness of Agile software teams through the lens of the CEM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our sample consisted of 1,118 team members representing 161 Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Our results show that age diversity contributes to more effective teamwork but not diversity in gender, role, or cultural background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This may reflect the value of having more varied levels of experience in teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Furthermore, the CEM also predicts a negative effect of diversity through social categorization and identity threat, which can surface through increased conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' While our results support this effect, we only found evidence for gender diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, the CEM predicts that task- and social moderators influence the impact of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' One such moderator that is frequently studied is psychological safety [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' While our results show that it contributes to more effective teamwork and less conflict in teams, it did not moderate the link between diversity and effectiveness or diversity and conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Thus, the presence of psychological safety in a team does not in itself allow teams to leverage their diversity better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Despite the strong focus on role diversity and cross-functional teamwork in Agile software methodologies [80], [2], we found no apparent effect on team effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' So while our results are broadly consistent with the CEM for age and gender diversity, it is surprising that heterogeneity in role or cultural background did not produce similar effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' One moderator that may be particularly relevant here is task interdependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Teams vary broadly in the degree to which members actually (need to) work together on tasks and, thus, the opportunities that arise to leverage the broader cognitive resources of diverse teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' This study has several implications for future studies of how diversity impacts the effectiveness of Agile software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' First, the role of task-related and social moderators should be investigated more thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The categorization- elaboration model [14] provides a valuable framework for such research because it integrates the opposing mechanisms of diversity proposed by cognitive resource diversity theory and the similarity-attraction paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' From a practical viewpoint, such research can also drive the development of training and methods to help teams and organizations to leverage their diversity on all sorts of dimensions, and not limited to gender, age, cultural background, and functional role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Second, more attention should be paid to the beliefs that teams have about diversity and its effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Such a “Diversity Mind-Set” [29] can act as a powerful moderator by making teams aware of their diversity and how it can expand their experience as a team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Finally, future research should investigate broader definitions of performance and effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' In this study, we mainly focused on stakeholder satisfaction and team morale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Since effectiveness is a multi-faceted construct [90], we likely missed aspects that are affected by diversity in teams, like speed, quality, or innovativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors would like to thank all participants in our study for their efforts and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' We would also like to thank The Liberators BV and its community of patrons for funding part of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' SUPPLEMENTARY MATERIALS A replication package for the sample study is available at the following DOI to support Open Science: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='7537784 under a CC-BY-NC-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' The package includes the model definitions (AMOS), syntaxes for SPSS, and a fully anonymized, cleaned, and aggregated dataset of the analyzed teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' RESPONSIBLE DISCLOSURE Data has been collected and stored according to the policy for research data management of Aalborg University, respecting the total anonymity of informants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNFOT4oBgHgl3EQf5jRf/content/2301.12954v1.pdf'} +page_content=' Christiaan Verwijs has a financial interest in The Liberators BV.' metadata={'source': 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b/V9FKT4oBgHgl3EQfnC5b/content/tmp_files/2301.11860v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bcbe5e93e7c75b91eb218d5d3bf89e5afcf68bcd --- /dev/null +++ b/V9FKT4oBgHgl3EQfnC5b/content/tmp_files/2301.11860v1.pdf.txt @@ -0,0 +1,4871 @@ +arXiv:2301.11860v1 [cs.LO] 27 Jan 2023 +The Cartesian Closed Bicategory +of Thin Spans of Groupoids +Pierre Clairambault +Aix Marseille Univ, Universit´e de Toulon, CNRS, LIS, Marseille +Email: Pierre.Clairambault@cnrs.fr +Simon Forest +Aix Marseille Univ, CNRS, I2M, Marseille, France +Email: Simon.Forest@univ-amu.fr +Abstract—Recently, there has been growing interest in bicat- +egorical models of programming languages, which are “proof- +relevant” in the sense that they keep distinct account of execution +traces leading to the same observable outcomes, while assigning +a formal meaning to reduction paths as isomorphisms. +In this paper we introduce a new model, a bicategory called +thin spans of groupoids. Conceptually it is close to Fiore et +al.’s generalized species of structures and to Melli`es’ homotopy +template games, but fundamentally differs as to how replication of +resources and the resulting symmetries are treated. Where those +models are saturated – the interpretation is inflated by the fact +that semantic individuals may carry arbitrary symmetries – our +model is thin, drawing inspiration from thin concurrent games: +the interpretation of terms carries no symmetries, but semantic +individuals satisfy a subtle invariant defined via biorthogonality, +which guarantees their invariance under symmetry. +We first build the bicategory Thin of thin spans of groupoids. +Its objects are certain groupoids with additional structure, its +morphisms are spans composed via plain pullback with identities +the identity spans, and its 2-cells are span morphisms making +the induced triangles commute only up to natural isomorphism. +We then equip Thin with a pseudocomonad !, and finally show +that the Kleisli bicategory Thin! is cartesian closed. +I. INTRODUCTION +The relational model [1] is one of the most basic and +elementary denotational models for linear logic or the λ- +calculus. At its heart, it is simply an interpretation of formulas +/ types as sets and proofs / programs as relations, i.e. in the +category Rel. Despite its simplicity the relational model is +ubiquitous: it is the basic substrate for the spectrum of so- +called web-based models of linear logic, including coherence +or finiteness spaces [2]. It faithfully predicts reduction time +[3]. It supports quantitative extensions such as in probabilistic +coherence spaces [4], the weighted relational model [5], and +even up to quantum computation [6] – quantitative extensions +which enjoy powerful full abstraction results [7], [8]. Presented +syntactically, the relational model exactly corresponds to non- +idempotent intersection types [9], a currently active research +topic in its own right (see e.g. [10], [11]) which enables +a syntactic methodology to addressing semantic questions. +Finally, it has a tight connection with game semantics [12], +[13], of which it appears as a desequentialization (see e.g. [8], +[14]–[16]). In short, it is at the crossroads of multiple topics, +past and current, of the denotational semantics universe. +Another recent trend in denotational semantics is the adop- +tion of bicategorical models [17] where the familiar categor- +ical laws hold only up to certain 2-cells satisfying coherence +conditions – in particular, Fiore and Saville have recently thor- +oughly explored cartesian closed bicategories [18]. In such +models, the denotation is no longer an invariant of reduction: +two convertible terms yield merely isomorphic objects, and +reduction paths have a genuine interpretation as specific iso- +morphisms [19] – thus bringing reduction into the categorical +model. There are still not many concrete bicategorical models, +and we are aware of only three (families of) such models that +can deal with non-linear computation, in chronological order: +firstly, Fiore, Gambino, Hyland and Winskel’s cartesian closed +bicategory of generalized species of structure [20]; secondly, +Castellan, Clairambault and Winskel’s thin concurrent games +[21] (as established by Paquet [22]); thirdly, Melli`es’ homo- +topy template games [23]. Of these three, the first is by far +the most studied with various works including generalizations +and application to semantics [24]–[26], links with intersection +types and Taylor expansion [27], [28], or applications to +the pure λ-calculus [29]. Beyond giving a non-degenerated +interpretation to reduction paths, those concrete bicategorical +models are “proof-relevant”, in the sense that they keep distinct +semantic witnesses for the possibly multiple evaluation traces +with the same observable behaviour and thus keep a clear, +branching account of non-determinism. +These models have something else in common: in their +construction, the main subtlety has to do with replication, i.e. +the modality ! of linear logic. In the relational model, !A is +the set M(A) of finite multisets of elements of A, or alter- +natively, the free monoid A∗ quotiented by permutations. In +bicategorical models, this is replaced by a categorification of +M(A): a category (or groupoid) whose objects keep separate +individual resource usages (e.g. A∗). Its morphisms are explicit +permutations, often called symmetries in this paper. Individuals +in the model must refer to specific resources (e.g. ai in +a1 . . . an ∈ A∗), but the categorical laws expected for mod- +els of programming languages requires that their behaviour +should still be invariant under symmetry. In both generalized +species of structure and template games, this is done by +saturating the set of witnesses with respect to symmetries: +intuitively, the behaviour of an individual cannot depend on +the specific identity of resources, because those resources +are seen through the “noise” of all possible symmetries – +this shall be reviewed gently in Section II. This saturation +complicates models and their construction, though for good +reasons. But this contrasts with thin concurrent games, which + +handles symmetry with a mechanism inspired by Abramsky- +Jagadeesan-Malacaria games [12] and Melli`es’ orbital games +[15]: strategies are not saturated, but their invariance under +“Opponent’s symmetries” is ensured by a subtle bisimulation- +like structure – we call this the thin approach. +We believe that the thin approach is helpful at least for appli- +cations to semantics: the absence of symmetries on witnesses +allow a more concrete flavour which may help when order- +ing individuals allowing continuous reasoning1, or simplify +quantitative extensions such as [24]. But more fundamentally, +there is a clear tension between these two worlds that deserves +investigation. Are proof-relevant relational models inherently +saturated? Is the thin approach only possible in games thanks +to the presence of time and causality? These fundamental +questions may be of interest beyond denotational semantics, as +the handling of symmetry in such models is deeply connected +to algebraic combinatorics [20] and homotopy theory [23]. +a) Contributions: We introduce the bicategory Thin of +thin spans of groupoids: its objects are certain groupoids with +additional structure, its morphisms certain spans, and its 2- +cells certain weak span morphisms, i.e. making the induced +triangles commute up to chosen natural isos. Identities are +identity spans, and composition of spans is by plain pullback. +Of course, plain pullbacks are too weak to support the +horizontal composition of weak span morphisms. To address +this, we first define uniform spans via a biorthogonality con- +struction, ensuring that the composition pullbacks also satisfy +the bipullback universal property. This allows us to compose +2-cells horizontally, but that horizontal composition is still not +canonically defined and fails to give a bicategory. +For the next step, we import from thin concurrent games and +from Melli`es’ orbital games a decomposition of symmetries +into positive symmetries (due to the program), and negative +symmetries (due to the environment). We then define thin +spans via a second biorthogonality construction, which ensures +that the horizontal composition of weak span morphisms are +canonically defined as long as we consider positive weak +span morphisms, where the chosen iso only involves positive +symmetries. We show this results in a bicategory Thin. +Furthermore, we equip Thin with a pseudocomonad !, and +show that the Kleisli bicategory Thin! is cartesian closed. +b) Outline: In Section II we start with a gentle introduc- +tion to the relational model and its proof-relevant extensions. +In Section III we introduce the bicategory Thin, deploying +first the uniform orthogonality and then the thin orthogonality. +In Section IV we introduce the pseudocomonad !, and show +that the Kleisli bicategory Thin! is cartesian closed. +II. RELATIONAL MODELS, SPANS, SPECIES +A. The Relational Model +The relational model is one of the simplest denotational +models of the λ-calculus, linear logic, or simple programming +1For instance, in [29], the generalization from finite to infinite computation +is not simply by continuity as per usual in denotational semantics, because +of the quotient involved in the management of saturation. +languages such as PCF. It consists in simply interpreting every +type A as a set �A�, and a program ⊢ M : A as a subset of +�A�. This set �A� is often called the web seeing that it is the +first component of the so-called web-based models of linear +logic such as coherence spaces and their extensions. One may +think of elements of �A� as completed executions (which is +straightforward enough for ground types such as booleans or +natural numbers but may be more complex for higher-order +types), and of �M� ⊆ �A� as simply the collection of all the +completed executions that M may achieve. +Example 1. The ground type for booleans is interpreted as +�B� = {tt, ff}, and the constant ⊢ tt : B as �tt� = {tt}. +The interpretation of a program M is computed composi- +tionally, following the methodology of denotational semantics, +organized by the categorical structure of sets and relations. +1) Basic categorical structure: There is a category Rel +with sets as objects, and as morphisms the relations from A +to B, i.e. subsets R ⊆ A×B. The identity on A is the diagonal +relation {(a, a) | a ∈ A} ⊆ A × A, and the composition of +R ⊆ A×B and S ⊆ B×C consists in all pairs (a, c) ∈ A×B +such that (a, b) ∈ R and (b, c) ∈ S for some b ∈ B. +Besides, Rel has a monoidal structure given by the +cartesian product on objects, and for Ri ∈ Rel(Ai, Bi), +R1 × R2 ∈ Rel(A1 × A2, B1 × B2) set as comprising all +((a1, a2), (b1, b2)) when (ai, bi) ∈ Ri – the unit I is a fixed +singleton set, say {∗}. Additionally, Rel is compact closed: +each set A has a dual A∗ defined simply as A itself, and there +are relations ηA ∈ Rel(I, A × A) and ǫA ∈ Rel(A × A, I), +both diagonal relations, satisfying coherence conditions [30]. +In particular, Rel is ⋆-autonomous and as such a model of +multiplicative linear logic, and the linear λ-calculus: the linear +arrow type is interpreted as �A ⊸ B� = �A� × �B�. Finally, +Rel has finite products, with the binary product of sets A and +B given by the disjoint union A + B = {1} × A ⊎ {2} × B. +2) The exponential modality: The exponential modality of +Rel is based on finite multisets. If A is a set, we write M(A) +for the set of finite multisets on A. To denote specific multisets +we use a list-like notation, as in e.g. [0, 1, 1] ∈ M(N) – we +write [] ∈ M(A) for the empty multiset. +For A a set, its bang !A is simply the set M(A). This ex- +tends to a comonad on Rel, satisfying the required conditions +to form a so-called Seely category – in particular, there is +M(A + B) ∼= M(A) × M(B) +a bijection providing the Seely isomorphism. Altogether, this +makes Rel a model of intuitionistic linear logic; and this +makes the Kleisli category Rel! cartesian closed so that we +may interpret (among others) the simply-typed λ-calculus. +Example 2. Considering the term ⊢ M : B → B of PCF +⊢ λxB. if x thenx +elseif x thenff elsett : B → B , +we have �M� = {([tt, tt], tt), ([tt, ff], ff), ([ff, ff], tt)}. +Here we can observe that the model is quantitative, in that +it records how many resources each execution consumes: one + +may observe output tt either with two evaluations of x to tt, +or with two evaluations of x to ff. One may observe output +ff with two evaluations of x, one to tt and one to ff. Recall +that in [tt, ff] = [ff, tt], the order is irrelevant. +The relational model also supports the interpretation of non- +determinism: if ⊢ choice : B is a new primitive evaluating +non-deterministically to tt or ff, then we may simply set +�choice� = {tt, ff} . +3) Extensions of the relational model: The relational model +is extremely flexible, and can be extended in multiple different +ways. In one direction one may add to the objects a coherence +relation and restrict to compatible morphisms – we obtain in +this way (multiset-based) coherence semantics. +Another extension is the weighted relational model [5], [31] +where a term ⊢ M : A, instead of denoting a subset of �A� – +i.e. a function �M� : �A� → {0, 1} – denotes a function +�M� : �A� → R +assigning to each point of the web a ∈ �A� a weight �M�a ∈ +R. The weight may be used to record additional information +about executions. One may record the number of distinct non- +deterministic branches leading to a certain result: for instance, +if R = N ∪ {+∞}, then �if choice thentt elsett�tt = 2. +With R = R+ = R+ ∪ {+∞}, we may track the probability +with which a certain result occurs, obtaining a model fully +abstract for probabilistic PCF [7]. The paper [5] contains other +examples: resource consumption, must convergence, etc. +It is natural to go one step further and make the relational +model “proof-relevant”. This means not merely recording a +weight or counting non-deterministic branches, but keeping +track of a set �M�a ∈ Set of witnesses of the execution of +M to a, for each ⊢ M : A and a ∈ �A�. There are well- +documented ways to do that which we shall review later on, +but for now let us attempt this naively. +B. The Bicategory of Spans +A first idea is to simply replace relations with spans. +1) Spans: Recall that if C is a category with pullbacks, then +we form Span(C) has having as objects those of C, and as +morphisms from A to B triples (S, ∂S +A, ∂S +B) forming a diagram +A +∂S +A +← +S +∂S +B +→ +B , +where intuitively S is a set of internal witnesses, projected to +A and B via the maps ∂S +A and ∂S +B. For C = Set one obtains a +relation by collecting the pairs (∂S +A(s), ∂S +B(s)) for s ∈ S, but +we have more: for each pair (a, b) ∈ A × B we have +witS(a, b) = {s ∈ S | ∂S +A(s) = a & ∂S +B(s) = b} , +a set of witnesses that a and b are related – hence this indeed +provides a notion of a proof-relevant relational model. +Example 3. Writing B = {tt, ff} and 1 = {∗}, we may +represent the program ⊢ if choice thentt elsett as +1 +∂l +← +{a, b} +∂r +→ +B +a span, where ∂l(a) = ∂l(b) = ∗, ∂r(a) = ∂r(b) = tt. +Thus, the evaluation of the program to tt has two witnesses. +2) A bicategory: The exact identity of S does not matter – +the same span above with S′ = {a′, b′} should not be treated +distinctly. A morphism between spans is f : S → S′ making +S +A +B +S′ +∂S +A +∂S +B +f +∂S′ +A +∂S′ +B +commute; an isomorphism of span is an invertible morphism. +The identity span on A is simply A ← A → A with +two identity maps. The composition of A ← S → B and +B ← T → C is obtained by first forming the pullback +T ⊙ S +S +T +A +B +C +l +r +∂S +A +∂S +B +∂T +B +∂T +C +(1) +and setting ∂T ⊙S +A += ∂S +A ◦ l and ∂T ⊙S +C += ∂T +C ◦ r – for +Span(Set), this means that T ⊙ S has elements all pairs +(s, t) such that ∂S +B(s) = ∂T +B(t), projected to A and C via +∂T ⊙S +A +((s, t)) = ∂S +A(s) and ∂T ⊙S +C +((s, t)) = ∂T +C(t). +This composition need not be associative on the nose, but +the universal property of pullbacks entails that it is associative +up to canonical isomorphism – forming a bicategory: +Theorem 1. If C has pullbacks, then Span(C) defined with +objects: +objects of C, +morphisms: +spans A ← S → B, +2-cells: +morphisms of spans, +forms a bicategory, denoted Span(C). +In fact, Span(C) is a compact closed bicategory [32], and +thus a model of the linear λ-calculus. In particular, Span(Set) +shares much structure with Rel: it has the same objects and +the operation sending a span A ← S → B to the pairs +(∂S +A(s), ∂S +B(s)) for s ∈ S is a functor, establishing Span(Set) +as a natural candidate for a proof-relevant relational model. +3) The exponential: However, the exponential of Rel does +not directly transport to Span. The operation M(−) does +yield a functor on Set obtained by setting, for f : A → B, +M(f)([a1, . . . , an]) = [f(a1), . . . , f(an)] +defining M(f) : M(A) → M(B). But M(f) does not lift +to Span(Set) as it does not preserve pullbacks. Indeed, the +diagram obtained by image of the composition pullback +M(T ⊙ S) +M(S) +M(T ) +M(B) +M(l) +M(r) +M(∂S +B) +M(∂T +B) +is no pullback: this would need a bijection of M(T ⊙S) with +{(µ, ν) ∈ M(S) × M(T ) | M(∂S +B)(µ) = M(∂T +B)(ν)} , +which fails in general. If S = T = B and B = 1, the pair +of multisets ([tt, ff], [tt, ff]) does not uniquely specify who is + +synchronized with whom: it may correspond to both multisets +[(tt, tt), (ff, ff)] and [(tt, ff), (ff, tt)] in M(T ⊙ S). +This might be expected: a finite multiset only remembers the +multiplicity of elements, but does not track distinct individual +occurrences. This is in tension with the goal of a proof-relevant +relational semantics, for which specific witnesses are naturally +associated with individual resource occurrences. +4) Categorifying objects: If the exponential is to track indi- +vidual resource occurrences, that means avoiding the quotient +of finite multisets: an element of !A may for instance be a +list, or a word a1 . . . an ∈ A∗ of elements of A. We must +of course still account for reorderings, which turn A∗ into a +groupoid – in fact, it is an instance of the construction of the +free symmetric monoidal category Sym(A) over a category +A: its objects are finite words a1 . . . an of objects of A, and a +morphism from a1 . . . an to a′ +1 . . . a′ +n consists of a permutation +π ∈ Sn, and a family (fi ∈ A(ai, aπ(i)))1≤i≤n. +Thus, objects are not mere sets but categories, which means +that we move from Span(Set) to Span(Cat). Indeed, Cat +also has pullbacks, and so the exact same construction as above +yields a bicategory Span(Cat) – except that now the functor +Sym : Cat → Cat preserves pullbacks and thus lifts to +Sym : Span(Cat) → Span(Cat) . +However, in this categorification, the Seely isomorphism +M(A + B) ∼= M(A) × M(B) is lost. Instead, we only get +Sym(A + B) ≃ Sym(A) × Sym(B) +an equivalence of categories. In order to lift it to spans, we +observe that given a functor F : A → B we get a span +ˆF += +A +A +B +F +idA +∈ +Span(Cat)(A, B) +so that lifting an equivalence F : A ≃ B : G to spans requires +us to provide a family of 2-cells, i.e. for each category A: +A +A +A +A +idA +GF +? +idA +idA +however whatever our choice for the mediating map is, one +of the triangles fails to commute on the nose but only up to +isomorphism, which the 2-cells of Span(Cat) are too strict +to accommodate. This invites weakening the 2-cells to: +Definition 1. A weak morphism from A ← S → B to A ← +S′ → B is a triple (F, F A, F B) where +S +A +F A ⇓ +⇓F B +B +S′ +∂S +A +∂S +B +F +∂S′ +A +∂S′ +B +with F A : ∂S +A ⇒ ∂S′ +A ◦F and F B : ∂S +B ⇒ ∂S′ +B ◦F natural isos. +We call this a strong morphism if F A and F B are identities. +Adopting weak morphisms seems to solve the problem +above, but only to run into a much more subtle one: in +P +S +T +B +l +r +u +µ=⇒ +v +Fig. 1. A bipullback +X +S +T +B +l′ +r′ +u +ν=⇒ +v +Fig. 2. Alternative square +Span(Cat), the horizontal composition of 2-cells F : S ⇒ +S′ and G : T ⇒ T ′ as required by the bicategorical structure +follows from the universal property of the pullback T ′ ⊙ S′: +T ⊙ S +S +T +A +B +C +S′ +T ′ +T ′ ⊙ S′ +G⊙F +F +G +(2) +but this universal property is powerless to compose horizon- +tally weak morphisms. We cannot have the cake and eat it +too: if our method to compose spans ignores the 2-categorical +nature of Cat, then we cannot hope composition to preserve +an equivalence between spans that relies on it, as required +for a model of linear logic. So it seems that this road to a +proof-relevant relational model is doomed – except that this +is exactly what we shall do in this paper! +Before we delve into that, we review existing solutions. +C. Proof-Relevant Relational Models, and Other Related Work +As plain pullbacks are “too 1-dimensional”, it is natural to +compose spans with a 2-dimensional version. +1) Bipullbacks: There are multiple variants for weakened +versions of pullbacks in a 2-category. In this paper, a central +notion will be that of a bipullback:2 +Definition 2. In a 2-category C, a bipullback of the cospan +S +u−→ B +v←− T is a square commuting up to an invertible 2-cell +as in Figure 1, such that for any square as in Figure 2: +(a) There is a morphism h : X → P and 2-cells α and β s.t.: +X +P +S +T +B +l′ +r′ +h +l +r +β=⇒ +α=⇒ +u +µ=⇒ +v += +X +S +T +B +l′ +r′ +u +ν=⇒ +v +(b) h, α, β are unique up to unique 2-cell – see Appendix A. +The important observation is that this alternative universal +property is sufficient to extend the definition of the horizontal +composition in (2) to weak morphisms – with the proviso that +this defines horizontal composition only up to iso; as (b) does +not guarantee uniqueness of h on the nose. +2According to the nlab, its proper name is a bi-iso-comma-object. + +2) Hoffnung’s monoidal tricategory: Hoffnung [33] con- +structs a categorification of Span(Cat) following this idea. +He exploits that Cat actually has pseudo-pullbacks3, which +are a special case of the definition above where α and β are +required to be identities and h is unique on the nose – making +horizontal composition of weak morphisms of spans a well- +defined function once a choice of pseudo-pullbacks is fixed. +Concretely, a pseudo-pullback of a cospan S +u−→ B +v←− T +may be constructed as a category with objects triples (s, θ, t) +where θ ∈ B(u(s), v(t)). So for instance, if S = T = +Sym(B) and B = Sym(1), the pseudo-pullback would have +two objects synchronizing [tt, ff] ∈ S and [tt, ff] ∈ T : +([tt, ff], id, [tt, ff]) and ([tt, ff, swap, [tt, ff]). The issue of +Section II-B3 is avoided by adding new witnesses carrying +all possible symmetries. This is a fundamental phenomenon +in models of linear logic, which we refer to as saturation. +Because saturation inflates the number of witnesses at each +composition, spans composed by pseudo-pullbacks no longer +form a bicategory. In particular, the post-composition of a span +A ← S → B with the identity span B ← B → B yields an +inflated S′ much bigger than S. So neutrality of identity no +longer holds up to isomorphism, but only up to equivalence +factoring in maps between maps of spans. Accordingly, Hoff- +nung actually constructs a monoidal tricategory of categorical +spans with weak morphisms, i.e. a one-object tetracategory! +3) Melli`es’ template games: Recently, Melli`es introduced +template games [34], in an attempt to unify various games +models. This is essentially a model of categorical spans where +categories are regarded as games and structured by a projection +to a category called the template, capturing the mechanisms of +synchronization and scheduling between players. Though [34] +was developped in a purely linear setting with spans related +by strong morphisms, Melli`es proposed a non-linear extension, +forming a model of differential linear logic [23]. +Melli`es’ contribution puts into play notions from homotopy +theory: he starts not with Cat, but from any 2-category +S equipped with a Quillen model structure (with additional +conditions). Spans are composed by mere pullbacks, but a span +A +u +←− S +v−→ B +must satisfy a fibration property to the effect that symmetries +in A and B can be lifted uniquely in S – in our terminology, S +is saturated. Saturation ensures that pullbacks between those +spans are actually homotopy pullbacks, and thus that they may +be used for the horizontal composition of weak morphisms. +The higher dimensional structure seen in Hoffnung [33] is then +managed by the homotopy-theoretic operation of localization, +formally inverting weak equivalences. This yields an actual +bicategory of objects of S related by homotopy spans. +This elegant construction gives a model of differential linear +logic, showing that the symmetries implicit in linear logic may +be naturally managed via the tools of homotopy theory. +3According to the nlab, its proper name is an iso-comma-object. +4) Generalized Species of Structures: Last but not least, +the most well-studied proof-relevant extension of Rel is defi- +nitely Fiore, Gambino, Hyland and Winskel’s cartesian closed +bicategory of generalized species of structure [20]. Relations +from A to B are replaced with distributors or profunctors: +F : Aop × B → Set +for A and B categories. This forms a (compact closed) +bicategory Dist of (small) categories, distributors and natural +transformations between them. The free symmetric monoidal +construction Sym(−) yields a pseudocomonad on Dist, +whose Kleisli bicategory Esp is cartesian closed. +As for the span-based approaches above, the way in which +Dist and Esp handle symmetries is saturated. This may first +be seen in the identity distributor which is defined to be +A[−, −] : Aop × A → Set +the Yoneda embedding, which associates as witnesses over a +pair (a, a) the homset A[a, a], including all symmetries on a. +Composition of distributors is via the coend formula +G ⊙ F = +� b∈B +F(−, b) × G(b, −) +which sets witnesses in (G ⊙ F)(a, c) to be pairs (s, t) ∈ +F(a, b) × G(b, C) quotiented by a relation identifying the +action of a morphism in B on s or on t. +Accordingly, when computing the interpretation of a pro- +gram ⊢ M : A in Esp, for every a ∈ �A� we get �M�(a) a set +of witnesses carrying around explicit symmetries, quotiented +by an equivalence relation letting symmetries flow around – +this is described syntactically elegantly by Olimpieri [28]. The +treatment of symmetry in Esp is, again, saturated. +5) Game semantics: To our knowledge, this saturation +phenomenon in models of linear logic first appears in Bail- +lot, Danos, Ehrhard and Regnier’s (BDER) version [35] of +Abramsky-Jagadeesan-Malacaria (AJM) games [12]. +In AJM games, the moves of a game !A are defined as pairs +(i, a) of i ∈ N a copy index, and a ∈ A a move in A – a +fundamental difficulty in setting up the games model, is that +of uniformity: ensuring that the behaviour of strategies does +not depend on the specific choice of copy indices (which is +the game semantics analogue of composition preserving weak +morphisms). In BDER, uniformity is guaranteed by requir- +ing strategies to be saturated: they are morally wrapped by +copycat processes exchanging non-deterministically all copy +indices. This “noise” prevents strategies from seeing specific +copy indices, let alone depending on them – this is analogous +to the saturation phenomenon above. +But in AJM games there is another choice: in the original +AJM setting [12], strategies carry a deterministic choice of +copy indices. Instead of saturation, uniformity is guaran- +teed by requiring that strategies satisfy a bisimulation-like +property, which ensures that whenever Opponent swaps their +copy indices, Player can swap theirs accordingly, leaving the +behaviour “up to copy indices” invariant. In contrast to the +“saturated” approach to uniformity, we refer to this as the + +“thin” approach. Similar ideas are at play in other game +models based on copy indices: in Melli`es’ orbital games [15], +and more recently in thin concurrent games4 [21]. +Thin concurrent games are a particularly striking related +work, because just as Esp, they also form a cartesian closed +bicategory as proved by Paquet [22], and also generalize the +relational model [37]. In thin concurrent games, strategies are +composed by pullback. But it is a theorem that this pullback is +also a bipullback, which can be used to compose horizontally +weak morphisms even though strategies are not saturated. +But this bipullback property follows from a subtle interactive +reindexing mechanism between strategies, relying crucially on +the fact that we have access to time – it seems hard to replicate +it purely statically as in a relational model. +III. THE BICATEGORY OF THIN SPANS +This long discussion lets us state the main question in this +paper: can we construct a thin version of categorical spans? +A. Pullbacks and Bipullbacks in Groupoids +For simplicity, we focus on spans of groupoids rather than +categories, which are sufficient for the interpretation of types +– we write Gpd for the small 2-category of groupoids. So +we aim to construct a bicategory whose objects are small +groupoids, whose morphisms are spans A ← S → B with +identity the identity span A ← A → A, whose composition is +plain pullback and yet, whose 2-cells are weak morphisms. +1) Notations and terminology: A span A ← S → B may +be presented as a functor S → A × B, so it is convenient not +to focus on spans, but on functors S → A over a groupoid +A. We refer to those with terminology inspired from game +semantics. A prestrategy on groupoid A is a pair (S, ∂F ) +where ∂F : S → A is called the display map. We often refer +to the prestrategy only with S, and write PreStrat(A) for the +set of prestrategies on A. A prestrategy from A to B is a +prestrategy on A × B – then, we write ∂F +A : S → A and +∂F +B : S → B for the two display maps. If S is a prestrategy +from A to B and T a prestrategy from B to C, we write T ⊙S +for the prestrategy from A to C obtained as in Section II-B2. +We often refer to morphisms in groupoids as symmetries and +write e.g. ϕ : s ∼=S s′ instead of ϕ ∈ S(s, s′). +We write 1 for the groupoid with one object ∗, and only the +identity morphism; and o for the groupoid with one object • +and only the identity morphism. If A, B are groupoids, then we +use A ⊢ B and A ⊸ B as synonyms for A × B, with objects +respectively denoted by a ⊢ b and a ⊸ b – likewise, their +morphisms have form θA ⊢ θB ∈ (A ⊸ B)(a ⊸ b, a′ ⊸ b′) +for θA ∈ A(a, a′) and θB ∈ B(b, b′) and likewise for θA ⊸ +θB. We find these purely notational distinctions useful to read +examples, since they coincide with familiar type constructors. +2) Indexed families: As explained earlier, types of λ-calculi +may be interpreted as groupoids – but in a linear language, +these groupoids remain discrete: only the exponential intro- +duces non-trivial morphisms. As those symmetries play a +4The first version of concurrent games with symmetry was saturated [36]. +crucial role, we introduce early our version of the exponential +construction. If X is a set, then we write Fam(X) the set +of families indexed by finite sets of natural numbers, i.e. of +(xi)i∈I where I ⊆f N and for all i ∈ I, xi ∈ X. +Definition 3. Consider A a (small) groupoid. The (small) +groupoid Fam(A) has: objects, the set Fam(A); morphisms +from (ai)i∈I to (bj)j∈J, pairs (π, (fi)i∈I) of a bijection +π : I ≃ J and for each i ∈ I, fi ∈ A(ai, bπ(i)). +This yields a functor Fam : Gpd → Gpd in the obvious +way. For (Ai)i∈I ∈ Fam(A), we call elements of I copy +indices. A family (ai)i∈I ∈ Fam(A) is more “intensional” +than A∗ (which is more intensional than M(A)): it gives +a presentation of a multiset in M(A) not only providing a +sequence, but assigning to each element a distinct “address”. +Just as multisets are connected to non-idempotent intersec- +tion types, families are connected with Vial’s sequence types +[38] – thus we often write families using Vial’s notation, e.g. +[2 · a2, 4 · a4, 12 · a12] ∈ Fam(A) +for (ai)i∈{2,4,12} – in the particular case where A = o, we +only write [i1, . . . , in] for [i1 · •, . . . , in · •]. +For any groupoid A, Fam(A) and Sym(A) are equivalent. +However, using Fam(A) is crucial in our model construction: +it allows the interpretation of programs to use copy indices as +identifiers for resource accesses, that are independent of other +concurrent resource accesses. We give a few examples: +Example 4. For a groupoid A, the dereliction span derA is +Fam(A) +derA +←−−− A +idA +−−→ A +where derA : A → Fam(A) sends a to [0 · a]. +In models of linear logic, the role of dereliction is to extract +a single instance of a replicable resource. In our model – as in +AJM games [12] and thin concurrent games [21] – dereliction +does so by picking a copy index (here 0), chosen in advance +once and for all. The specific choice is irrelevant; in fact for +any n the span using n instead of 0 will be turn out to be +isomorphic to derA. But, the span must comprise a choice. +Example 5. The interpretation of the term M of Example 2 +in thin spans shall have head groupoid that with four objects +[0 · tt, 1 · tt] ⊸ tt , +[0 · ff, 1 · ff] ⊸ tt , +[0 · tt, 1 · ff] ⊸ ff , +[0 · ff, 1 · tt] ⊸ tt , +morphisms reduced to identities, and display map the identity. +The use of specific copy indices allows one to observe which +occurrence of x evaluates to tt or ff, hence associating distinct +points to the two evaluations leading to ff. +3) Bipullbacks of groupoids: If composition-by-pullback is +to allow us to compose horizontally weak morphisms, we must +ensure that every composition pullback is also a bipullback. +It is useful to understand a bit better the shape of bipullbacks +in Gpd. A first useful fact is that condition (b) of Definition 2 +(uniqueness up to iso) automatically holds in the case of Gpd; + +furthermore, we can characterise those pullbacks that are also +bipullbacks (see Appendix B): +Lemma 1. A pullback square in Gpd, of the form +P +S +T +B +l +r +f +g +is a bipullback if and only if it satisfies the following property: +for all s ∈ S, t ∈ T and θ ∈ B(fs, gt), there is ϕ ∈ S(s, s′) +and ψ ∈ T (t′, t) such that fs′ = gt′ and θ = fψ ◦ gϕ. +Let us comment on this. We regard triples of the form +s ∈ S , +θ ∈ B(fs, gt) , +t ∈ T +as pairs of states (s, t) that match up to symmetry – we call +this a reindexing problem. The lemma above says that given a +reindexing problem, we can always find s′ symmetric to s and +t′ symmetric to t matching on the nose, in a way compatible +with θ – called a solution to the reindexing problem. Thus, +the lemma above may be reformulated to say that a pullback +is a bipullback iff all its reindexing problems have a solution. +We show a concrete example of this reindexing process: +Example 6. Take B = Fam(o) ⊸ Fam(o), with objects +[i1, . . . , in] ⊸ [j1, . . . , jk] . +Take S the sub-groupoid of B with objects [i1, . . . , in] ⊸ +[i1, . . . , in] and morphisms all θ ⊸ θ for θ in Fam(o); and +T the full sub-groupoid of B with objects [j1, . . . , jn] ⊸ [0]. +The pullback of S → B ← T is a bipullback. For instance, +θ ∈ B([2] ⊸ [2], [1] ⊸ [0]) +is a reindexing problem that may be solved by first applying +ϕ ∈ S([2] ⊸ [2], [0] ⊸ [0]) +in S. We are reduced to finding morphisms in S and T w.r.t. +θ′ ∈ B([0] ⊸ [0], [1] ⊸ [0]) +Now, applying ψ ∈ T ([0] ⊸ [0], [1] ⊸ [0]) in T , we have +ϕ ∈ S([2] ⊸ [2], [0] ⊸ [0]) , +ψ ∈ T ([0] ⊸ [0], [1] ⊸ [0]) +a solution to the reindexing problem, as in Lemma 1. +That the pullback of two prestrategies forms a bipullback +is not a property of either: in this example neither strategy +is a fibration as in [23], and solving the reindexing problem +requires reindexing in both groupoids. So it is a property +emerging from the harmonious interaction between two pre- +strategies. In an appropriate game semantics setting [21], one +can prove that under reasonable assumptions, such interactive +reindexing always succeeds. However, this is a gradual process +progressing over time – which we do not have access to here. +B. Orthogonality and Uniform Groupoids +1) Definition: In the literature on models of linear logic, +there is a technique for choreographing models where one only +composes pairs of morphisms satisfying a given interactive +property: biorthogonality. The first step is to specify the +desired interactive property via an orthogonality relation: +Definition 4. Take (S, ∂S) and (T, ∂T ) prestrategies on B. +We say they are uniformly orthogonal, written S ⊥ T , iff +the pullback of the cospan S → B ← T is also a bipullback. +If S ⊆ PreStrat(B), then its uniform orthogonal is set to: +S⊥ = {T ∈ PreStrat(B) | ∀S ∈ S, S ⊥ T }. +As usual with orthogonality, this automatically entails a +number of properties: for all S ⊆ PreStrat(B), we have +S ⊆ S⊥⊥, and S⊥ = S⊥⊥⊥. We are particularly interested in +sets of the form S⊥, which are invariant under biorthogonal: +Definition 5. A uniform groupoid is a pair (A, UA) where +A is a groupoid and UA ⊆ PreStrat(A) is s.t. U⊥⊥ +A += UA. +We often refer to a uniform groupoid (A, UA) just with A +when it is clear from the context that it is a uniform groupoid. +2) Constructions: The uniform groupoid 1 is the terminal +groupoid equipped with U1 = PreStrat(1). If A and B are +uniform groupoids, their tensor A⊗ B is the groupoid A× B +equipped with the set UA⊗B = (UA ⊗ UB)⊥⊥, writing +UA ⊗ UB = {(S × T, ∂S × ∂T ) | S ∈ UA, T ∈ UB} +with ∂S × ∂T : S × T → A × B. The dual A⊥ of A is +(A, UA⊥) with UA⊥ = U⊥ +A. The par of A and B has +UA`B = (U⊥ +A ⊗ U⊥ +B)⊥ +yielding the De Morgan duality (A⊗ B)⊥ = A⊥ `B⊥. From +this we derive the linear arrow A ⊸ B = A⊥ ` B. +A uniform prestrategy on uniform groupoid A is simply +any S ∈ UA. If A, B are uniform groupoids, then a uniform +prestrategy from A to B is a uniform prestrategy on A ⊸ B. +3) Uniform composition: We claim that whenever compos- +ing S ∈ UA⊸B with T ∈ UB⊸C, we have the orthogonality +(S, ∂S +B) ⊥ (T, ∂T +B) +so that the composition pullback is a bipullback. +If S is a prestrategy on A and T is a prestrategy from A to +B, we write T @S from the prestrategy on B obtained by +T ⊙ S +S +T +A +B +called the application of T to S. This lets us state: +Proposition 1. Consider (A, UA) and (B, UB) uniform +groupoids, and T a prestrategy from A to B; consider fur- +thermore a class S ⊆ UA s.t. (A, idA) ∈ S and UA = S⊥⊥. +Then T ∈ UA⊸B iff the following two conditions hold: +(1) for all S ∈ S, T @S ∈ UB, + +(2) (T, ∂T +A) ∈ U⊥ +A. +Proof. Unfolding the definitions, one encounters a few dia- +gram chasing lemmas on pullbacks that are also bipullbacks +– themselves proved via Lemma 1. See Appendix C. +The apparent asymmetry is intriguing: by definition A⊥ ` +B = A⊥`B⊥⊥, so that T ∈ UA⊸B iff the span B ← T → A +denoted by T ⋆ obtained by reversing the two legs, is in +UB⊥⊸A⊥. A similar phenomenon appears in the orthogonal- +ity used by Ehrhard for his extensional collapse [39]. +Now, observe that (A, idA) ∈ UA always – not the identity +span, but the identity functor regarded as a prestrategy on A. +Indeed, if S ∈ U⊥ +A, then the pullback of A → A ← S is +clearly a bipullback, so (A, idA) ∈ U⊥⊥ +A += UA. But now this +lets us instantiate Proposition 1 with S = UA. Then given +S ∈ UA⊸B, the application S@(A, idA) is (up to iso) the +right leg (S, ∂S +B), which must by (1) be in UB. Likewise, if +T ∈ UB⊸C, the left leg (T, ∂T +B) is in U⊥ +B. Hence, +(S, ∂S +B) +⊥ +(T, ∂T +B) +and thus the composition pullback of S and T is a bipullback. +Proposition 1 has more consequences, all obtained in the +particular case where S = UA: we saw above that (A, idA) ∈ +UA, but the same argument goes to show (A, idA) ∈ U⊥ +A as +well – so the identity span satisfies condition (2). Since it also +satisfies (1), we have (A ← A → A) ∈ UA⊸A as expected. +Likewise, if A ← S → B and B ← T → C are uniform +prestrategies, then it follows fairly easily that the composition +A ← T ⊙ S → C is indeed in UA⊸C (see Appendix C). +4) Horizontal composition of 2-cells: We have an identity +uniform prestrategy in UA⊸A, and a well-defined composition +of S ∈ UA⊸B and T ∈ UB⊸C such that the composition +pullback is always a bipullback. So given weak morphisms +S +A +F A⇓ +⇓F B +B +S′ +∂S +A +∂S +B +F +∂S′ +A +∂S′ +B +T +B +GB⇓ +⇓GC +C +T ′ +∂T +B +∂T +C +G +∂T ′ +B +∂T ′ +C +by the bipullback property of T ′ ⊙ S′ there are a functor H +and natural isos α and β such that we have the equality +T ⊙ S +S +T +S′ +T ′ +B += +(FB)−1 +====⇒ +GB +==⇒ += +T ⊙ S +S +T ′ ⊙ S′ +T +S′ +T ′ +B +H +α=⇒ +β=⇒ += +altogether yielding a weak morphism as in the diagram: +S +T ⊙ S +T +A +C +S′ +T ′ ⊙ S′ +T ′ +∂S +A +F +H +G +∂T +C +⇓F A +⇓α +⇓β−1 +⇓GC +∂S′ +A +∂T ′ +C +. +However, H, α, β are not unique: though Lemma 1 guar- +antees the existence of solutions to all reindexing problems, +those may not be unique. We only know by condition (b) of +Definition 2 that different choices of H, α, β yield isomorphic +weak morphisms of uniform prestrategies, by which we mean +isomorphic morphisms of the 2-category Unif(A): +Definition 6. Consider A a uniform groupoid. +The 2-category Unif(A) has: objects UA, i.e. uniform pre- +strategies on A; morphisms from S to T the weak morphisms, +i.e. pairs (F : S → T, φ : ∂S ⇒ ∂T F); 2-cells from (F, φ) to +(G, ψ) the natural transformations µ : F ⇒ G such that: +S +T +A +G +ψ=⇒ += +S +T +A +G +F +⇑µ +φ=⇒ +. +Thus, although bipullbacks guarantee the existence of a +fitting weak morphism for horizontal composition, there is +a priori no canonical choice. One could pick a choice of +horizontal composition, but there is no reason why an arbitrary +choice would satisfy the coherence conditions for a bicategory. +C. Thin Spans of Groupoids +In fact, if formulated in the adequate way, the reindexing +problems that arise from the interpretation of programming +languages do have a unique solution – as in Example 6. But +to prove that, we shall need to add more structure to uniform +groupoids, starting with polarized sub-groupoids: +1) Polarized sub-groupoids: Consider the groupoid +Fam(o) ⊸ Fam(o) +of Example 6, interpreting the formula !o ⊸ !o of intuitionis- +tic linear logic. Here, the two occurrences of ! are intuitively +very different: on the left-hand side, as in Example 4 the +program performs the copying – in game semantics the copy +index would be carried by a Player move. In contrast, for the +right hand side exponential, the environment does the copying +– in game semantics, the copy index would be carried by an +Opponent move. This assigns a polarity to certain symmetries, +very clear in game semantics: those reindexing copy indices +only for exponentials in covariant position (resp. contravariant +position) are negative (resp. positive). We enrich the groupoids +interpreting types to keep track of these special symmetries: +Definition 7. A polarized groupoid is a groupoid A with two +sub-groupoids A− and A+, with the same objects as A. +It would be natural to require additional conditions for +this structure (in particular, see conditions (a) and (b) in +Lemma 3). We omit them here, as they shall hold automat- +ically once we introduce the more complete notion of a thin +groupoid. +If θ ∈ A−(a1, a2), we write θ : a1 ∼=− +A a2 and likewise +for positive symmetries. Usual constructions on groupoids +extend to polarized groupoids componentwise. The dual of +(A, A−, A+) is defined as (A, A+, A−), exchanging the two +sub-groupoids. Finally, we set (!A)− += +Fam(A−) and +(!A)+ = Famid(A+), which has morphisms from (ai)i∈I +to (bj)j∈J those (π, (θi)i∈I) such that I = J and π = idI +– thus we see indeed that Player cannot reindex copy indices +from the outer ! in !A, as it appears in covariant position. + +2) Thinness: Solutions to reindexing problems may be +computed interactively as in Example 6. Intuitively, the +uniqueness of the solution relies on the fact that at each stage, +there is a unique choice of reindexing. This is captured by the +definition of thin below, imported from thin concurrent games: +Definition 8. Consider A a polarized groupoid, and S a +prestrategy on A. We say that S is thin iff for all ϕ : s ∼=S s′, +if ∂Sϕ is positive then s = s′ and ϕ = ids. +Intuitively, this captures that positive copy indices are se- +lected deterministically from negative ones – so a non-trivial +symmetry ϕ : s ∼=S s′ cannot display to a purely positive +symmetry on A. This is in contrast with the saturated case, +where spans must be able to reach all positive symmetries. +We show how thinness addresses uniqueness for the reso- +lution of reindexing problems. Call a solution to a reindexing +problem ϕ, ψ as in Lemma 1 positive if writing ∂Sϕ = ϕA ⊢ +ϕB and ∂T ψ = ψB ⊢ ψC, we have ϕA ⊢ ψC positive. +Lemma 2. Consider A, B, C polarized uniform groupoids, +S ∈ UA⊸B and T ∈ UB⊸C s.t. T ⊙ S ∈ UA⊸C is thin. +Then, any reindexing problem in the composition pullback +of S and T has at most one positive solution. +Proof. Consider a reindexing problem s ∈ S, t ∈ T, θ : +∂S +Bs ∼=B ∂T +Bt with solutions ϕ1 : s ∼=S s′ +1 and ψ1 : t′ +1 ∼=T t +with ∂S +Bs′ +1 = ∂T +Bt′ +1 and ∂T +Bψ1 ◦ ∂S +Bϕ1 = θ, and ϕ2 : s ∼=S s′ +2 +and ψ2 : t′ +2 ∼=T t with ∂S +Bs′ +2 = ∂T +Bt′ +2 and ∂T +Bψ2 ◦ ∂S +Bϕ1 = θ. +Then, ∂S(ϕ2 ◦ ϕ−1 +1 ) = ∂T (ψ2 ◦ ψ−1 +1 ), so that we have +Ω = (ϕ2 ◦ ϕ−1 +1 , ψ2 ◦ ψ−1 +1 ) : (s′ +1, t′ +1) ∼=T ⊙S (s′ +2, t′ +2) , +whose display to A ⊢ C is positive since ϕ1, ψ1 and ϕ2, ψ2 +are positive solutions. Hence, by thin, Ω is an identity map +which entails ϕ1 = ϕ2 and ψ1 = ψ2 as required. +Thus, thinness allows us to find canonical solutions to +reindexing problems by insisting on finding positive solutions. +However, this relies on thinness not of S and T , but of T ⊙S. +Again in thin concurrent games, this follows by induction on +the causal structure. In the absence of a handle on causality, +we must as for uniformity treat the fact that T ⊙ S is thin as +an interactive property, again handled by biorthogonality. +D. Thin Spans +1) The thin orthogonality: We observe that for A a polar- +ized groupoid, a prestrategy S on A is thin iff the pullback +P +S +A+ +A +id+ +A +(3) +is discrete, i.e. all the morphisms in P are identities. We shall +base our orthogonality on this observation, and set: +Definition 9. For A a polarized uniform groupoid, S ∈ UA, +and T ∈ U⊥ +A, we say S and T are thinly orthogonal, written +S ⊥⊥ T +iff the pullback T ⊙ S is discrete. +Note that this is already assuming that S and T are +uniformly orthogonal. If S ⊆ UA, then its thin orthogonal is +S⊥⊥ = {T ∈ U⊥ +A | ∀S ∈ S, S ⊥⊥ T } , +and as before we have S ⊆ S⊥⊥⊥⊥ (note that this typechecks +only because U⊥⊥ +A += UA) and S⊥⊥ = S⊥⊥⊥⊥⊥⊥ for all S ⊆ UA. +2) Thin groupoids: As before, we are interested in sets of +uniform prestrategies closed under bi-thin-orthogonal: +Definition 10. A thin groupoid is a polarized uniform +groupoid with a set TA ⊆ UA of strategies s.t. T⊥⊥⊥⊥ +A += TA, +and such that (A−, idA) ∈ TA and (A+, idA) ∈ T⊥⊥ +A . +If S ∈ TA then S is automatically thin in the sense of +Definition 8: as (A+, idA) ∈ T⊥⊥ +A the pullback (3) is discrete. +This also entails properties of the polarized symmetries: +Lemma 3. Consider A a thin groupoid. Then we have: +(a) if θ : a ∼=− +A a′ and θ : a ∼=+ +A a′, then a = a′ and θ = ida. +(b) if θ : a ∼=A a′, then there are unique a′′ along with θ− : +a ∼=− +A a′′ and θ+ : a′′ ∼=+ +A a′ such that θ = θ+ ◦ θ−. +Proof. For (a), this follows from A− ⊥⊥ A+, as then the +pullback of the cospan A− ֒→ A ←֓ A+ is discrete. +For (b), A− ∈ TA ⊆ UA and A+ ∈ T⊥⊥ +A ⊆ U⊥ +A, we also +have A− ⊥ A+. Hence, the pullback of the cospan A− ֒→ +A ←֓ A+ is a bipullback. But then any θ : a ∼=A a′ forms +a reindexing problem, whose solution is exactly the seeked +reindexing. Uniqueness follows immediately from (a). +Thus, we get from the definition of thin groupoids some +of the expected properties of the polarized sub-groupoids: if +a symmetry is both positive and negative then it must be an +identity, and any symmetry can be obtained by first “reindex- +ing Opponent moves”, then ”reindexing Player moves”. +3) Further structure: Constructions on uniform groupoids +extend to thin groupoids in the expected way. The thin +groupoid 1 has T1 = PreStrat(1). If A and B are thin +groupoids, their tensor is the uniform groupoid A ⊗ B ex- +tended with TA⊗B = (TA ⊗ TB)⊥⊥⊥⊥. The dual of A has +TA⊥ = T⊥⊥ +A . The par of A and B has TA`B = (T⊥⊥ +A ⊗T⊥⊥ +B)⊥⊥, +and the linear arrow is A ⊸ B = A⊥ ` B. +To establish the compositional properties of strategies, we +rely on the following analogue of Proposition 1: +Proposition 2. Consider T ∈ UA⊸B for A, B thin groupoids, +along with a class S ⊆ TA such that S⊥⊥⊥⊥ = TA. +Then, T ∈ TA⊸B iff T @S ∈ TB for all S ∈ S. +This follows from diagram chasing lemmas on situations +where the pullbacks are discrete, see Appendix C. Interest- +ingly, this is also equivalent to T ⋆@S ∈ T⊥⊥ +A for all S ∈ T⊥⊥ +B. +It is a direct consequence of Proposition 2 that the identity +span on A is in TA⊸A for any thin groupoid A, and that if S ∈ +TA⊸B and T ∈ TB⊸C then T ⊙ S ∈ TA⊸C. Together with +Lemma 2, we have thus identified a compositional situation +where the composition pullback of spans is a bipullback, and +where all arising reindexing problems have a unique solution +if one insists on this solution being positive. + +4) Positive weak morphisms: Insisting on positive solutions +amounts to relating strategies not via arbitrary weak mor- +phisms, but with positive weak morphisms: +Definition 11. Consider A a thin groupoid, S, T ∈ TA, and +(F, φ) a weak morphism from S to T , i.e. F : S → T and +φ : ∂S ⇒ ∂T ◦ F. Then, (F, φ) is positive if φ is positive, that +is, if ∀s ∈ S, φs : ∂Ss ∼=+ +A ∂T F(s) is a positive symmetry. +Intuitively, comparing strategies with positive morphisms +amounts to relating them only via maps that do not reindex +Opponent moves. This has the effect of making everything +stricter, and cutting the higher dimension. More precisely: +Proposition +3. +Let +A +be +a +thin +groupoid. +Consider +PreThin(A) the sub-2-category of Unif(A) with objects +TA, and Thin(A) where additionally morphisms are positive. +Then, Thin(A) is locally discrete, i.e. all 2-cells are identi- +ties. Moreover, PreThin(A) and Thin(A) are biequivalent. +Proof. The first is a direct consequence of thinness: if µ : +(F, φ) ⇒ (G, ψ) : S → T for φ and ψ positive, then by defi- +nition of 2-cells of Unif(A), for all s ∈ S, µs ∈ T (Fs, Gs) +is such that ψs = ∂T µs ◦ φs, i.e. ∂T µs = ψs ◦ φ−1 +s +positive. +Thus, µs is an identity morphism by thinness. +For the biequivalence, the crux is that if (F, φ) : S → T is +a weak morphism, then there is a unique (F+, φ+) : S → T +positive isomorphic to (F, φ), and a unique 2-cell µ between +them. Uniqueness follows from thinness. For existence, note +that if s ∈ S and θ : ∂Ss ∼=A a, then there exist unique +ϕ : s ∼=S s′ and θ+ : ∂Ss ∼=+ +A a such that θ = θ+ ◦ ∂Sϕ – +this exploits thinness, and the reindexing problem from the fact +that the pullback of the cospan S ֒→ A ←֓ A+ is a bipullback. +We obtain (F+, φ+) by applying this lemma pointwise. +This proposition illustrates the situation well: thanks to the +thin biorthogonality, the 2-category PreThin(A) is repre- +sented up to biequivalence as a mere category Thin(A). The +higher dimensional structure simply vanishes. +5) The bicategory Thin: With this in place, we may finally +define the components of our bicategory Thin. Its objects +are thin groupoids. Its morphisms from A to B are strategies +from A to B, i.e. elements of TA⊸B – recall that they are +(S, ∂S : S → A × B), in particular spans from A to B +A +∂S +A +←−− S +∂S +B +−−→ B . +The identities are identity spans, and composition is via the +pullback (1). If S and T are strategies from A to B, the 2- +cells from S to T are the positive morphisms (F, φ) : S → T . +As φ : ∂S ⇒ ∂T ◦ F is a family of positive morphisms on +A⊥ ` B with underlying plain groupoid A × B, it may be +equivalently presented as pair of F A : ∂S +A ⇒ ∂T +A ◦F and F B : +∂S +B ⇒ ∂T +B ◦ F, as in Definition 1. For horizontal composition +of positive morphisms, we first proceed as in Section III-B4 +and obtain a connected groupoid of (non necessarily positive) +horizontal compositions – which must all have the same image +through the biequivalence of Proposition 3, providing our +unique positive horizontal composition. Altogether, we have: +Theorem 2. Those components form Thin, a bicategory. +Proof. See details in Appendix D. +Next, we develop the further structure of Thin. +IV. CARTESIAN CLOSED STRUCTURE +To construct a cartesian closed bicategory, we intend to +follow [20]. We first turn the construction Fam – thereafter +denoted by ! – into a pseudocomonad, and then equip the +Kleisli bicategory Thin! with the cartesian closed structure. +A. The Pseudocomonad +We first develop the action of ! on objects of Thin. +1) The bang of thin groupoids: First, ! is defined on uniform +groupoids via U!A = (!UA)⊥⊥, where we have used +!UA = {(!S, !∂S) | S ∈ UA} +using the functorial action !∂S : !S → !A. For thin groupoids, +the positive and negative symmetries of !A were defined in +Section III-C1. The thin structure is set as T!A = (!TA)⊥⊥⊥⊥ +– it is a direct verification that this is a thin groupoid. +2) The bang of strategies: If S ∈ TA⊸B, we have ∂S = +⟨∂S +A, ∂S +B⟩ for ∂S +A : S → A and ∂S +B : S → B – its bang is +!A +!∂S +A +←−− !S +!∂S +B +−−→ !B +packaged as (!S, ⟨!∂S +A, !∂S +B⟩). That this is in T!A⊸!B relies on: +Lemma 4. Consider A, B thin groupoids, and T a prestrategy +from !A to B. Then, the following two properties hold: +(1) +T ∈ U!A⊸B iff (T, ∂T +!A) ∈ U⊥ +!A and +for all S ∈ UA, T @!S ∈ UB, +(2) +T ∈ T!A⊸B iff for all S ∈ TA, T @!S ∈ TB. +This is an immediate application of Propositions 1 and 2. +Since U!A = (!UA)⊥⊥ and T!A = (!TA)⊥⊥⊥⊥. From this +lemma, it is a rather direct verification that for any S ∈ +TA⊸B, we have !S ∈ T!A⊸!B as required. +3) A pseudofunctor: Since ! is a functor, it preserves the +identity span on the nose. Since ! preserves pullbacks, for +S ∈ TA⊸B and T ∈ TB⊸C, the universal property gives us +mS,T : !(T ⊙ S) ∼= !T ⊙ !S +a strong invertible 2-cell in Thin. As expected, this 2-cell +is natural in S and T (with respect to positive morphisms). +Altogether, we obtain a pseudofunctor ! : Thin → Thin. +See Appendix F for details. +4) A pseudomonad on groupoids: In fact we first turn ! +into a pseudomonad on Gpd, from which its pseudocomonad +structure on Thin shall be derived. We noted earlier that we +have a functor Fam : Gpd → Gpd – in fact, it is extended +to a 2-endofunctor on the 2-category of small groupoids, noted +! : Gpd → Gpd , +defined on a 2-cell α : F ⇒ G : A → B as the natural +transformation !α : !F ⇒ !G with components all pairs +(!α)(Ai)i∈I = (idI, (αAi)i∈I) ∈ !B((FAi)i∈I, (GAi)i∈I) . + +!!A +!A +!!A +!A +µA +η!A +!ηA +id!A +αA +⇒ +µA +βA +⇐ +Fig. 3. Unit natural isomorphisms +!!!A +!!A +!!A +!A +µ!A +!µA +µA +γA +⇒ +µA +Fig. 4. Associativity natural isomorphism +To turn this into a pseudomonad, we must adjoin a multipli- +cation and a unit. The components of the unit are the functors +ηA +: +A +→ +!A +a +�→ +(a){0} = [0 · a] +with the obvious functorial action. The intuition is that the +unit transports a single resource usage from A to !A, arriving +at a singleton family. In doing so, it must select a copy index. +Any natural number will do – the rest of the paper does not +depend on this choice – but for definiteness and compatibility +with the traditional convention from AJM games, we pick 0. +For the multiplication µA : !!A → !A, we must flatten a +family of families into a family. For this purpose, we fix an +injective function ⟨−, −⟩ : N2 → N – again, the results of +this paper do not depend on that choice. Given I ⊆f N and a +family (Ji)i∈I where Ji ⊆f N for all i ∈ I, let us write +Σi∈IJi = {⟨i, j⟩ | i ∈ I, j ∈ Ji} , +which is by definition still a finite subset of N. Then we set +µA +: +!!A +→ +!A +((ai,j)j∈Ji)i∈I +�→ +(ai,j)⟨i,j⟩∈Σi∈IJi +for any groupoid A, along with the obvious functorial action. +Altogether this yields η : idGpd ⇒ ! and µ : !! ⇒ !, two +(strict 2-) natural transformations. The monad laws, if they +were to hold on the nose, would mean that ⟨0, i⟩ = ⟨i, 0⟩ = i +and ⟨⟨i, j⟩, k⟩ = ⟨i, ⟨j, k⟩⟩ for all i, j, k ∈ N; and it is clear +that no injection satisfying those laws exists. Nevertheless, for +every groupoid A the coherence laws for a monad hold up to +natural isomorphisms: we have αA, βA and γA as indicated +in Figures 3 and 4. For instance, for any (aj)j∈J ∈ !A: +(αA)(aj)j∈J : (aj)j∈J ∼=!A (aj)⟨0,j⟩∈Σi∈{0}J +reindexing along the bijection J +≃ Σi∈{0}J. The other +components act similarly – note that they are all negative +symmetries. The associated families (αA)A∈Gpd, (βA)A∈Gpd +and (γA)A∈Gpd satisfy the conditions for modifications, and +the additional coherence laws for a pseudomonad: +Proposition 4. The 2-functor ! : Gpd → Gpd along with +the components above yield a pseudomonad on Gpd. +5) Lifting functors to spans: We shall turn ! into a pseudo- +comonad on Thin by lifting the components above to spans. +In general, if F : B → A is a functor, then there is a span ˇF +A +F +←− B +idB +−−→ B , +called the lifting of F – but we need sufficient conditions on +F for this construction to yield morphisms in Thin. For that +purpose, if A and B are thin groupoids, we say that a functor +F : A → B is a renaming iff the following conditions hold: +(1) +for all θ : a ∼=A a′, if θ is positive then so is Fθ, +(2) +for all (T, ∂T ) ∈ U⊥ +B, (T, F ◦ ∂T ) ∈ U⊥ +A, +(3) +for all (T, ∂T ) ∈ T⊥⊥ +B , (T, F ◦ ∂T ) ∈ T⊥⊥ +A. +Clearly, renamings compose – we consider the 2-category +Ren whose objects are thin groupoids, whose morphisms +are renamings, and whose 2-cells are negative natural trans- +formations. As expected, lifting renamings yields thin spans +(see Appendix E). Lifting can be extended to 2-cells: if +α : F ⇒ G : A → B is a negative natural transformation, +then ˇα is the positive morphism described by the diagram: +B +A +B +B +F +idB +idB +α⇒ +G +idB +, +noting that this is positive as negative α is in contravariant +position. Altogether, we get (see details in Appendix E): +Proposition 5. There is a pseudofunctor ˇ− : Renop → Thin. +Here, Renop is Ren with the morphisms reversed, but +the 2-cells unchanged. It can be checked that for any thin +groupoid A, the functors ηA : A → !A and µA : !!A → !A are +renamings, in particular for every thin groupoid A we get +ˇ +ηA ∈ Thin(!A, A) +ˇ +µA ∈ Thin(!A, !!A) +the main components to turn ! into a pseudocomonad. Unlike +in Gpd, the families ˇη and ˇµ are not strict 2-natural transfor- +mations but only pseudonatural transformations, with 2-cells +ηS +: +ˇηB ⊙ !S +⇒ +S ⊙ ˇηA +µS +: +ˇµB ⊙ !S +⇒ +!!S ⊙ ˇµA , +positive isomorphisms obtained for S ∈ Thin(A, B) from +the universal property of pullbacks, via the observation that +η : idGpd ⇒ ! and µ : !! ⇒ ! are cartesian natural +transormations. It may be checked that ηS and µS are natural +in S and satisfy the coherence conditions of pseudonatural +transformations. Finally, the modifications α, β, γ involved in +the pseudomonad structure of ! on Gpd lift to the modifica- +tions required for the pseudocomonad structure of ! on Thin. +Theorem 3. We have a pseudocomonad ! on Thin. +Proof. See details in Appendix G. +We move on to studying the Kleisli bicategory Thin! whose +horizontal composition, denoted ⊙!, is defined as expected. + +B. Cartesian Closed Structure +1) Finite products: First, we show that Thin! has finite +products, i.e. is a fp-bicategory in the sense of Fiore and +Saville [18] – unlike them, we work with binary products. +a) Terminal object: Write ⊤ for the empty groupoid, +made a thin groupoid with U⊤ = T⊤ = {id∅}. For any +thin groupoid A, Thin!(A, ⊤) has exactly one element – the +empty groupoid. Thus, Thin! has a (strict) terminal object. +b) Binary product: If A and B are thin groupoids, then +the with A & B has underlying groupoid A + B the disjoint +union, with (A+B)− = A−+B− and (A+B)+ = A++B+. +We adjoin UA&B = (UA + UB)⊥⊥ and TA&B = (TA + +TB)⊥⊥⊥⊥, where as usual, UA + UB comprises the set of all +(S +T, ∂S +∂T) for (S, ∂S) ∈ UA and (T, ∂T) ∈ UB, using +the functorial action of + (and likewise for TA + TB). +c) Pairing and projections: The projections are simply +set as L! = +̌ +(ηA+B ◦ ¯l) ∈ Thin!(A & B, A) and R! = +̌ +(ηA+B ◦ ¯r) ∈ Thin!(A & B, B) for ¯l : A → A + B and +¯r : B → A + B the obvious coprojections/renamings. The +pairing of S ∈ Thin!(Γ, A) and T ∈ Thin!(Γ, B) is +(S + T, ∂!Γ : S + T → !Γ, ∂A&B : S + T → A + B) +with ∂!Γ the co-pairing and ∂A&B = ∂S +A + ∂T +B. We have: +Proposition 6. For any thin groupoids Γ, A and B, there is +Thin!(Γ, A & B) +⊥ +Thin!(Γ, A) × Thin!(Γ, B) +(L! ⊙!−,R! ⊙!−) +⟨−,−⟩ +an adjoint equivalence. +Proof. If S ∈ Thin!(Γ, A) and T ∈ Thin!(Γ, B) there are +ωA +S,T : L! ⊙!⟨S, T ⟩ ∼= S +ωB +S,T : R! ⊙!⟨S, T ⟩ ∼= T +positive isos, and for U ∈ Thin!(Γ, A & B) there is +¯ωU : U ∼= ⟨L! ⊙!U, R! ⊙!T ⟩ +a positive iso, defined in the obvious way. Those are all natural +in S, T, U, and satisfy the required triangle identities. +See Appendix H for more details. Altogether, this estab- +lishes that Thin! is a fp-bicategory in the sense of [18]. +2) Cartesian closure: If A and B are thin groupoids, then +we set A ⇒ B = !A ` B. Before we describe the additional +components, we must observe the Seely equivalence: +!A ⊗ !B +!(A & B) +sA,B +¯sA,B +where sA,B sends (ai)i∈I, (bj)j∈J to (ck)k∈I⊲⊳J, with I⊲⊳J = +̟(I ⊔ J) for some chosen bijection ̟ = [̟l, ̟r] between +N ⊔ N and N, and where c̟l(i) = ai and c̟r(j) = bj; and +¯sA,B sends (ck)k∈K to (ai)i∈I, (bj)j∈J where I ⊆ K is the +subset of those i ∈ K such that ci = ai ∈ A, and likewise +for bj. Both functors are renamings, and the isomorphisms +witnessing the equivalence are negative. +Via the Seely equivalence, we first define the evaluation as +the span with basic groupoid !A× B, with left leg the functor +!A × B → (!A × B) × !A → !(!A × B) × !A +sA,B +−−−→ !((A ⇒ B) & A) +and right leg the projection !A × B → B. This yields a thin +span evA,B ∈ Thin!((A ⇒ B) & A, B). Now, we need +Λ(−) : Thin!(Γ & A, B) → Thin!(Γ, A ⇒ B) +the currying functor: given S ∈ Thin!(Γ&A, B), its currying +Λ(S) is simply S, with display map post-composed with +!(Γ + A) × B +¯sΓ,A +≃ (!Γ × !A) × B ∼= !Γ × (!A × B) . +With this data in place, we may finally prove: +Proposition 7. For any groupoids Γ, A, B, there is +Thin!(Γ, A ⇒ B) +⊥ +Thin!(Γ & A, B) +evA,B⊙!(−&A) +Λ(−) +an adjoint equivalence. +Proof. One can first show the existence of adjoint equivalence +between the currying operation Λ(−), and a symmetric uncur- +rying operation ¯Λ(−). The unit and counit of this adjunction +can be derived from the ones of the Seely (adjoint) equiva- +lence. One can then prove that ¯Λ(−) is in fact isomorphic to +evA,B ⊙! (−&A) in order to get the wanted equivalence. +See Appendix I for details. Altogether, we have: +Theorem 4. We have Thin!, a cartesian closed bicategory. +This entails that we can interpret types of the simply-typed +λ-calculus as thin groupoids, morphisms as thin spans and +rewrites between terms as certain positive isomorphisms [19]. +V. CONCLUSION +This paper focuses on the construction of Thin!, leaving +for later its application to semantics of λ-calculi and program- +ming languages. We believe this opens multiple perspectives +for further research: firstly, we may explore the obtained +interpretation of the λ-calculus, which syntactically should +correspond to the sequence typing system of Vial [38] and to +the non-uniform λ-calculus of Melli`es [15]. We should explore +links with other models of the literature, notably with the +weighted relational model recasting ideas from [37], and with +generalized species of structures and template games. Another +related direction consists in accommodating another feature +of template games, the mechanism to capture scheduling and +synchronization [34], into thin spans. +In more semantic directions, we believe that with respect to +generalized species of structures, the fact that operations on +thin spans involve no quotient may be helpful in two ways: (1) +individuals may be ordered concretely, and the model should +support continuous reasoning allowing one to deal easily with +infinite computation; and (2) adding “typed” weights coming +from an SMCC as in [24] should be a lot simpler, since those +weigths no longer have to themselves be saturated. + +ACKNOWLEDGMENT +Work supported by the ANR projects DyVerSe (ANR- +19-CE48-0010-01) and PPS (ANR-19-CE48-0014); and by +the Labex MiLyon (ANR-10-LABX-0070) of Universit´e de +Lyon, within the program “Investissements d’Avenir” (ANR- +11-IDEX-0007), operated by the French National Research +Agency (ANR). +REFERENCES +[1] J. 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Bipullbacks +In order to complete the equational definition of bipull- +backs of Definition 2, it is useful to consider the intensional +definition first: given a 2-category C with invertible 2-cells +(i.e., a (2, 1)-category) and a cospan S +u−→ B +v←− T in C, +a bipullback is a pseudocone (P, l, r, µ) as in Figure 1 such +that, for every X ∈ C, the precomposition of the pseudocone +(P, l, r, µ) by morphisms X → P induces an equivalence of +categories between C(X, P) and the category of pseudocones +over the cospan S +u−→ B +v←− T and of vertex X and pseudocone +morphisms. The essential surjectiveness of this precomposition +corresponds exactly to the condition (a) of Definition 2. Its full +faithfulness can be expressed as the following condition: +(b’) given a 2-cell equality +X +P +S +T +B +h +l +r +u +µ=⇒ +v += +X +P +S +T +B +l◦h +r◦h +h′ +l +r +β=⇒ +α=⇒ +u +µ=⇒ +v +for some h, h′ : X → P and 2-cells α: l ◦ h ⇒ l ◦ h′ and +β : r ◦ h′ ⇒ r ◦ h, there is a unique θ: h ⇒ h′ such that +α = lθ and β = rθ−1. +It is not too difficult to show that the latter property is +equivalent to the one asserting that, given two decompositions +of a pseudocone ν +X +S +T +B +l′ +r′ +u +ν=⇒ +v += +X +P +S +T +B +l′ +r′ +h +l +r +β=⇒ +α=⇒ +u +µ=⇒ +v += +X +P +S +T +B +l′ +r′ +h′ +l +r +β′ +==⇒ +α′ +==⇒ +u +µ=⇒ +v +there exists a unique θ: h ⇒ h′ such that lθ = α′ ◦ α−1 +and rθ = β′−1 ◦ β, or equivalently α′ = (lθ) ◦ α and β′ = +β ◦ (rθ−1), which is the complete form of the condition (b). +B. Pullbacks in Gpd +It happens that pullbacks in Gpd are well-behaved w.r.t +2-cells: +Proposition 8. A pullback of a cospan S +u−→ B +v←− T in +Gpd is a strict 2-pullback, that is, also admits a universal +factorization property w.r.t. morphisms of cones. +Proof. Let I be the groupoid consisting of a walking isomor- +phism u between two objects 0 and 1. Given two functors F +and G between two groupoids C and D, a 2-cell +α: F ⇒ G: C → D ∈ Gpd +is then exactly the data of a functor H : I × C +→ D +such that H(0, −) = F and H(1, −) = G. Using this +correspondence, the property that a pullback (P, l, r) over the +cospan is a 2-pullback easily reduces to the one that (P, l, r) +is a 1-pullback. +Note that a pullback is a cone, which is in particular a +pseudocone with identity as inner 2-cell. One might then ask +when a pullback is a bipullback, in which case we have the +following characterization: +Proposition 9. Let (P, l, r) be a pullback over a cospan S +u−→ +B +v←− T in Gpd. Then the pseudocone induced by (P, l, r) +is a bipullback if and only if it satisfies the condition (a) of +Definition 2. +Proof. By Proposition 8, (P, l, r) is a 2-pullback, so that it +satisfies a universal property w.r.t. cone morphisms. This con- +dition is in fact exactly (b’) which is equivalent to (b). Thus, +only (a) is left to check for (P, l, r) to be a bipullback. +We can then refine the previous proposition into a “point- +wise” characterization in the form of already stated Lemma 1 +for which we now provide a proof. +Proof of Lemma 1. The implication is immediate, since the +data of an isomorphism θ: f(s) → g(t) is equivalent to the +one of a pseudocone on S +B +T +f +g +of vertex the terminal +groupoid, whose factorization in the form of the condition of +Proposition 9 is exactly the wanted property. +We now show the converse property. So let +Z +S +T +B +h +k +f +θ=⇒ +g +be a pseudocone. By hypothesis, for every z ∈ Z, there exist +sz ∈ S, φz : h(z) → sz, tz ∈ T , ψz : tz → k(z) such +that f(sz) = g(tz) and θz = g(ψz) ◦ f(φz). The collection +of isomorphisms (φz)z∈Z induces a functor h′ defined by +h′(z) = sz for z ∈ Z, and h′(w) = φz′ ◦ h(w) ◦ φ−1 +z +for w: z → z′ ∈ Z. Similarly, we get a functor k′ defined +by h′(z) = tz for z ∈ Z, and k′(w) = ψ−1 +z′ ◦ k(w) ◦ ψz +for w: z → z′ ∈ Z. Given w: z → z′ ∈ Z, we check +that f ◦ h′(w) and g ◦ k′(w) are equal. Since there are only + +isomorphisms involved, it is enough to check that the equality +holds when in the context g(ψz′) ◦ (−) ◦ f(φz): +g(ψz′) ◦ f(h′(w)) ◦ f(φz) = g(ψz′) ◦ f(h′(w) ◦ φz) += g(ψz′) ◦ f(φz′ ◦ h(w)) += g(ψz′) ◦ f(φz′) ◦ f(h(w)) += θz′ ◦ f(h(w)) += g(k(w)) ◦ θz += g(k(w)) ◦ g(ψz) ◦ f(φz) += g(ψz′) ◦ g(k′(w)) ◦ f(φz). +Thus, (Z, h′, k′) is a cone on S +B +T +f +g +, so there exists +m: Z +→ P which factors h′ and k′ through l and r. +The collections (φz)z∈Z and (ψz)z∈Z defines natural iso- +morphisms φ: h ⇒ l ◦m and ψ: r ◦m ⇒ k which satisfy +(gψ) ◦ (fφ) = θ. Hence, the condition of Proposition 9 is +satisfied and (P, l, r) is a bipullback. +We also have the following criterion for rectangles of +bipullbacks: +Lemma 5. Given a rectangle made of two squares which are +pullbacks in Gpd as in +L +M +R +A +B +C +⌜ +πL +M +πL +A +⌜ +πM +R +πM +B +h +f +g +, +the following hold: +(i) if the whole rectangle is a bipullback, then the left square +is too; +(ii) if the left and right square are bipullback, then the whole +rectangle is a bipullback. +Proof. We +first +prove +(i). +For +this +purpose, +we +use +Proposition 9. So let a ∈ A, m ∈ M and θ: f(a) → πM +B (m). +We have that g(θ) is a morphism from g◦f(a) to g◦πM +B (m) = +h ◦ πM +R (m). Since the outer rectangle is assumed to be a +bipullback, there exist a′ ∈ A, r′ ∈ R, uA: a → a′ ∈ A, +vR : r′ → πM +R (m) such that g(θ) = h(vR) ◦ (g ◦ f)(uA). +Thus, we have g(θ ◦ (f(uA))−1) = h(vR). Since the right +square is a pullback, there exists a unique wM such that +πM +B (wM) = θ ◦ (f(uA))−1 and πM +R (wM) = vR. Moreover, +by the right pullback again, the target of wM is m; its source +is some m′ such that πM +B (m′) = f(a′). Then, we have that +θ = πM +B (wM) ◦ f(uA). We can conclude with Proposition 9 +that the left pullback is a bipullback. +We now prove (ii) using Proposition 9 again. So let a ∈ A, +r ∈ R and θ: (g ◦ f)(a) → h(r). Since the right square is a +bipullback, there exist b′ ∈ B, r′ ∈ R, uB : f(a) → b′ and +vR : r′ → r such that θ = h(vR)◦g(uB). Since b′ and r′ have +the same projection in C through g and h respectively, there +exists m′ ∈ M such that πM +B (m′) = b′ and πM +R (m′) = r′. +Thus, we have uB : f(a) → πM +B (m′). Since the left square +is a bipullback, there exist a′′ ∈ A, m′′ ∈ M, ˜uA : a → a′′, +˜vM : m′′ → m′ such that uB = πM +B (˜vM) ◦ f(˜uA). We thus +have +θ = h(vR) ◦ g(uB) = h(vR) ◦ g(πM +B (˜vM) ◦ f(˜uA)) += h(vR) ◦ g(πM +B (˜vM)) ◦ g(f(˜uA)) += h(vR) ◦ h(πM +R (˜vM)) ◦ g(f(˜uA)) += h(vR ◦ πM +R (˜vM)) ◦ (g ◦ f)(˜uA) +which is precisely the factorization required by Proposition 9 +to conclude that the whole rectangle is a bipullback. +C. Uniformity and thinness +Several arguments concerning uniformity requires some +sort of diagram chasing relative to bipullbacks. An important +lemma for this is the following: +Lemma 6. Consider the diagram in Gpd +S +P +Q +L +M +R +A +B +lS +P +rS +Q +2 +lP +L +rP +M +1 +l +Q +M +rQ +R +3 +f L +A +f M +A +f M +B +f R +B +where the square 1, 2 and 3 are pullbacks, and derive from it +the following diagram using the product structure: +S +M +L × R +A × B +r P +M ◦ lS +P +(lP +L ◦ lS +P ,rQ +R ◦ r S +Q) +4 +(f M +A ,f M +B ) +f L +A×f R +B +Then 4 is a pullback. Moreover, the following hold: +(i) if 1 and the rectangle made of 2 and 3 are bipullbacks, +then 4 is a bipullback; +(ii) if 3 and the rectangle made of 1 and 2 are bipullbacks, +then 4 is a bipullback; +(iii) if 4 is a bipullback, then the rectangle made of 1 and 2 +(resp. 2 and 3) is a bipullback. +Proof. The fact that 4 is a pullback is an easy consequence +of the fact that 1,2,3 are pullbacks. +One can then use Lemma 1 without too much trouble on +the different bipullback hypotheses in order to deduce that the +wanted pullbacks are bipullbacks. +With the above tool, we can now prove Proposition 1. +Proof of Proposition 1. We first prove the first implication, +and start by showing (1). So let (S, ∂S +A) ∈ S. Given (U, ∂U +B) ∈ +U⊥ +B, we must show that T @S ⊥ U. By hypothesis, we have +that T ⊥ S × U, i.e., the pullback of ∂T +A×B and ∂S +A × ∂U +B +is a bipullback. Thus, we conclude by Lemma 6(iii) that the + +pullback of ∂T @S +B +and ∂U +B is a bipullback, i.e., T @S ⊥ U. +Hence, T @S ∈ UB. +We now show (2). Since idB is an isofibration, we have +that the pullback of ∂T +B and idB is a bipullback. Thus, given +U ∈ UA, by Lemma 6, we have that ∂T +A ⊥ ∂U +A if and only if +∂T +A×B ⊥ ∂U +A × idB. But the latter holds, since T ∈ UA⊸B. +Hence, ∂T +A ∈ U⊥ +A. +We now show the converse implication. So assume that T +satisfies (1) and (2). First note that, since (A, idA) ∈ UA, +we have that ∂T +B ∈ U⊥ +B by (1). Given V ∈ UB, we must +show that, for every U ∈ UA, we have T ⊥ U × V . Since +we have ∂T +B ⊥ ∂V +B, by Lemma 6, this is equivalent to have +U ⊥ T ⋆@V for every U ∈ UA. By hypothesis, it is equivalent +to only check the previous condition for U ∈ S. By Lemma 6 +again, it is equivalent to check that T ⊥ U × V for every +U ∈ S. Since U ⊥ ∂T +A, by Lemma 6 again, it is equivalent +to check that T @U ⊥ V , but the latter holds by (1). Thus, +T ∈ UA⊸B. +Using Proposition 1, we can prove the compatibility of +uniformity with composition: +Proposition 10. Given uniform groupoids (A, UA) and +(B, UB) and prestrategies S ∈ UA⊸B and T ∈ UB⊸C, +we have T ⊙ S ∈ UA⊸C. +Proof. Recall that the composition of the two spans S and T +is formed as in Equation (1). We show the uniformity of T ⊙S +using Proposition 1 with UA taken as generating class of UA. +The fact that (1) is satisfied for T ⊙ S is immediate from its +validity for both S and T . The fact that (2) holds, that is, that +∂S +A ◦ l ∈ U⊥ +A, is a consequence of the fact that ∂T +B ∈ U⊥ +B +by (2) on T , and the dual of Proposition 1 for S, asserting in +particular that S⋆ maps elements of U⊥ +B to U⊥ +A. +We +handle +thinness +similarly, +and +start +by +proving +Proposition 2: +Proof of Proposition 2. Assume that T ∈ TA⊸B and let S ∈ +S. By Proposition 1, we already have T @S ∈ UA⊸B. Next, +we use the fact that TB = T⊥⊥⊥⊥ +B +to show that T @S ∈ TB. So +let U ∈ T⊥⊥ +B . By Lemma 6, we have T @S ⊥⊥U iff T ⊥⊥S ×U. +But the latter holds since T ∈ TA⊸B and S ∈ S ⊆ TA. +Thus, T @S ∈ TB. +Conversely, assume that T @S ∈ TB for every S ∈ S. First +observe that TA⊸B = (TA ⊗ T⊥⊥ +B)⊥⊥. So we must show that, +for every S ∈ TA and U ∈ T⊥⊥ +B , T ⊥⊥ S × U. By Lemma 6, +for a given U, the latter is equivalent to S ⊥⊥ T ⋆@U for every +S ∈ TA, i.e., T ⋆@U ∈ T⊥⊥ +A . But since T⊥⊥ +A = S⊥⊥, for a +given U, it is equivalent to S ⊥⊥ T ⋆@U for every S ∈ S, itself +equivalent to T ⊥⊥S×U for every S ∈ S, and finally equivalent +to T @S ⊥⊥ U for every S ∈ S, which amounts to our initial +assumption. +Using Proposition 2, we can then prove the compatibility +of thinness with composition: +Proposition 11. Given thin groupoids A and B and strategies +S ∈ UA⊸B and T ∈ UB⊸C, we have T ⊙ S ∈ TA⊸C. +Proof. The proof is similar to (in fact simpler than) the one +of Proposition 10 and follows from the criterion given by +Proposition 2. +D. Details about the bicategory Thin +We have the following convenient characterization of +0-composition of 2-cells of Thin: +Proposition +12 +(Paved Characterization of Composition +(PCC)). Given thin groupoids A, B, C, strategies R, R′ : A ⇸ +B, S, S′ : B ⇸ C and weak morphisms F : R ⇒ R′ and +G: S ⇒ S′ of Thin, if there exist a functor H : S ⊙ R → +S′ ⊙ R′ and two natural transformations Hl and Hr as in +S ⊙ R +S′ ⊙ R′ +R +R′ +l +H +Hl +==⇒ +l +F +and +S ⊙ R +S′ ⊙ R′ +S +S′ +r +H +Hr +==⇒ +r +G +such that +S ⊙ R +S′ ⊙ R′ +R +R′ +B +B +l +H +Hl +==⇒ +l +F +∂R +B +F B +==⇒ +∂R′ +B += +S ⊙ R +S′ ⊙ R′ +S +S′ +B +B +r +H +Hr +==⇒ +r +G +∂S +B +GB +==⇒ +∂S′ +B +and such that the natural transformations +HA ˆ= +S ⊙ R +S′ ⊙ R′ +R +R′ +A +A +l +H +Hl +==⇒ +l +F +∂R +A +F A +==⇒ +∂R′ +A +HC ˆ= +S ⊙ R +S′ ⊙ R′ +S +S′ +C +C +r +H +Hr +==⇒ +r +G +∂S +C +GC +==⇒ +∂S′ +C +are positive over A⊥ and C respectively, we have that H ˆ= +(H, HA, HC) is a positive morphism S ⊙ R ⇒ S′ ⊙ R′ of +Thin and that H = G ⊙ F. +Proof. The fact that H is a 2-cell S⊙R ⇒ S′⊙R′ of Thin is +immediate by the polarity assumption. The equality of 2-cells +given by the hypothesis can be rewritten as +S ⊙ R +R +S +S′ ⊙ R′ +R′ += +S′ +B +H +F +Hl +==⇒ +G +(Hr)−1 +=====⇒ +∂R′ +B +∂S′ +B += +S ⊙ R +R += +S +R′ +S′ +B +F +∂R +B +G +∂S +B +∂R′ +B +∂S′ +B +(F B)−1 +=====⇒ +GB +==⇒ +so that Hl and Hr provide a factorization of the pseudocone +on the right, and define an object of the groupoid of com- +positions mentioned in Section III-D5. The actual horizontal + +composition in Thin is then obtained by applying the biequiv- +alence of Proposition 3. But since ⟨HA, HC⟩ is already a +positive natural transformation on A ⊸ C, this biequivalence +does nothing on this object and H = G ⊙ F. +Lemma 7. Given thin groupoids A, B, C, we have a functor +(−) ⊙ (−): Thin(B, C) × Thin(A, B) → Thin(A, C). +Proof. By the definition we took for the composition of the +2-cells of Thin, we already have that (−) ⊙ (−) respects the +sources and targets of weak morphisms, so that we are left to +verify functoriality. +Given R ∈ Thin(A, B) and S ∈ Thin(B, C), a solution +in H, Hl and Hr for the equation +S ⊙ R +S ⊙ R +R +R +B +B +H +l +Hl +==⇒ +l +idR +∂R +B += +∂R +B += +S ⊙ R +S ⊙ R +S +S +B +B +H +r +Hr +==⇒ +r +idS +∂S +B += +∂S +B +is given by H = idR⊙S, Hl = idl and Hr = idr. Thus, since +identities are member of A− and C+, the polarity condition +of Proposition 12 is satisfied so that +idS ⊙ idR += +(idS⊙R, id∂R +A◦l, id∂S +C◦r) += +idS⊙R. +Now, given four positive morphisms organized as +R +F−→ R′ +F ′ +−→ R′′ ∈ Thin(A, B) , +S +G +−→ S′ +G′ +−→ S′′ ∈ Thin(B, C) , +the procedure to compute F ⊙ G gives us H, Hl and Hr s.t. +S ⊙ R +S′ ⊙ R′ +R +R′ +B +B +l +H +Hl +==⇒ +l +F +∂R +B +F B +==⇒ +∂R′ +B += +S ⊙ R +S′ ⊙ R′ +S +S′ +B +B +r +H +Hr +==⇒ +r +G +∂S +B +GB +==⇒ +∂S′ +B +and with ∂R′ +A Hl and ∂S′ +C Hr respectively negative on A and +positive on C; similarly, the procedure to compute F ′ ⊙ G′ +gives us H′, H′l and H′r such that +S′ ⊙ R′ +S′′ ⊙ R′′ +R′ +R′′ +B +B +l +H′ +H′l +==⇒ +l +∂R′ +B +F ′ +F ′B +===⇒ +∂R′′ +B += +S′ ⊙ R′ +S′′ ⊙ R′′ +S′ +S′′ +B +B +r +H′ +H′r +===⇒ +r +∂S′ +B +G′ +G′B +===⇒ +∂S′′ +B +and with ∂R′′ +A H′l and ∂S′′ +C H′r respectively negative on A and +positive on C. On the one hand, we thus have that (G′ ⊙ +F ′) ◦ (G ⊙ F) is the span morphism K = (K, Kl, Kr) with +K = H′H and +Kl = +S ⊙ R +S′ ⊙ R′ +S′′ ⊙ R′′ +R +R′ +R′′ +A +A +A +l +H +Hl +==⇒ +l +H′ +H′l +==⇒ +l +F +∂R +A +F A +==⇒ +∂R′ +A +F ′ +F ′A +===⇒ +∂R′′ +A +Kr = +S ⊙ R +S′ ⊙ R′ +S′′ ⊙ R′′ +S +S′ +S′′ +C +C +C +r +H +Hr +==⇒ +r +H′ +H′r +===⇒ +r +G +∂S +C +GC +==⇒ +∂S′ +C +G′ +G′C +===⇒ +∂S′′ +C +. +On the other hand, we have +S ⊙ R +S′ ⊙ R′ +S′′ ⊙ R′′ +R +R′ +R′′ +B +B +B +l +H +Hl +==⇒ +l +H′ +H′l +==⇒ +l +F +∂R +B +F B +==⇒ +∂R′ +B +F ′ +F ′B +===⇒ +∂R′′ +B += +S ⊙ R +S′ ⊙ R′ +S′′ ⊙ R′′ +S +S′ +S′′ +B +B +B +r +H +Hr +==⇒ +r +H′ +H′r +===⇒ +r +G +∂S +B +GB +==⇒ +∂S′ +B +G′ +G′B +===⇒ +∂S′′ +B +. +Moreover, the two natural transformations obtained as the +horizontal pasting of Hl and H′l along l and the horizontal +pasting of Hr and H′r along r satisfy the polarity condition +of the PCC. +Hence, by considering again the diagrammatic definition of +K, the PCC tells us that K is also (G′ ◦ G)⊙ (F ′ ◦ F), which +concludes functoriality. +We now show the unitality of the horizontal composition. +Given a thin groupoid A, we write ccA for the identity span +A +idA +←−− A +idA +−−→. We have +Lemma 8. Given a thin groupoid A, we have ccA ∈ TA⊸A. +Proof. This is an easy consequence of Proposition 1 and +Proposition 2. +We also write ccA for the corresponding functor 1 → +Thin(A, A). Given an additional thin groupoid B, there is +a transformation R between the functors +Thin(A, B) +∼ +−→ Thin(A, B) × 1 → · · · +id×ccA +−−−−−→ Thin(A, B) × Thin(A, A) +(−)⊙(−) +−−−−−→ Thin(A, B) +and +Thin(A, B) +id +−→ Thin(A, B) + +whose component at S ∈ Thin(A, B) is defined as follows. +Recall that the span S ⊙ ccA is defined by the pullback +S ⊙ ccA +A +S +A +A +B +l +r +idA +idA +∂S +A +∂S +B +. +Then, RS +ˆ= r is an isomorphism (as the pullback of an +isomorphism) which moreover induces a strong isomorphism +of strategies RS : S ⊙ ccA ⇒ S ∈ Thin. +Lemma 9. R ˆ= (RS)S∈Thin is a natural isomorphism. +Proof. Let A, B, S, S′ : A +⇸ +B and F : S +→ +S′ in +Thin(A, B). We first picture the two compositions T +ˆ= +ccA ⊙S and T ′ ˆ= ccA ⊙S′ on the diagram +T +A +S +A +A +B +A +S′ +T ′ +lT +rT +idA +idA +∂S +A +∂S +B +idA +idA +∂S′ +A +∂S′ +B +lT ′ +rT ′ +. +We now compute the composition F ⊙ idccA. By the PCC, it +is the morphism ˜F : S ⊙ ccA ⇒ S′ ⊙ ccA defined by +˜F += +T +rT +−−→ S +F−→ S′ +(rT ′)−1 +−−−−−→ T ′ +and +˜F A = +T +S +S′ +T ′ +A +A +A +A +A +A +A +A +lT +r T += +∂S +A +F +F A +==⇒ ∂S′ +A +(r T ′ )−1 += +lT ′ +idA += +idA += +idA += +idA +˜F B = +T +S +S′ +T ′ +S +S′ +B +B +rT +rT += +F +(rT ′ )−1 +rT ′ +∂S +B +F +F B +==⇒ +∂S′ +B +. +The naturality of R is then expressed by the equation +RS′ ◦(F ⊙ idccA) = F ◦ RS +that we now check. We first have +RS′ ◦(F ⊙ idccA) += +T +rT +−−→ S +F−→ S′ += +F ◦ RS . +Moreover, +(RS′ ◦(F ⊙ idccA))A += +T +S +S′ +A +A +A +A +A +A +lT +rT += +∂S +A +F +F A +==⇒ +∂S′ +A +idA += +idA += +idA += +(F ◦ RS)A +and +(RS′ ◦(F ⊙ idccA))B += +T +S +S′ +S +S′ +B +B +r T +r T += +F +∂S +B +F +F B +==⇒ +∂S′ +B += +(F ◦ RS)B. +Which concludes the proof that R defines a natural iso. +Similarly, there is a transformation L between +Thin(A, B) +∼ +−→ 1 × Thin(A, B) → · · · +ccB ×id +−−−−−→ Thin(B, B) × Thin(A, B) +(−)⊙(−) +−−−−−→ Thin(A, B) +and +Thin(A, B) +id +−→ Thin(A, B) +whose component at a strategy S ∈ Thin(A, B) is defined as +follows. Recall that the span ccB ⊙S is defined by the pullback +ccB ⊙S +S +B +A +B +B +l +r +∂S +B +∂S +A +idB +idB +. +Then, LS +ˆ= l is an isomorphism (as the pullback of an +isomorphism) which moreover induces a strong isomorphism +of thin spans LS : ccB ⊙S ⇒ S ∈ Thin. As before, we have +Lemma 10. L ˆ= (LS)S∈Thin is a natural isomorphism. +Given thin groupoids A, B, C, D, there is a transformation +A: ((−) ⊙ (−)) ⊙ (−) ⇒ (−) ⊙ ((−) ⊙ (−)) +: Thin(C, D) × Thin(B, C) × Thin(A, B) +→ Thin(A, D) +whose component at S ∈ Thin(A, B), T ∈ Thin(B, C) and +U ∈ Thin(C, D) is given by a strong morphism +AS,T,U : (U ⊙ T ) ⊙ S → U ⊙ (T ⊙ S) + +defined as expected between the two compositions using the +different pullbacks involved, as in +(U ⊙ T ) ⊙ S +U ⊙ T +S +T +U +A +B +C +D +S +T +U +T ⊙ S +U ⊙ (T ⊙ S) +r(U⊙T )⊙S +l(U⊙T )⊙S +AS,T,U +lU⊙T +rU⊙T +lT ⊙S +rT ⊙S +lU⊙(T ⊙S) +rU⊙(T ⊙S) +. +An inverse for AS,T,U is defined symmetrically, so that A is +an isomorphic transformation. +Lemma 11. The transformation A is a natural isomorphism. +Proof. Let F : S ⇒ S′ : A ⇸ B, G: T ⇒ T ′: B ⇸ C +and H : U ⇒ U ′ : C ⇸ D be weak morphisms in Thin. +We compute (U ⊙ T ) ⊙ S as usual but moreover factor the +projection l (U⊙T )⊙S canonically through T ⊙ S by a unique +morphism ˜l +(U⊙T )⊙S so that we get a diagram +(U ⊙ T ) ⊙ S +T ⊙ S +U ⊙ T +S +T +U +A +B +C +D +˜l +(U⊙T )⊙S +and we get a similar diagram for (U ⊙ T ) ⊙ S. Symmetri- +cally, the projection r U⊙(T ⊙S) can be factored canonically +through U ⊙ T by a morphism ˜r U⊙(T ⊙S), and the projection +r U′⊙(T ′⊙S′) through U ′ ⊙ T ′ by a morphism ˜r U′⊙(T ′⊙S′). +Note that l U⊙(T ⊙S) ◦ AS,T,U = ˜l +(U⊙T )⊙S and other similar +equalities hold. By computing K ˆ= G ⊙ F, we get natural +transformations ˜KA and ˜KC such that +KA = +T ⊙ S +T ′ ⊙ S′ +S +S′ +A +A +K +lT ⊙S +˜ +KA +==⇒ +lT ′⊙S′ +∂S +A +F +F A +==⇒ +∂S′ +A +and +KC = +T ⊙ S +T ′ ⊙ S′ +T +T ′ +C +C +K +rT ⊙S +˜ +KC +==⇒ +rT ′⊙S′ +∂T +C +G +GC +==⇒ +∂T ′ +C +which satisfy moreover that +T ⊙ S +T ′ ⊙ S′ +S +S′ +B +B +K +lT ⊙S +˜ +KA +==⇒ +lT ′⊙S′ +∂S +B +F +F B +==⇒ +∂S′ +B += +T ⊙ S +T ′ ⊙ S′ +T +T ′ +B +B +K +r T ⊙S +˜ +KC +==⇒ +r T ′⊙S′ +∂T +B +G +GB +==⇒ +∂T ′ +B +. +Similarly, by computing L ˆ= H ⊙ G, we get natural transfor- +mations ˜LB and ˜LD such that +LB = +U ⊙ T +U ′ ⊙ T ′ +T +T ′ +B +B +L +lU⊙T +˜LB +==⇒ +lU′⊙T ′ +∂T +B +G +GB +==⇒ +∂T ′ +B +and +LD = +U ⊙ T +U ′ ⊙ T ′ +U +U ′ +D +D +L +rU⊙T +˜LD +==⇒ +rU′⊙T ′ +∂U +D +H +HD +===⇒ +∂U′ +D +which satisfy moreover that +U ⊙ T +U ′ ⊙ T ′ +T +T ′ +C +C +L +lU⊙T +˜LB +==⇒ +lU′⊙T ′ +∂T +C +G +GC +==⇒ +∂T ′ +C += +U ⊙ T +U ′ ⊙ T ′ +U +U ′ +C +C +L +r U⊙T +˜LD +==⇒ +r U′⊙T ′ +∂U +C +H +HC +==⇒ +∂U′ +C +. +Since +(U ′ ⊙ T ′) ⊙ S′ +T ′ ⊙ S′ +U ′ ⊙ T ′ +T ′ +r (U′⊙T ′)⊙S′ +˜l +(U′⊙T ′)⊙S′ +r T ′⊙S′ +lU′⊙T ′ +is a bipullback by Lemma 5, and that the natural transforma- +tions ˜KC and ˜LB define a pseudocone of vertex (U ⊙ T ) ⊙S + +on the associated cospan, we get M, ˜ +M A and ˜ +M D such that +(U ⊙ T ) ⊙ S +(U ′ ⊙ T ′) ⊙ S′ +T ⊙ S +T ′ ⊙ S′ +T +T ′ +M +˜l +(U⊙T )⊙S +˜ +MA +===⇒ +˜l +(U′⊙T ′)⊙S′ +K +rT ⊙S +˜ +KC +==⇒ +rT ′⊙S′ +G += +(U ⊙ T ) ⊙ S +(U ′ ⊙ T ′) ⊙ S′ +U ⊙ T +U ′ ⊙ T ′ +T +T ′ +M +r(U⊙T )⊙S +˜ +MD +===⇒ +r(U′⊙T ′)⊙S′ +L +rT ⊙S +˜LB +==⇒ +rT ′⊙S′ +G +. +We thus get a weak morphism M = (M, M A, M D) between +(U ⊙ T ) ⊙ S and (U ′ ⊙ T ′) ⊙ S′ in Span, where +M A = +(U ⊙ T ) ⊙ S +(U ′ ⊙ T ′) ⊙ S′ +T ⊙ S +T ′ ⊙ S′ +S +S′ +A +A +M +˜l +(U⊙T )⊙S +˜ +MA +===⇒ +˜l +(U′⊙T ′)⊙S′ +K +lT ⊙S +˜ +KA +==⇒ +lT ′⊙S′ +∂S +A +F +F A +==⇒ +∂S′ +A +and +M D = +(U ⊙ T ) ⊙ S +(U ′ ⊙ T ′) ⊙ S′ +U ⊙ T +U ′ ⊙ T ′ +U +U ′ +D +D +M +r(U⊙T )⊙S +˜ +MD +===⇒ +r(U′⊙T ′)⊙S′ +L +rU⊙T +˜LD +==⇒ +rU′⊙T ′ +∂U +D +H +HD +===⇒ +∂U′ +D +. +Using the biequivalence of Proposition 3, we can suppose +that +˜ +M A and +˜ +M B where chosen so that M A and M D are +respectively negative on A and positive on D. By the PCC, +we can then verify directly that M = (H ⊙ G) ⊙ F. +Now +consider +the +positive +weak +morphism +¯ +M += +AS′,T ′,U′ ◦M ◦ A−1 +S,T,U: we have that ¯ +M A and ¯ +M D are as on +Figure 5 and Figure 6. By using the PCC to characterize the +composition of G⊙F with H, we have that H⊙(G⊙F) = ¯ +M, +so that +AS′,T ′,U′ ◦((H ⊙ G) ⊙ F) = (H ⊙ (G ⊙ F)) ◦ AS,T,U +which was the wanted naturality. +We can now prove Theorem 2: +Proof of Theorem 2. By Lemmas 10 to 11, the 0-composition +is naturally left unital, right unital and associative. Moreover, +the coherence conditions on the natural isomorphisms, re- +quired by the definition of bicategories, directly follow from +their pullback definitions. +E. Renamings +Proposition 13. Given F : A → B ∈ Ren, ˇF ∈ Thin. +Proof. We first prove that ˇF ∈ UB⊸A and we use the dual +version Proposition 1 for this purpose. We already have that +idA ∈ UA since it is an isomorphism. We are left to show +that ˇF ⋆@S ∈ UB⊥ for every S ∈ UA. Up to isomorphism of +domain, ˇF ⋆@S is the composition F ◦ ∂S and, by hypothesis +on F, the latter is in UB⊥. So ˇF ∈ UB⊸A by Proposition 1. +We are left to show that ˇF ∈ TB⊸A. But it follows from +Proposition 2 by the same arguments as for uniformity. +Given F : A → B and G: B → C in Ren, there is +mF,G : +ˇ +(GF) ⇒ ˇF ⊙ ˇG +a strong morphism of Thin, defined by the universal property +of the pullback as +mF,G = ⟨F, idA⟩: A → ˇF ⊙ ˇG. +Lemma 12. Let A, B, C be thin groupoids, and φ: F ⇒ +F ′ : A → B and ψ: G ⇒ G′ : B → C be two 2-cells of +Ren. The composition ˇφ ⊙ ˇψ: ˇF ⊙ ˇG ⇒ ˇF ′ ⊙ ˇG′ is given by +(H, χC, χA) where χC and χA are respectively +ˇF ⊙ ˇG +ˇF ′ ⊙ ˇG′ +B +B +C +C +H +l +¯φ=⇒ +l +G +ψ=⇒ +G′ +and +ˇF ⊙ ˇG +ˇF ′ ⊙ ˇG′ +A +A +A +A +H +r += +r +idA += +idA +with +¯φ = +ˇF ⊙ ˇG +A +A +ˇF ′ ⊙ ˇG′ +B +B +B +B +r +l += +F +φ=⇒ +r−1 +F ′ += +l +and H as on the top of ¯φ. +Proof. This is a consequence of the PCC. +Proposition 14. Given thin groupoids A, B, C, the 2-cells +mF,G for F : A → B and G: B → C in Thin define a +natural iso m of type +ˇ +((−)(2) ◦ (−)(1)) ⇒ +ˇ +(−)(1) ⊙ ˇ +(−)(2) +: Ren(A, B) × Ren(B, C) → Thin(C, A) . +Proof. Let φ: F ⇒ F ′ : A → B and ψ: G ⇒ G′ : B → C be +two 2-cells of Ren. +We must show that +ˇ +(GF) +ˇ +(G′F ′) +ˇF ⊙ ˇG +ˇF ′ ⊙ ˇG′ +ˇ +(ψφ) +mF,G += +mF ′,G′ +ˇφ⊙ ˇ +ψ + +¯ +M A = +U ⊙ (T ⊙ S) +(U ⊙ T ) ⊙ S +(U ′ ⊙ T ′) ⊙ S′ +U ′ ⊙ (T ′ ⊙ S′) +T ⊙ S +T ⊙ S +T ′ ⊙ S′ +T ′ ⊙ S′ +S +S +S′ +S′ +A +A +A +A +lU⊙(T ⊙S) +A−1 +S,T,U += +M +˜l +(U⊙T )⊙S +˜ +MA +===⇒ +˜l +(U′⊙T ′)⊙S′ +AS′,T ′,U′ += +lU′⊙(T ′⊙S′) +lT ⊙S += +K +lT ⊙S +˜ +KA +==⇒ +lT ′⊙S′ += +lT ′⊙S′ +∂S +A += +∂S +A +F +F A +==⇒ +∂S′ +A += +∂S′ +A +Fig. 5. The natural transformation ¯ +MA +¯ +M D = +U ⊙ (T ⊙ S) +(U ⊙ T ) ⊙ S +(U ′ ⊙ T ′) ⊙ S′ +U ′ ⊙ (T ′ ⊙ S′) +U ⊙ T +U ⊙ T +U ′ ⊙ T ′ +U ′ ⊙ T ′ +U +U +U ′ +U ′ +D +D +D +D +˜rU⊙(T ⊙S) +A−1 +S,T,U += +M +r(U⊙T )⊙S +˜ +MD +===⇒ +r(U′⊙T ′)⊙S′ +AS′,T ′,U′ += +˜rU′⊙(T ′⊙S′) +rU⊙T += +L +rU⊙T +˜LD +==⇒ +rU′⊙T ′ += +rU′⊙T ′ +∂U +D += +∂U +D +H +HD +===⇒ +∂U′ +D += +∂U′ +D +Fig. 6. The natural transformation ¯ +MD +in Thin(C, A). But this equation can easily be deduced from +Lemma 12, whose statement implies that +ˇφ ⊙ ˇψ = mF ′,G′ ◦ +ˇ +(ψφ) ◦ m−1 +F,G +We can now finish the proof of Proposition 5: +Proof. For every A and B, +ˇ +(−) can easily be seen to define a +functor ˇ +(−)A,B : Ren(A, B) → Thin(B, A). We are just left +to show that the usual coherence conditions for pseudofunctors +are satisfied by m. But the required coherence conditions +follow directly from the universal property of the pullback. +F. The ! functor and its structure +We +now +finish +the +definition +of +the +pseudofunctor +!: Thin → Thin. First, while we described weak morphisms +between two spans S, S′ : A ⇸ B as triples (F, F A, F B), +often identifying F with the whole triple, we will in the +following often refer to the first element of the triple F by +F, for disambiguation. We now start by proving the naturality +of the coherence m. +Lemma 13. Let F = (F , F A, F B): S ⇒ S′ : A ⇸ B and +G = (G, GB, GC): T ⇒ T ′: B ⇸ C. Let χS and χT be +two 2-cells given by the definition of horizontal composition +so that G ⊙ F is given by the two 2-cells +T ⊙ S +T ′ ⊙ S′ +S +S′ +A +A +G⊙F +l +χS +==⇒ +l +F +∂S +A +F A +==⇒ +∂S′ +A +idA +and +T ⊙ S +T ′ ⊙ S′ +T +T ′ +C +C +G⊙F +r +χT +==⇒ +r +G +∂T +C +GC +==⇒ +∂T ′ +C +idC +. +The composition !G ⊙ !F is then given by the two 2-cells +!T ⊙ !S +!(T ⊙ S) +!(T ′ ⊙ S′) +!T ′ ⊙ !S′ +!S +!S +!S′ +!S′ +!A +!A +!A +!A +m−1 +S,T +l += +!(G⊙F ) +! l +!χS +==⇒ +mS′,T ′ +! l += +l +id!S +!∂S +A += +!F +!∂S +A +!F A +===⇒ +id!S′ +!∂S′ +A += +!∂S′ +A +!idA +!idA +!idA +and +!T ⊙ !S +!(T ⊙ S) +!(T ′ ⊙ S′) +!T ′ ⊙ !S′ +!T +!T +!T ′ +!T ′ +!C +!C +!C +!C +m−1 +S,T +r += +!(G⊙F ) +! r +!χT +==⇒ +mS′,T ′ +! r += +r +id!T +!∂T +C += +!G +!∂T +C +!GC +===⇒ +id!T ′ +!∂T ′ +C += +!∂T ′ +C +!idC +!idC +!idC +. + +Proof. We use the PCC. The respective positivity and nega- +tivity of the proposed 2-cells follow from the positivity of the +vertical composition of χS and F A and the negativity of the +vertical composition of χT and GC. We are left to show the top +row of the two proposed 2-cells satisfy the equality required +by the PCC, but it follows from that satisfied by χS and χT +by the functoriality of !: Gpd → Gpd on 2-cells. +We can now conclude naturality: +Lemma 14. The morphisms mA,B,C +S,T +define a natural iso +mA,B,C : !((−) ⊙ (−)) ⇒ !(−) ⊙ !(−) +: Thin(A, B) × Thin(B, C) → Thin(A, C). +Proof. Let F : S ⇒ S′ : A ⇸ B and G: T ⇒ T ′: B ⇸ C. +We must show that +mA,B,C +S′,T ′ ◦ !(G ⊙ F) = (!G ⊙ !F) ◦ mA,B,C +S,T +. +But it directly follows from Lemma 13, whose conclusion +states in particular that +mA,B,C +S′,T ′ ◦ !(G ⊙ F) ◦ (mA,B,C +S,T +)−1 = !G ⊙ !F. +We can thus conclude that ! is a pseudofunctor: +Proposition 15. The functor !: Gpd → Gpd induces a +pseudofunctor !: Thin → Thin. +Proof. By Lemma 14, we have an adequate natural isomor- +phism expressing the functoriality of ! on Thin. The coher- +ence laws for pseudofunctors can be directly verified by the +universal properties of the pullbacks involved in the horizontal +compositions appearing in these laws. +G. The ! pseudocomonad +We are going to derive the ! pseudocomonad on Thin +from the ! pseudomonad on Gpd through functoriality. Before +using this functoriality argument, we need to describe what are +the (higher) categories we are going to apply it to. The domain +(bi)category will be the one of endofunctors on Gpd with +properties similar to the ones of !: Gpd → Gpd, while the +codomain (bi)category will be the one of endopseudofunctors +on Thin. We shall first discuss how to relate some functors +on Gpd to pseudofunctors on Thin. +Given a functor H: Gpd → Gpd, there is a canonical +uniform groupoid HA = (HA, UHA) associated to any +uniform groupoid A, where UHA = {HS | S ∈ UA}⊥⊥. +Proposition 16. If H preserves pullbacks, and pullbacks +which are bipullbacks, then given uniform groupoids A and +B, and S ∈ UA⊸B, we have HS ∈ UHA⊸HB. +Proof. This is a direct consequence of Proposition 1. +We call bicartesian functors the functors H which satisfy +the hypothesis of the above property. +A ±-functor is a tuple (H, H+, ι) with H, H+ being +functors Gpd → Gpd where H and H+ are bicartesian +and preserve functors (between groupoids) that are bijective on +objects (of the groupoids), and such that H+ preserves discrete +groupoids, and ι: H+ ⇒ H being a natural transformation +which is pointwise (that is, such that each ιX is) monomorphic +and surjective on objects (of the groupoids), satisfying more- +over that it is bicartesian, meaning that its naturality squares +are both pullbacks and bipullbacks. Intuitively, the definition +of ±-functor is an abstraction of the case of !: Gpd → Gpd, +from which we derive a functor Thin → Thin. In the case +of !, given a thin groupoid A, a positive sub-groupoid (!A)+ +is defined from a construction which is not derivable from the +definition of ! and the data of A and A+, so that we have to +take it into account in our definition of ±-functor, in the form +of a functor H+ and a natural transformation ι: H+ ⇒ H. +We should a priori also require similar data for the negative +side, but it so happens that, in the case of !, (!A)− = !(A−), so +that it is in fact not necessary. In order to show that ! induces +a pseudocomonad on Thin, the ±-functors we will consider +will only be iterated compositions of !. +Given a ±-functor (H, H+, ι) and a thin groupoid A, there +is a canonical thin groupoid HA whose underlying uniform +groupoid is defined as earlier, whose class of thin prestrategies +is THA = {HS | S ∈ TA}⊥⊥⊥⊥, and whose negative and +positive sub-groupoids are (HA)− = HA− and (HA)+ = +H+A+ with embeddings given by the compositions +HA− +H(id− +A) +−−−−−→ HA +and +H+A+ +H+(id+ +A) +−−−−−→ H+A +ιA +−→ HA. +By the conditions of ±-functors, they are elements of THA +and T⊥⊥ +HA as required (exercise to the reader). +Proposition 17. Given a ±-functor (H, H+, ι) and thin +groupoids A and B, and S ∈ TA⊸B, HS ∈ THA⊸HB. +Proof. This is an easy consequence of the hypotheses on a +±-functor and Propositions 1 and 2. +Proposition 18. Given a ±-functor (H, H+, ι) and a thin +groupoid A, H preserves negative (resp. positive) 2-cells. +Proof. Given a negative 2-cell φ: F ⇒ F ′ : X → A, writing +X(0) for the discrete groupoid with the same object as X, we +have a commutative diagram +X(0) +X +A− +A +e +φ− +==⇒ +¯ +F +¯ +F ′ +φ=⇒ +F +F ′ +id− +A +for some 2-cell φ− : ¯F ⇒ ¯F ′, where the top arrow e is the +canonical embedding. By hypothesis on H, the image of e by +H is bijective on objects of X. Moreover, by functoriality, +we have H(id− +A) ◦ H(φ−) = H(φ) ◦ H(e). Thus, all the +components of the natural transformation H(φ) are in the +image of H(id− +A). Thus, it is negative. + +Now, given a positive 2-cell φ: F ⇒ F ′ : X → A, we have +a similar commutative diagram +X(0) +X +A+ +A +e +φ+ +==⇒ +¯ +F +¯ +F ′ +φ=⇒ +F +F ′ +id+ +A +for some 2-cell φ+ : ¯F ⇒ ¯F ′. We then have the commutative +diagram +H+X(0) +H+X +HX +H+A+ +H+A +HA +H+(e) +H+(φ+) +=====⇒ +H+( ¯ +F ) +H+( ¯ +F ′) +ιX +H+(φ) +====⇒ +H+(F ) +H+(F ′) +H(φ) +===⇒ +H(F ) +H(F ′) +H+(id+ +A) +ιA +where the top arrow is bijective on objects by assumptions. +Thus, all the components of H(φ) are in the image of ιA ◦ +H+(id+ +A), so that H(φ) is positive. +Given two ±-functors H += +(H, H+, ι) and K += +(K, K+, κ), a ±-transformation is a pair (α, α+) of bi- +cartesian natural transformations where α: H ⇒ K and +α+ : H+ ⇒ K+ are such that +κ ◦ α+ = α ◦ ι. +Now, a ±-modification between two such ±-transformations +(α, α+) and (β, β+) is the data of a modification m: α ⇛ β +in the 3-category of 2-categories. +Proposition 19. Given a thin modification m: α ⇛ β : H ⇒ +K and a thin groupoid A, the 2-cell +mA : αA ⇒ βA : HA → KA +is negative as a 2-cell on KA. +Proof. Since m is a modification, we have that mA◦H(id− +A) = +K(id− +A) ◦ mA−. Since H(id− +A) is bijective on objects by +hypotheses on H and id− +A, we have that all the components of +mA are in the image of K(id− +A), so that they are negative. +±-functors, ±-transformations and ±-modifications can be +equipped with the evident operations in order to form a +strict 3-category ±-Funct with one object (which, morally, +is Gpd, the domain and codomain of each ±-functor). The +important point to note here is that the data of !, η, µ, α, β +and γ induces the expected way a pseudomonad in ±-Funct. +We shall now describe an operation +ˇ +(−) relating ±-Funct +and Thin. First, a ±-functor (H, H+, ι) induces an endofunc- +tor ˇH: Thin → Thin mapping a thin groupoid A to the thin +groupoid HA defined as earlier, and mapping spans and their +weak morphisms to their images by H, which is well-defined +by Propositions 17 and 18. +Now, given a ±-transformation (α, α+): (H, H+, ι) ⇒ +(K, K+, κ), +we +define +a +pseudonatural +transformation +ˇα: ˇK ⇒ ˇH whose component at a thin groupoid A is +ˇ +(αA), +that is, the image of αA : HA → KA by the pseudofunctor +ˇ +(−) : Renop → Thin. +We can then extend the pseudofunctor ˇ− : Renop → Thin +to some sort of 3-dimensional functor +ˇ +(−) : ±-Functco → Bicat +sending the unique 0-cell to Thin and the higher cells in +the hom-bicategory Bicat(Thin, Thin) (here, ±-Functco +denotes ±-Funct with 2-cells reversed). +While this is probably a trifunctor, it would be very tiresome +to prove. Instead, we will only rely on the simpler proposition +stating that +Proposition 20. Considering ±-Funct as a strict 2-category +by forgetting the dimension 0, +ˇ +(−) induces a pseudofunctor +ˇ +(−) : ±-Functco → Bicat(Thin, Thin) +between bicategories. +Proof. By checking the axioms of pseudofunctors. +We can now briefly describe a proof of Theorem 3: +Proof of Theorem 3. The most satisfying proof of this state- +ment would rely on the fact that +ˇ +(−) : ±-Functco → Bicat(Thin, Thin) +is a trifunctor and that a trifunctor sends any pseudocomonad +to a pseudocomonad, but we do not know a proof for the latter +fact (though it is probably true) and deem a full proof of the +former tedious. +Instead, we can rely on the weaker Proposition 20 to prove +the required coherences. Following [40], we are required to +prove the equations of modifications of Figure 7 are verified. +The idea is to relate each of these equations to the equations +satisfied by the pseudomonad !: Gpd → Gpd, and this +is done through paving. For example, we use the following +pavings for the two first modifications of the left hand-side of +the first equation of Figure 7: +!!ˇµ ⊙ (!ˇµ ⊙ ˇµ) +(!!ˇµ ⊙ !ˇµ) ⊙ ˇµ +̌ +(!!µ) ⊙ (̌ +(!µ) ⊙ ˇµ) +(̌ +(!!µ) ⊙ ̌ +(!µ)) ⊙ ˇµ +̌ +(!!µ) ⊙ ̌ +(µ ◦ !µ) +̌ +(!µ ◦ !!µ) ⊙ ˇµ +̌ +((µ ◦ !µ) ◦ (!!µ)) +̌ +(µ ◦ (!µ ◦ (!!µ))) += += + +!!ˇµ ⊙ (!ˇµ ⊙ ˇµ) ⇛ (!!ˇµ ⊙ !ˇµ) ⊙ ˇµ ⇛ !(!ˇµ ⊙ ˇµ) ⊙ ˇµ ⇛ !(ˇµ! ⊙ ˇµ) ⊙ ˇµ ⇛ (!ˇµ! ⊙ !ˇµ) ⊙ ˇµ ⇛ !ˇµ! ⊙ (!ˇµ ⊙ ˇµ) +⇛ !ˇµ! ⊙ (ˇµ! ⊙ ˇµ) ⇛ (!ˇµ! ⊙ ˇµ!) ⊙ ˇµ = (!ˇµ ⊙ ˇµ)! ⊙ ˇµ ⇛ (ˇµ! ⊙ ˇµ)! ⊙ ˇµ = (ˇµ!! ⊙ ˇµ!) ⊙ ˇµ += +!!ˇµ ⊙ (!ˇµ ⊙ ˇµ) ⇛ !!ˇµ ⊙ (ˇµ! ⊙ ˇµ) ⇛ (!!ˇµ ⊙ ˇµ!) ⊙ ˇµ +ex +≡⇛ (ˇµ!! ⊙ !ˇµ) ⊙ ˇµ ⇛ ˇµ!! ⊙ (!ˇµ ⊙ ˇµ) ⇛ ˇµ!! ⊙ (ˇµ! ⊙ ˇµ) ⇛ (ˇµ!! ⊙ ˇµ!) ⊙ ˇµ +and +!ˇη! ⊙ (!ˇµ ⊙ ˇµ) ⇛ !ˇη! ⊙ (ˇµ! ⊙ ˇµ) ⇛ (!ˇη! ⊙ ˇµ!) ⊙ ˇµ = (!ˇη ⊙ ˇµ)! ⊙ ˇµ ⇛ cc! ! ⊙ ˇµ = cc!! ⊙ˇµ ⇛ ˇµ += +!ˇη! ⊙ (!ˇµ ⊙ ˇµ) ⇛ (!ˇη! ⊙ !ˇµ) ⊙ ˇµ ⇛ !(ˇη! ⊙ ˇµ) ⊙ ˇµ ⇛ ! cc! ⊙ˇµ = cc!! ⊙ˇµ ⇛ ˇµ +Fig. 7. The two required equations for ! to be a pseudocomonad +and +(!!ˇµ ⊙ !ˇµ) ⊙ ˇµ +!(!ˇµ ⊙ ˇµ) ⊙ ˇµ +(!̌ +(!µ) ⊙ !ˇµ) ⊙ ˇµ +!(̌ +(!µ) ⊙ ˇµ) ⊙ ˇµ +(̌ +(!!µ) ⊙ ̌ +(!µ)) ⊙ ˇµ +̌ +(!µ ◦ !!µ) ⊙ ˇµ +̌ +(!(µ ◦ !µ)) ⊙ ˇµ +!(̌ +µ ◦ !µ) ⊙ ˇµ +̌ +(!(µ ◦ !µ)) ⊙ ˇµ +̌ +(!µ ◦ !!µ) ⊙ ˇµ +̌ +(µ ◦ (!µ ◦ (!!µ))) +̌ +(µ ◦ (!µ ◦ !!µ)) += += += +. +The other elementary modifications of Figure 7 are paved +similarly, so that the first equation of Figure 7 is reduced to +the equation +̌ +(µ ◦ !µ ◦ !!µ) ⇛ +̌ +(µ ◦ !µ ◦ !µ!) +⇛ +̌ +(µ ◦ µ! ◦ !µ!) ⇛ +̌ +(µ ◦ µ! ◦ µ!!) += +̌ +(µ ◦ !µ ◦ !!µ) ⇛ +̌ +(µ ◦ µ! ◦ !!µ) += +̌ +(µ ◦ !µ ◦ µ!!) ⇛ +̌ +(µ ◦ µ! ◦ µ!!) +which is the image by +ˇ +(−) of an equation satisfied by the +pseudomonad (!, η, µ) on Gpd, and the second equation of +Figure 7 to the equation +̌ +((µ ◦ !µ) ◦ !η!) ⇛ +̌ +((µ ◦ µ!) ◦ !η!) += +̌ +(µ ◦ (µ! ◦ !η!)) ⇛ +̌ +(µ ◦ id!!) = ˇµ += +̌ +((µ ◦ !µ) ◦ !η!) = +̌ +(µ ◦ (!µ ◦ !η!)) ⇛ +̌ +(µ ◦ id!!) = ˇµ +also an image of an equation satisfied by the pseudomonad +(!, η, µ) on Gpd. So that (!, ˇη, ˇµ) is indeed a pseudocomonad +on Thin. +H. The cartesian product +Given thin groupoids A, B, we write¯l +A,B and ¯r A,B, simply +denoted ¯l and ¯r as earlier when A, B can be deduced from +the context, for the coprojections +¯l : A ֒→ A + B +and +¯r : B ֒→ A + B. +By applying the functor ˇ +(−), we get thin spans ˇ¯l: A&B ⇸ A +and ˇ¯r: A & B ⇸ B. Given thin groupoids Γ, A, B, we define +a functor +⟨−, −⟩Γ,A,B : Thin(Γ, A)×Thin(Γ, B) → Thin(Γ, A&B), +often abbreviated ⟨−, −⟩, as follows. Given S ∈ Thin(Γ, A) +and T ∈ Thin(Γ, B), we define ⟨S, T ⟩ as the span +⟨S, T ⟩ += +S + T +Γ +A + B +[∂S +Γ ,∂T +Γ ] +∂S +A+∂T +B +. +Proposition +21. Given +S +∈ +Thin(Γ, A) +and T +∈ +Thin(Γ, B), we have ⟨S, T ⟩ ∈ Thin(Γ, A & B). +Proof. By an adequate use of Propositions 1 and 2. +Given morphisms F : S → S′ ∈ Thin(Γ, A) and G: T → +T ′ ∈ Thin(Γ, B), ⟨F, G⟩ is defined as the morphism H with +H = F + G and +HΓ = +S + T +S′ + T ′ +Γ +Γ +F +G +[∂S +Γ ,∂T +Γ ] +[F Γ,GΓ] +=====⇒ +[∂S′ +Γ ,∂T ′ +Γ ] +and +HA+B = +S + T +S′ + T ′ +A + B +A + B +F +G +∂S +A+∂T +B +F A+GB +======⇒ +∂S′ +A +∂T ′ +B . + +One immediately verifies that these two 2-cells have the ade- +quate polarities, so that ⟨F, G⟩ ∈ Thin(Γ, A& B). Moreover, +the functoriality of ⟨−, −⟩Γ,A,B is immediately verified. +Given S ∈ Thin(Γ, A & B), we write SA for the span +SA = +SA +S +Γ +A +∂ +SA +A +∂S +Γ +where SA is the submonoid of S whose image by ∂S +A+B is in +A, and where ∂SA +A +is the induced map SA → A from ∂S +A+B. +We define a span SB similarly. +Proposition 22. Given S ∈ Thin(Γ, A & B), we have SA ∈ +Thin(Γ, A) and SB ∈ Thin(Γ, B). +Proof. As the result of the composition of two thin spans, we +know that ˇ¯l ⊙ S is in Thin(Γ, A). It is the span +ˇ¯l ⊙ S +S +A +Γ +A + B +A +l +r +∂S +Γ +∂S +A+B +¯l +idA +which, by an isomorphism of pullbacks, is isomorphic to +SA +S +A +Γ +A + B +A +∂ +SA +A +∂S +Γ +∂S +A+B +¯l +idA +which is exactly SA. Thus, the latter is in Thin(Γ, A). A +similar argument holds for SB. +The mapping S �→ SA extends to a functor +(−)A : Thin(Γ, A & B) → Thin(Γ, A) +the +expected +way. +Similarly, +we +obtain +a +functor +(−)B : Thin(Γ, A & B) → Thin(Γ, B). +Proposition 23. The functors (−)A and (−)B are isomorphic +to the functors ˇ¯l ⊙ (−) and ˇ¯r ⊙ (−) respectively. +Proof. Using the PCC, one can show the naturality of the +family of isomorphisms of thin spans ˇ¯l ⊙ S ∼= SA described +in the proof of Proposition 22, obtaining an isomorphism of +functors. A similar argument holds for (−)B and ˇ¯l ⊙(−). +Proposition 24. Given thin groupoids Γ, A, B, there is an +adjoint equivalence +Thin(Γ, A & B) +⊥ +Thin(Γ, A) × Thin(Γ, B) +((−)A,(−)B) +⟨−,−⟩ +. +Proof. Given S ∈ Thin(Γ, A & B), write ιS +A : SA ֒→ S and +ιS +B : SB ֒→ S for the canonical inclusions of Gpd. There is +a canonical γS : S → ⟨SA, SB⟩ defined as the inverse of +SA + SB +[ιS +A,ιS +B] +−−−−→ S +and it induces a strong morphism of thin span γS : S ⇒ +⟨SA, SB⟩ ∈ Thin(Γ, A&B). The naturality of γ can be shown +using the PCC, so that we obtain a natural isomorphism +γ : idThin(Γ,A&B) ⇒ ⟨(−)A, (−)B⟩. +Given (R, T ) ∈ Thin(Γ, A) × Thin(Γ, B), we have a +canonical isomorphism δA +R,T : ⟨R, T ⟩A → R defined as the +pullback isomorphism between +⟨R, T ⟩A +⟨R, T ⟩ +A +A + B +ι⟨R,T ⟩ +A +⌜ +∂⟨R,T ⟩ +A+B +¯l +and +R +⟨R, T ⟩ +A +A + B +¯l +∂R +A +⌜ +∂⟨R,T ⟩ +A+B +¯l +. +It extends to a morphism δA +R,T : ⟨R, T ⟩A → R ∈ Thin(Γ, A). +We define similarly a morphism δB +R,T : ⟨R, T ⟩B → T +∈ +Thin(Γ, B). Writing δR,T = ⟨δA +R,T , δB +R,T ⟩, the naturality of +δR,T with respect to R and T can be checked using the PCC, +so that we obtain a natural isomorphism +δ: (⟨(−)(1), (−)(2)⟩A, ⟨(−)(1), (−)(2)⟩B) +⇒ +idThin(Γ,A)×Thin(Γ,B). +We thus have an equivalence, and we verify that it is adjoint. +We check the first zigzag equation, namely +(δ((−)A, (−)B)) ◦ (((−)A, (−)B)γ) = id((−)A,(−)B). +(4) +In order to verify the above equality, by symmetry, we +just need to check its projection on Thin(Γ, A). So let +S ∈ Thin(Γ, A & B). The component of the left-hand side +of (4) at S is then +SA +(γS)A +−−−−→ (⟨SA, SB⟩)A +δA +⟨SA,SB⟩ +−−−−−−→ SA. +By unfolding the definition of γ and δ, we compute that +SA +(δA +⟨SA,SB⟩)−1 +−−−−−−−−−→ (SA + SB)A +((γS)A)−1 +−−−−−−→ SA +ιS +A +−→ S +is precisely ιS +A, which happens to be a monomorphism, so that +((γS)A)−1 ◦ (δA +⟨SA,SB⟩)−1 = idSA, which is, up to inverses, +what we wanted to show. Thus, the first zigzag equation holds. + +We now verify the second zigzag equation, namely +(⟨−, −⟩δ) ◦ (γ⟨−, −⟩) = id⟨−,−⟩. +(5) +So let (R, T ) ∈ Thin(Γ, A) × Thin(Γ, B). The component +of the left-hand side of (5) at (R, T ) is +⟨R, T ⟩ +γ⟨R,T ⟩ +−−−−→ ⟨⟨R, T ⟩A, ⟨R, T ⟩B⟩ +⟨δA +R,T ,δB +R,T ⟩ +−−−−−−−→ ⟨R, T ⟩ +By unfolding the definition of γ and δ, we compute that +R +¯l−→ ⟨R, T ⟩ +⟨δA +R,T ,δB +R,T ⟩−1 +−−−−−−−−−→ ⟨⟨R, T ⟩A, ⟨R, T ⟩B⟩ +γ−1 +⟨R,T ⟩ +−−−−→ ⟨R, T ⟩ +reduces to R +¯l−→ ⟨R, T ⟩ and similarly, +T +¯r−→ ⟨R, T ⟩ +⟨δA +R,T ,δB +R,T ⟩−1 +−−−−−−−−−→ ⟨⟨R, T ⟩A, ⟨R, T ⟩B⟩ +γ−1 +⟨R,T ⟩ +−−−−→ ⟨R, T ⟩ +reduces to T +¯r−→ ⟨R, T ⟩ so that, since ¯l and ¯r are jointly +surjective, γ−1 +⟨R,T ⟩ ◦ ⟨δA +R,T , δB +R,T ⟩−1 = id⟨R,T ⟩, which is, up to +inverses, what we wanted. So the second zigzag holds. +Proposition 25. Given thin groupoids Γ, A, B, there is an +adjoint equivalence +Thin(Γ, A & B) +⊥ +Thin(Γ, A) × Thin(Γ, B) +(ˇ¯l⊙(−),ˇ¯r⊙(−)) +⟨−,−⟩ +. +Proof. This is a consequence of Propositions 23 and 24. +Note that, given (R, T ) ∈ Thin(Γ, A) × Thin(Γ, B), the +component at (R, T ) of the counit associated to the adjoint +equivalence of Proposition 25 is the composite +(ˇ¯l ⊙ ⟨R, T ⟩, ˇ¯r ⊙ ⟨R, T ⟩) +∼ += +−→ (⟨R, T ⟩A, ⟨R, T ⟩B) +δ−→ (R, T ). +Now, we can conclude the proof of Proposition 6: +Proof of Proposition 6. We have the equalities Thin!(Γ, A& +B) = Thin(!Γ, A & B) and Thin!(Γ, A) × Thin!(Γ, B) = +Thin(!Γ, A) × Thin(!Γ, B). Moreover, it is quite immediate +that, up to these identifications, there is an isomorphism of +functors (L! ⊙!(−), R! ⊙!(−)) ∼= (ˇ¯l ⊙(−), ˇ¯r ⊙ (−)). Thus, the +unit/counit pair of the adjoint equivalence of Proposition 25 +can be adjusted to get a unit/counit pair witnessing that we +have an adjoint equivalence as in the statement. +I. The evaluation adjunction +We give here some additional details for the proof of +Proposition 7, stating the existence of an adjoint equivalence +between the currying operation and the evaluation one. This +adjoint equivalence will be derived from the Seely equivalence +already introduced. +1) Properties of the Seely equivalence: +Proposition 26. The family of functors sA,B for groupoids +A, B form a 2-natural transformation +s: !(−) × !(−) ⇒ !((−) + (−)): Gpd × Gpd → Gpd. +Proof. The naturality w.r.t 2-cells is checked by direct point- +wise computation. +Proposition 27. The natural transformation s is bicartesian. +Proof. By a direct use of the point-wise characterization of +pullbacks and Lemma 1 on the naturality squares of s. +Similarly, we have the same kind of properties for ¯s: +Proposition 28. The family of functors ¯sA,B for groupoids +A, B form a 2-natural transformation +¯s: !((−) + (−)) ⇒ !(−) × !(−): Gpd × Gpd → Gpd. +Proposition 29. The natural transformation ¯s is bicartesian. +2) The Seely coherence 2-cell: While the Seely isomor- +phisms of 1-categorical models of linear logic are required to +satisfy an equality, in our 2-categorical setting we only have +the following 2-cell +!!A × !!B +!!A × !!B +!(!A + !B) +!A × !B +!!(A + B) +!(A + B) +!(A + B) +s!A,!B +SeeA,B +=====⇒ +µA×µB +![!(¯l),!(¯r)] +sA,B +µA+B +We first compute the action of the two vertical morphisms +on objects of !!A × !!B. So let a = ((ai,j)j∈JA +i )i∈IA ∈ !!A +and b = ((bi,j)j∈JB +i )i∈IB ∈ !!B. The mappings associated +with the left vertical morphism are +(a, b) +�→(¯l((ai,j)j∈JA +i ))̟l(i)∈̟l(IA) ∪ (¯r((bi,j)j∈JB +i ))̟r(i)∈̟r(IB) +�→((¯l(ai,j))j∈JA +i )̟l(i)∈̟l(IA) ∪ ((¯r(bi,j))j∈JB +i )̟r(i)∈̟r(IB) +�→(¯l(ai,j))⟨̟l(i),j⟩∈� +i∈̟l(IA) JA +i ∪ (¯r(bi,j))⟨̟r(i),j⟩∈� +i∈̟r(IB ) JB +i +and the mappings associated with the right are +(a, b) �→ ((ai,j)⟨i,j⟩∈� +i∈IA JA +i , (bi,j)⟨i,j⟩∈� +i∈IB JB +i ) +�→ (¯l(ai,j))̟l(⟨i,j⟩)∈̟l(� +i∈IA JA +i ) +∪ (¯l(bi,j))̟r(⟨i,j⟩)∈̟r(� +i∈IB JB +i ) +where, given D ∈ Gpd and (di)i∈I, (d′ +j)j∈J ∈ !D with I ∩ +J = ∅, we write (di)i∈I ∪ (d′ +j)j∈J for the evident family +indexed by I ∪ J. We thus define (SeeA,B)a,b as the map +(π, (id)k∈K) where π is the bijection K → K′ with +K = � +i∈̟l(IA)JA +i ∪ � +i∈̟r(IB)JB +i + +and +K′ = ̟l(� +i∈IAJA +i ) ∪ ̟r(� +i∈IBJB +i ) +such that π maps [̟l(i), j] ∈ K to ̟l([i, j]) ∈ K′, and +[̟r(i), j] ∈ K to ̟r([i, j]) ∈ K′. The naturality of SeeA,B +with respect to morphisms (a, b) → (a′, b′) of !!A × !!B can +be readily checked. So SeeA,B is indeed a 2-cell of Gpd. +We moreover verify that +Proposition 30. The family of 2-cells SeeA,B for A, B ∈ +Gpd is natural with respect to functors F : A +→ +A′ +and G: B +→ +B′ of Gpd. In other words, See += +(SeeA,B)A,B∈Gpd is a modification. +Proof. A direct point-wise computation of naturality. +3) Properties of the evaluation span: Given thin groupoids +A, B, recall that we introduced a span +evA,B : (A ⇒ B) & A ⇸ B +defined by +evA,B += +!A × B +!A × B × !A +!(!A × B) × !A +!((!A × B) & A) +B +(l,r,l) +r +η!A×B×!A +s!A×B,A +. +We have that +Proposition 31. We have evA,B ∈ T!((A⇒B)&A)⊸B. +Proof. By an adequate use of Propositions 1 and 2, using +Proposition 27, the bicartesianness of η and Lemma 5 to get +the required bipullbacks. +4) The +currying +operation: +Recall +that, +given +thin +groupoids Γ, A, B and S ∈ Thin!(Γ & A, B), we defined +Λ(S) as the span +Λ(S) += +S +!Γ +!A × B +l ◦¯sΓ,A◦∂S +!(Γ+A) +(r ◦¯sΓ,A◦∂S +!(Γ+A),∂S +B) . +Proposition 32. Given thin groupoids Γ, A, B and S +∈ +Thin(Γ & A, B), we have Λ(S) ∈ T!Γ⊸(A⇒B). +Proof. While more involved than the other instances, this +still relies on Propositions 1 and 2, using Proposition 29 +and Lemma 5 to get the required intermediate bipullbacks. +The operation Λ(−) can be extended to weak morphisms +the expected way, and it is compatible with the polarities, and +we moreover have +Proposition 33. Given thin groupoids Γ, A, B, the operation +Λ(−): Thin!(Γ & A, B) → Thin!(Γ, A ⇒ B) +is functorial. +5) The uncurrying operation: We can define conversely +an uncurrying operation. Given S ∈ Thin!(Γ, A ⇒ B), we +define ¯Λ(S) as the span +¯Λ(S) += +S +!(Γ & A) +B +sΓ,A◦(∂S +!Γ,∂S +!A) +∂S +B +. +As before, using similar methods, we can verify that +Proposition 34. Given thin groupoids Γ, A, B and σ +∈ +Thin!(Γ, A ⇒ B), we have ¯Λ(σ) ∈ T!(Γ&A)⊸B. +Moreover, we can extend this uncurrying operation to the +weak morphisms the expected way, and this operation is +compatible with the polarities, and we moreover have +Proposition 35. Given thin groupoids Γ, A, B, the operation +¯Λ(−): Thin!(Γ, A ⇒ B) → Thin!(Γ & A, B) +is functorial. +6) The adjoint equivalences: We have a first adjoint equiv- +alence between the currying and uncurrying operations: +Proposition 36. There is an adjoint equivalence +Thin!(Γ, A ⇒ B) +⊥ +Thin!(Γ & A, B) +¯Λ(−) +Λ(−) +. +Proof. The application of ¯Λ(−) and Λ(−) to spans essentially +amounts to adequately postcompose the display maps of these +spans by sΓ,A and ¯sΓ,A. The unit/counit pair witnessing the ad- +joint equivalence are then easily derived from a unit/counit pair +(ΣΓ,A, ¯ΣΓ,A) witnessing the adjoint equivalence sΓ,A ⊣ ¯sΓ,A. +For example, given S ∈ Thin!(Γ, A ⇒ B), the component of +the unit of ¯Λ(−) ⊣ Λ(−) at S is the weak morphism (idS, φ) +with +φ!Γ = +S +S +S +!Γ × !A +!Γ × !A +!(Γ + A) +!Γ × !A +!Γ × !A +!Γ +!Γ +!Γ +∂S +!Γ += +(∂S +!Γ,∂S +!A) += +(∂S +!Γ,∂S +!A) +ΣΓ,A +===⇒ +sΓ,A +¯sΓ,A +l += +l +and +φ!A×B = +S +S +!A × B +!A × B +∂S +!A×B +(φ!A,φB) +======⇒ +∂S +!A×B + +where +φ!A = +S +S +S +!Γ × !A +!Γ × !A +!(Γ + A) +!Γ × !A +!Γ × !A +!A +!A +!A +∂S +!A += +(∂S +!Γ,∂S +!A) += +(∂S +!Γ,∂S +!A) +ΣΓ,A +===⇒ +sΓ,A +¯sΓ,A +r += +r +and +φB = +S +S +B +B +∂S +B += +∂S +B +where we write ∂S +A and ∂S +B for l ◦∂S +!A×B and r ◦∂S +!A×B re- +spectively (and similarly for ∂S +A and ∂S +B). The counit is defined +similarly. The fact that this unit/counit pair satisfies the zigzag +equations is a consequence of the fact that (ΣΓ,A, ¯ΣΓ,A) +satisfies the same equations, by vertical pasting of 2-cells. +In order to get another adjoint equivalence, we prove that the +uncurrying functor is isomorphic to the uncurrying-through- +evaluation operation: +Proposition 37. The functor ev ⊙! (− & A): Thin!(Γ, A ⇒ +B) → Thin!(Γ & A, B) is isomorphic to the uncurrying +functor ¯Λ(−). +Proof. Let S ∈ Thin!(Γ, A ⇒ B). We compute ev⊙!(S&A). +It is the composition of the spans +!(S + A) +!(!Γ + !A) +!!(Γ + A) +!(Γ + A) +!((!A × B) + A) +!(∂S +!Γ+ηA) +!(∂S +!A×B+A) +![!(¯l),!(¯r)] +µΓ+A +and +!A × B +!A × B × !A +!(!A × B) × !A +!((!A × B) + A) +B +(l,r,l) +r +η!A×B×!A +s!A×B,A +. +We compute the inner pullback of this composition as the +rectangle of pullbacks shown in Figure 8. Thus, up to a +canonical isomorphism of pullbacks, ev ⊙! (S & A) is ¯S ∈ +Thin!(Γ & A, B) with S as the support of ¯S and +∂ +¯S +!(Γ+A) = +S +⟨S,∂S +!A⟩ +−−−−→ S × !A +ηS×!A +−−−−→ !S × !A +sS,A +−−−→ !(S + A) +!(∂S +!Γ+ηA) +−−−−−−→ !(!Γ + !A) +![!(¯l),!(¯r)] +−−−−−−→ !!(Γ + A) +µΓ+A +−−−→ !(Γ + A) +and +∂ +¯S +B += +S +∂S +!A×B +−−−−→ !A × B +r−→ B. +The operation S +�→ +¯S can be shown to extend nat- +urally to weak morphisms, so that we obtain a functor +¯ +(−): Thin!(Γ, A ⇒ B) → Thin!(Γ & A, B) which is natu- +rally isomorphic to ev⊙! (−&A). So we are left to show that +¯ +(−) ∼= ¯Λ(−). For this purpose, for S ∈ Thin!(Γ, A ⇒ B), we +define an isomorphism θS = (θS, θ!(Γ+A) +S +, θB +S ): ¯S ⇒ ¯Λ(S). +We take θS = idS and θB +S += id∂S +B, and define θ!(Γ+A) +S +as the (vertically expressed) 2-cell of Figure 9. We directly +observe that θ!(Γ+A) +S +and θB +S have the adequate polarities, so +that θS ∈ Thin!(Γ & A, B) and it is an isomorphism. The +naturality of θS w.r.t. S can be checked diagrammatically, by +pasting. Thus, θ defines an isomorphism +¯ +(−) ∼= ¯Λ(−), so that +we have +ev ⊙! ((−) & A) +∼ += +=⇒ +¯ +(−) +θ=⇒ ¯Λ(−). +We can now conclude the proof of Proposition 7: +Proof of Proposition 7. By Proposition 36, we have an ad- +joint equivalence between the currying and uncurrying op- +erations. By Proposition 37, we can replace the uncurrying +operation by ev⊙! ((−)&A): by adjusting the unit and counit +the expected way, we keep the adjoint equivalence. + +!(S + A) +!S × !A +S × !A +S +!((!A × B) + A) +!(!A × B) × !A +!A × B × !A +!A × B +!(∂S +!A×B+A) +!(∂S +!A×B)×!A +sS,A +⌝ +∂S +!A×B×!A +ηS×!A +⌝ +∂S +!A×B +(S,∂S +!A) +⌝ +s!A×B,A +η!A×B×!A +(l,r,l) +. +Fig. 8. The inner rectangle of the composition of the two spans +S +!Γ × !A +!!Γ × !!A +!(!Γ + !A) +!!(Γ + A) +!(Γ + A) +S +!Γ × !A +!!Γ × !!A +!Γ × !A +!(Γ + A) +S +!Γ × !A +!Γ × !A +!(Γ + A) +(∂S +!Γ,∂S +!A) += +η!Γ×!ηA +s!Γ,!A +⇓ SeeΓ,A +![!(¯l),!(¯r)] +µΓ+A +(∂S +!Γ,∂S +!A) += +η!Γ×!ηA +⇓ αΓ × βA +µΓ×µA +sΓ,A += +(∂S +!Γ,∂S +!A) +sΓ,A +Fig. 9. The θ!(Γ+A) +S +2-cell +Recall the definition of α and β from Figure 3. + diff --git a/V9FKT4oBgHgl3EQfnC5b/content/tmp_files/load_file.txt b/V9FKT4oBgHgl3EQfnC5b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2af66b0202f9cac3694d899a881300b0e32751fe --- /dev/null +++ b/V9FKT4oBgHgl3EQfnC5b/content/tmp_files/load_file.txt @@ -0,0 +1,2410 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf,len=2409 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='11860v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='LO] 27 Jan 2023 The Cartesian Closed Bicategory of Thin Spans of Groupoids Pierre Clairambault Aix Marseille Univ, Universit´e de Toulon, CNRS, LIS, Marseille Email: Pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Clairambault@cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='fr Simon Forest Aix Marseille Univ, CNRS, I2M, Marseille, France Email: Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Forest@univ-amu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='fr Abstract—Recently, there has been growing interest in bicat- egorical models of programming languages, which are “proof- relevant” in the sense that they keep distinct account of execution traces leading to the same observable outcomes, while assigning a formal meaning to reduction paths as isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In this paper we introduce a new model, a bicategory called thin spans of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Conceptually it is close to Fiore et al.’s generalized species of structures and to Melli`es’ homotopy template games, but fundamentally differs as to how replication of resources and the resulting symmetries are treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Where those models are saturated – the interpretation is inflated by the fact that semantic individuals may carry arbitrary symmetries – our model is thin, drawing inspiration from thin concurrent games: the interpretation of terms carries no symmetries, but semantic individuals satisfy a subtle invariant defined via biorthogonality, which guarantees their invariance under symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We first build the bicategory Thin of thin spans of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Its objects are certain groupoids with additional structure, its morphisms are spans composed via plain pullback with identities the identity spans, and its 2-cells are span morphisms making the induced triangles commute only up to natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We then equip Thin with a pseudocomonad !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', and finally show that the Kleisli bicategory Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' is cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' INTRODUCTION The relational model [1] is one of the most basic and elementary denotational models for linear logic or the λ- calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' At its heart, it is simply an interpretation of formulas / types as sets and proofs / programs as relations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' in the category Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Despite its simplicity the relational model is ubiquitous: it is the basic substrate for the spectrum of so- called web-based models of linear logic, including coherence or finiteness spaces [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It faithfully predicts reduction time [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It supports quantitative extensions such as in probabilistic coherence spaces [4], the weighted relational model [5], and even up to quantum computation [6] – quantitative extensions which enjoy powerful full abstraction results [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Presented syntactically, the relational model exactly corresponds to non- idempotent intersection types [9], a currently active research topic in its own right (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [10], [11]) which enables a syntactic methodology to addressing semantic questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Finally, it has a tight connection with game semantics [12], [13], of which it appears as a desequentialization (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [8], [14]–[16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In short, it is at the crossroads of multiple topics, past and current, of the denotational semantics universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Another recent trend in denotational semantics is the adop- tion of bicategorical models [17] where the familiar categor- ical laws hold only up to certain 2-cells satisfying coherence conditions – in particular, Fiore and Saville have recently thor- oughly explored cartesian closed bicategories [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In such models, the denotation is no longer an invariant of reduction: two convertible terms yield merely isomorphic objects, and reduction paths have a genuine interpretation as specific iso- morphisms [19] – thus bringing reduction into the categorical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' There are still not many concrete bicategorical models, and we are aware of only three (families of) such models that can deal with non-linear computation, in chronological order: firstly, Fiore, Gambino, Hyland and Winskel’s cartesian closed bicategory of generalized species of structure [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' secondly, Castellan, Clairambault and Winskel’s thin concurrent games [21] (as established by Paquet [22]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' thirdly, Melli`es’ homo- topy template games [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Of these three, the first is by far the most studied with various works including generalizations and application to semantics [24]–[26], links with intersection types and Taylor expansion [27], [28], or applications to the pure λ-calculus [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Beyond giving a non-degenerated interpretation to reduction paths, those concrete bicategorical models are “proof-relevant”, in the sense that they keep distinct semantic witnesses for the possibly multiple evaluation traces with the same observable behaviour and thus keep a clear, branching account of non-determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' These models have something else in common: in their construction, the main subtlety has to do with replication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' the modality !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' of linear logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In the relational model, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A is the set M(A) of finite multisets of elements of A, or alter- natively, the free monoid A∗ quotiented by permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In bicategorical models, this is replaced by a categorification of M(A): a category (or groupoid) whose objects keep separate individual resource usages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Its morphisms are explicit permutations, often called symmetries in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Individuals in the model must refer to specific resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ai in a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' an ∈ A∗), but the categorical laws expected for mod- els of programming languages requires that their behaviour should still be invariant under symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In both generalized species of structure and template games, this is done by saturating the set of witnesses with respect to symmetries: intuitively, the behaviour of an individual cannot depend on the specific identity of resources, because those resources are seen through the “noise” of all possible symmetries – this shall be reviewed gently in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This saturation complicates models and their construction, though for good reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But this contrasts with thin concurrent games, which handles symmetry with a mechanism inspired by Abramsky- Jagadeesan-Malacaria games [12] and Melli`es’ orbital games [15]: strategies are not saturated, but their invariance under “Opponent’s symmetries” is ensured by a subtle bisimulation- like structure – we call this the thin approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We believe that the thin approach is helpful at least for appli- cations to semantics: the absence of symmetries on witnesses allow a more concrete flavour which may help when order- ing individuals allowing continuous reasoning1, or simplify quantitative extensions such as [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But more fundamentally, there is a clear tension between these two worlds that deserves investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Are proof-relevant relational models inherently saturated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Is the thin approach only possible in games thanks to the presence of time and causality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' These fundamental questions may be of interest beyond denotational semantics, as the handling of symmetry in such models is deeply connected to algebraic combinatorics [20] and homotopy theory [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' a) Contributions: We introduce the bicategory Thin of thin spans of groupoids: its objects are certain groupoids with additional structure, its morphisms certain spans, and its 2- cells certain weak span morphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' making the induced triangles commute up to chosen natural isos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Identities are identity spans, and composition of spans is by plain pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Of course, plain pullbacks are too weak to support the horizontal composition of weak span morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' To address this, we first define uniform spans via a biorthogonality con- struction, ensuring that the composition pullbacks also satisfy the bipullback universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This allows us to compose 2-cells horizontally, but that horizontal composition is still not canonically defined and fails to give a bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For the next step, we import from thin concurrent games and from Melli`es’ orbital games a decomposition of symmetries into positive symmetries (due to the program), and negative symmetries (due to the environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We then define thin spans via a second biorthogonality construction, which ensures that the horizontal composition of weak span morphisms are canonically defined as long as we consider positive weak span morphisms, where the chosen iso only involves positive symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We show this results in a bicategory Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Furthermore, we equip Thin with a pseudocomonad !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', and show that the Kleisli bicategory Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' is cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' b) Outline: In Section II we start with a gentle introduc- tion to the relational model and its proof-relevant extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In Section III we introduce the bicategory Thin, deploying first the uniform orthogonality and then the thin orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In Section IV we introduce the pseudocomonad !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', and show that the Kleisli bicategory Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' is cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' RELATIONAL MODELS, SPANS, SPECIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The Relational Model The relational model is one of the simplest denotational models of the λ-calculus, linear logic, or simple programming 1For instance, in [29], the generalization from finite to infinite computation is not simply by continuity as per usual in denotational semantics, because of the quotient involved in the management of saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' languages such as PCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It consists in simply interpreting every type A as a set �A�, and a program ⊢ M : A as a subset of �A�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This set �A� is often called the web seeing that it is the first component of the so-called web-based models of linear logic such as coherence spaces and their extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One may think of elements of �A� as completed executions (which is straightforward enough for ground types such as booleans or natural numbers but may be more complex for higher-order types), and of �M� ⊆ �A� as simply the collection of all the completed executions that M may achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The ground type for booleans is interpreted as �B� = {tt, ff}, and the constant ⊢ tt : B as �tt� = {tt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The interpretation of a program M is computed composi- tionally, following the methodology of denotational semantics, organized by the categorical structure of sets and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 1) Basic categorical structure: There is a category Rel with sets as objects, and as morphisms the relations from A to B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' subsets R ⊆ A×B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The identity on A is the diagonal relation {(a, a) | a ∈ A} ⊆ A × A, and the composition of R ⊆ A×B and S ⊆ B×C consists in all pairs (a, c) ∈ A×B such that (a, b) ∈ R and (b, c) ∈ S for some b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Besides, Rel has a monoidal structure given by the cartesian product on objects, and for Ri ∈ Rel(Ai, Bi), R1 × R2 ∈ Rel(A1 × A2, B1 × B2) set as comprising all ((a1, a2), (b1, b2)) when (ai, bi) ∈ Ri – the unit I is a fixed singleton set, say {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Additionally, Rel is compact closed: each set A has a dual A∗ defined simply as A itself, and there are relations ηA ∈ Rel(I, A × A) and ǫA ∈ Rel(A × A, I), both diagonal relations, satisfying coherence conditions [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In particular, Rel is ⋆-autonomous and as such a model of multiplicative linear logic, and the linear λ-calculus: the linear arrow type is interpreted as �A ⊸ B� = �A� × �B�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Finally, Rel has finite products, with the binary product of sets A and B given by the disjoint union A + B = {1} × A ⊎ {2} × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) The exponential modality: The exponential modality of Rel is based on finite multisets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If A is a set, we write M(A) for the set of finite multisets on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' To denote specific multisets we use a list-like notation, as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [0, 1, 1] ∈ M(N) – we write [] ∈ M(A) for the empty multiset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For A a set, its bang !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A is simply the set M(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This ex- tends to a comonad on Rel, satisfying the required conditions to form a so-called Seely category – in particular, there is M(A + B) ∼= M(A) × M(B) a bijection providing the Seely isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Altogether, this makes Rel a model of intuitionistic linear logic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and this makes the Kleisli category Rel!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' cartesian closed so that we may interpret (among others) the simply-typed λ-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Considering the term ⊢ M : B → B of PCF ⊢ λxB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' if x thenx elseif x thenff elsett : B → B , we have �M� = {([tt, tt], tt), ([tt, ff], ff), ([ff, ff], tt)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Here we can observe that the model is quantitative, in that it records how many resources each execution consumes: one may observe output tt either with two evaluations of x to tt, or with two evaluations of x to ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One may observe output ff with two evaluations of x, one to tt and one to ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Recall that in [tt, ff] = [ff, tt], the order is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The relational model also supports the interpretation of non- determinism: if ⊢ choice : B is a new primitive evaluating non-deterministically to tt or ff, then we may simply set �choice� = {tt, ff} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3) Extensions of the relational model: The relational model is extremely flexible, and can be extended in multiple different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In one direction one may add to the objects a coherence relation and restrict to compatible morphisms – we obtain in this way (multiset-based) coherence semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Another extension is the weighted relational model [5], [31] where a term ⊢ M : A, instead of denoting a subset of �A� – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' a function �M� : �A� → {0, 1} – denotes a function �M� : �A� → R assigning to each point of the web a ∈ �A� a weight �M�a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The weight may be used to record additional information about executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One may record the number of distinct non- deterministic branches leading to a certain result: for instance, if R = N ∪ {+∞}, then �if choice thentt elsett�tt = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' With R = R+ = R+ ∪ {+∞}, we may track the probability with which a certain result occurs, obtaining a model fully abstract for probabilistic PCF [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The paper [5] contains other examples: resource consumption, must convergence, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It is natural to go one step further and make the relational model “proof-relevant”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This means not merely recording a weight or counting non-deterministic branches, but keeping track of a set �M�a ∈ Set of witnesses of the execution of M to a, for each ⊢ M : A and a ∈ �A�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' There are well- documented ways to do that which we shall review later on, but for now let us attempt this naively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The Bicategory of Spans A first idea is to simply replace relations with spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 1) Spans: Recall that if C is a category with pullbacks, then we form Span(C) has having as objects those of C, and as morphisms from A to B triples (S, ∂S A, ∂S B) forming a diagram A ∂S A ← S ∂S B → B , where intuitively S is a set of internal witnesses, projected to A and B via the maps ∂S A and ∂S B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For C = Set one obtains a relation by collecting the pairs (∂S A(s), ∂S B(s)) for s ∈ S, but we have more: for each pair (a, b) ∈ A × B we have witS(a, b) = {s ∈ S | ∂S A(s) = a & ∂S B(s) = b} , a set of witnesses that a and b are related – hence this indeed provides a notion of a proof-relevant relational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Writing B = {tt, ff} and 1 = {∗}, we may represent the program ⊢ if choice thentt elsett as 1 ∂l ← {a, b} ∂r → B a span, where ∂l(a) = ∂l(b) = ∗, ∂r(a) = ∂r(b) = tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, the evaluation of the program to tt has two witnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) A bicategory: The exact identity of S does not matter – the same span above with S′ = {a′, b′} should not be treated distinctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A morphism between spans is f : S → S′ making S A B S′ ∂S A ∂S B f ∂S′ A ∂S′ B commute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' an isomorphism of span is an invertible morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The identity span on A is simply A ← A → A with two identity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The composition of A ← S → B and B ← T → C is obtained by first forming the pullback T ⊙ S S T A B C l r ∂S A ∂S B ∂T B ∂T C (1) and setting ∂T ⊙S A = ∂S A ◦ l and ∂T ⊙S C = ∂T C ◦ r – for Span(Set), this means that T ⊙ S has elements all pairs (s, t) such that ∂S B(s) = ∂T B(t), projected to A and C via ∂T ⊙S A ((s, t)) = ∂S A(s) and ∂T ⊙S C ((s, t)) = ∂T C(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This composition need not be associative on the nose, but the universal property of pullbacks entails that it is associative up to canonical isomorphism – forming a bicategory: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If C has pullbacks, then Span(C) defined with objects: objects of C, morphisms: spans A ← S → B, 2-cells: morphisms of spans, forms a bicategory, denoted Span(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In fact, Span(C) is a compact closed bicategory [32], and thus a model of the linear λ-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In particular, Span(Set) shares much structure with Rel: it has the same objects and the operation sending a span A ← S → B to the pairs (∂S A(s), ∂S B(s)) for s ∈ S is a functor, establishing Span(Set) as a natural candidate for a proof-relevant relational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3) The exponential: However, the exponential of Rel does not directly transport to Span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The operation M(−) does yield a functor on Set obtained by setting, for f : A → B, M(f)([a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , an]) = [f(a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , f(an)] defining M(f) : M(A) → M(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But M(f) does not lift to Span(Set) as it does not preserve pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Indeed, the diagram obtained by image of the composition pullback M(T ⊙ S) M(S) M(T ) M(B) M(l) M(r) M(∂S B) M(∂T B) is no pullback: this would need a bijection of M(T ⊙S) with {(µ, ν) ∈ M(S) × M(T ) | M(∂S B)(µ) = M(∂T B)(ν)} , which fails in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If S = T = B and B = 1, the pair of multisets ([tt, ff], [tt, ff]) does not uniquely specify who is synchronized with whom: it may correspond to both multisets [(tt, tt), (ff, ff)] and [(tt, ff), (ff, tt)] in M(T ⊙ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This might be expected: a finite multiset only remembers the multiplicity of elements, but does not track distinct individual occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is in tension with the goal of a proof-relevant relational semantics, for which specific witnesses are naturally associated with individual resource occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 4) Categorifying objects: If the exponential is to track indi- vidual resource occurrences, that means avoiding the quotient of finite multisets: an element of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A may for instance be a list, or a word a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' an ∈ A∗ of elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We must of course still account for reorderings, which turn A∗ into a groupoid – in fact, it is an instance of the construction of the free symmetric monoidal category Sym(A) over a category A: its objects are finite words a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' an of objects of A, and a morphism from a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' an to a′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' a′ n consists of a permutation π ∈ Sn, and a family (fi ∈ A(ai, aπ(i)))1≤i≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, objects are not mere sets but categories, which means that we move from Span(Set) to Span(Cat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Indeed, Cat also has pullbacks, and so the exact same construction as above yields a bicategory Span(Cat) – except that now the functor Sym : Cat → Cat preserves pullbacks and thus lifts to Sym : Span(Cat) → Span(Cat) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' However, in this categorification, the Seely isomorphism M(A + B) ∼= M(A) × M(B) is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Instead, we only get Sym(A + B) ≃ Sym(A) × Sym(B) an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In order to lift it to spans, we observe that given a functor F : A → B we get a span ˆF = A A B F idA ∈ Span(Cat)(A, B) so that lifting an equivalence F : A ≃ B : G to spans requires us to provide a family of 2-cells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' for each category A: A A A A idA GF ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' idA idA however whatever our choice for the mediating map is, one of the triangles fails to commute on the nose but only up to isomorphism, which the 2-cells of Span(Cat) are too strict to accommodate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This invites weakening the 2-cells to: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A weak morphism from A ← S → B to A ← S′ → B is a triple (F, F A, F B) where S A F A ⇓ ⇓F B B S′ ∂S A ∂S B F ∂S′ A ∂S′ B with F A : ∂S A ⇒ ∂S′ A ◦F and F B : ∂S B ⇒ ∂S′ B ◦F natural isos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We call this a strong morphism if F A and F B are identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Adopting weak morphisms seems to solve the problem above, but only to run into a much more subtle one: in P S T B l r u µ=⇒ v Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A bipullback X S T B l′ r′ u ν=⇒ v Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Alternative square Span(Cat), the horizontal composition of 2-cells F : S ⇒ S′ and G : T ⇒ T ′ as required by the bicategorical structure follows from the universal property of the pullback T ′ ⊙ S′: T ⊙ S S T A B C S′ T ′ T ′ ⊙ S′ G⊙F F G (2) but this universal property is powerless to compose horizon- tally weak morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We cannot have the cake and eat it too: if our method to compose spans ignores the 2-categorical nature of Cat, then we cannot hope composition to preserve an equivalence between spans that relies on it, as required for a model of linear logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So it seems that this road to a proof-relevant relational model is doomed – except that this is exactly what we shall do in this paper!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Before we delve into that, we review existing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof-Relevant Relational Models, and Other Related Work As plain pullbacks are “too 1-dimensional”, it is natural to compose spans with a 2-dimensional version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 1) Bipullbacks: There are multiple variants for weakened versions of pullbacks in a 2-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In this paper, a central notion will be that of a bipullback:2 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In a 2-category C, a bipullback of the cospan S u−→ B v←− T is a square commuting up to an invertible 2-cell as in Figure 1, such that for any square as in Figure 2: (a) There is a morphism h : X → P and 2-cells α and β s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : X P S T B l′ r′ h l r β=⇒ α=⇒ u µ=⇒ v = X S T B l′ r′ u ν=⇒ v (b) h, α, β are unique up to unique 2-cell – see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The important observation is that this alternative universal property is sufficient to extend the definition of the horizontal composition in (2) to weak morphisms – with the proviso that this defines horizontal composition only up to iso;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' as (b) does not guarantee uniqueness of h on the nose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2According to the nlab, its proper name is a bi-iso-comma-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) Hoffnung’s monoidal tricategory: Hoffnung [33] con- structs a categorification of Span(Cat) following this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' He exploits that Cat actually has pseudo-pullbacks3, which are a special case of the definition above where α and β are required to be identities and h is unique on the nose – making horizontal composition of weak morphisms of spans a well- defined function once a choice of pseudo-pullbacks is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Concretely, a pseudo-pullback of a cospan S u−→ B v←− T may be constructed as a category with objects triples (s, θ, t) where θ ∈ B(u(s), v(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So for instance, if S = T = Sym(B) and B = Sym(1), the pseudo-pullback would have two objects synchronizing [tt, ff] ∈ S and [tt, ff] ∈ T : ([tt, ff], id, [tt, ff]) and ([tt, ff, swap, [tt, ff]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The issue of Section II-B3 is avoided by adding new witnesses carrying all possible symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is a fundamental phenomenon in models of linear logic, which we refer to as saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Because saturation inflates the number of witnesses at each composition, spans composed by pseudo-pullbacks no longer form a bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In particular, the post-composition of a span A ← S → B with the identity span B ← B → B yields an inflated S′ much bigger than S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So neutrality of identity no longer holds up to isomorphism, but only up to equivalence factoring in maps between maps of spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Accordingly, Hoff- nung actually constructs a monoidal tricategory of categorical spans with weak morphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' a one-object tetracategory!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3) Melli`es’ template games: Recently, Melli`es introduced template games [34], in an attempt to unify various games models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is essentially a model of categorical spans where categories are regarded as games and structured by a projection to a category called the template, capturing the mechanisms of synchronization and scheduling between players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Though [34] was developped in a purely linear setting with spans related by strong morphisms, Melli`es proposed a non-linear extension, forming a model of differential linear logic [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Melli`es’ contribution puts into play notions from homotopy theory: he starts not with Cat, but from any 2-category S equipped with a Quillen model structure (with additional conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Spans are composed by mere pullbacks, but a span A u ←− S v−→ B must satisfy a fibration property to the effect that symmetries in A and B can be lifted uniquely in S – in our terminology, S is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Saturation ensures that pullbacks between those spans are actually homotopy pullbacks, and thus that they may be used for the horizontal composition of weak morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The higher dimensional structure seen in Hoffnung [33] is then managed by the homotopy-theoretic operation of localization, formally inverting weak equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This yields an actual bicategory of objects of S related by homotopy spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This elegant construction gives a model of differential linear logic, showing that the symmetries implicit in linear logic may be naturally managed via the tools of homotopy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3According to the nlab, its proper name is an iso-comma-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 4) Generalized Species of Structures: Last but not least, the most well-studied proof-relevant extension of Rel is defi- nitely Fiore, Gambino, Hyland and Winskel’s cartesian closed bicategory of generalized species of structure [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Relations from A to B are replaced with distributors or profunctors: F : Aop × B → Set for A and B categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This forms a (compact closed) bicategory Dist of (small) categories, distributors and natural transformations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The free symmetric monoidal construction Sym(−) yields a pseudocomonad on Dist, whose Kleisli bicategory Esp is cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As for the span-based approaches above, the way in which Dist and Esp handle symmetries is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This may first be seen in the identity distributor which is defined to be A[−, −] : Aop × A → Set the Yoneda embedding, which associates as witnesses over a pair (a, a) the homset A[a, a], including all symmetries on a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Composition of distributors is via the coend formula G ⊙ F = � b∈B F(−, b) × G(b, −) which sets witnesses in (G ⊙ F)(a, c) to be pairs (s, t) ∈ F(a, b) × G(b, C) quotiented by a relation identifying the action of a morphism in B on s or on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Accordingly, when computing the interpretation of a pro- gram ⊢ M : A in Esp, for every a ∈ �A� we get �M�(a) a set of witnesses carrying around explicit symmetries, quotiented by an equivalence relation letting symmetries flow around – this is described syntactically elegantly by Olimpieri [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The treatment of symmetry in Esp is, again, saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 5) Game semantics: To our knowledge, this saturation phenomenon in models of linear logic first appears in Bail- lot, Danos, Ehrhard and Regnier’s (BDER) version [35] of Abramsky-Jagadeesan-Malacaria (AJM) games [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In AJM games, the moves of a game !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A are defined as pairs (i, a) of i ∈ N a copy index, and a ∈ A a move in A – a fundamental difficulty in setting up the games model, is that of uniformity: ensuring that the behaviour of strategies does not depend on the specific choice of copy indices (which is the game semantics analogue of composition preserving weak morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In BDER, uniformity is guaranteed by requir- ing strategies to be saturated: they are morally wrapped by copycat processes exchanging non-deterministically all copy indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This “noise” prevents strategies from seeing specific copy indices, let alone depending on them – this is analogous to the saturation phenomenon above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But in AJM games there is another choice: in the original AJM setting [12], strategies carry a deterministic choice of copy indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Instead of saturation, uniformity is guaran- teed by requiring that strategies satisfy a bisimulation-like property, which ensures that whenever Opponent swaps their copy indices, Player can swap theirs accordingly, leaving the behaviour “up to copy indices” invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In contrast to the “saturated” approach to uniformity, we refer to this as the “thin” approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Similar ideas are at play in other game models based on copy indices: in Melli`es’ orbital games [15], and more recently in thin concurrent games4 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thin concurrent games are a particularly striking related work, because just as Esp, they also form a cartesian closed bicategory as proved by Paquet [22], and also generalize the relational model [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In thin concurrent games, strategies are composed by pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But it is a theorem that this pullback is also a bipullback, which can be used to compose horizontally weak morphisms even though strategies are not saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But this bipullback property follows from a subtle interactive reindexing mechanism between strategies, relying crucially on the fact that we have access to time – it seems hard to replicate it purely statically as in a relational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' THE BICATEGORY OF THIN SPANS This long discussion lets us state the main question in this paper: can we construct a thin version of categorical spans?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Pullbacks and Bipullbacks in Groupoids For simplicity, we focus on spans of groupoids rather than categories, which are sufficient for the interpretation of types – we write Gpd for the small 2-category of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So we aim to construct a bicategory whose objects are small groupoids, whose morphisms are spans A ← S → B with identity the identity span A ← A → A, whose composition is plain pullback and yet, whose 2-cells are weak morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 1) Notations and terminology: A span A ← S → B may be presented as a functor S → A × B, so it is convenient not to focus on spans, but on functors S → A over a groupoid A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We refer to those with terminology inspired from game semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A prestrategy on groupoid A is a pair (S, ∂F ) where ∂F : S → A is called the display map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We often refer to the prestrategy only with S, and write PreStrat(A) for the set of prestrategies on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A prestrategy from A to B is a prestrategy on A × B – then, we write ∂F A : S → A and ∂F B : S → B for the two display maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If S is a prestrategy from A to B and T a prestrategy from B to C, we write T ⊙S for the prestrategy from A to C obtained as in Section II-B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We often refer to morphisms in groupoids as symmetries and write e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ϕ : s ∼=S s′ instead of ϕ ∈ S(s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We write 1 for the groupoid with one object ∗, and only the identity morphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and o for the groupoid with one object • and only the identity morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If A, B are groupoids, then we use A ⊢ B and A ⊸ B as synonyms for A × B, with objects respectively denoted by a ⊢ b and a ⊸ b – likewise, their morphisms have form θA ⊢ θB ∈ (A ⊸ B)(a ⊸ b, a′ ⊸ b′) for θA ∈ A(a, a′) and θB ∈ B(b, b′) and likewise for θA ⊸ θB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We find these purely notational distinctions useful to read examples, since they coincide with familiar type constructors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) Indexed families: As explained earlier, types of λ-calculi may be interpreted as groupoids – but in a linear language, these groupoids remain discrete: only the exponential intro- duces non-trivial morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As those symmetries play a 4The first version of concurrent games with symmetry was saturated [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' crucial role, we introduce early our version of the exponential construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If X is a set, then we write Fam(X) the set of families indexed by finite sets of natural numbers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' of (xi)i∈I where I ⊆f N and for all i ∈ I, xi ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider A a (small) groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The (small) groupoid Fam(A) has: objects, the set Fam(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' morphisms from (ai)i∈I to (bj)j∈J, pairs (π, (fi)i∈I) of a bijection π : I ≃ J and for each i ∈ I, fi ∈ A(ai, bπ(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This yields a functor Fam : Gpd → Gpd in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For (Ai)i∈I ∈ Fam(A), we call elements of I copy indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A family (ai)i∈I ∈ Fam(A) is more “intensional” than A∗ (which is more intensional than M(A)): it gives a presentation of a multiset in M(A) not only providing a sequence, but assigning to each element a distinct “address”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Just as multisets are connected to non-idempotent intersec- tion types, families are connected with Vial’s sequence types [38] – thus we often write families using Vial’s notation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [2 · a2, 4 · a4, 12 · a12] ∈ Fam(A) for (ai)i∈{2,4,12} – in the particular case where A = o, we only write [i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , in] for [i1 · •, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , in · •].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For any groupoid A, Fam(A) and Sym(A) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' However, using Fam(A) is crucial in our model construction: it allows the interpretation of programs to use copy indices as identifiers for resource accesses, that are independent of other concurrent resource accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We give a few examples: Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For a groupoid A, the dereliction span derA is Fam(A) derA ←−−− A idA −−→ A where derA : A → Fam(A) sends a to [0 · a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In models of linear logic, the role of dereliction is to extract a single instance of a replicable resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In our model – as in AJM games [12] and thin concurrent games [21] – dereliction does so by picking a copy index (here 0), chosen in advance once and for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The specific choice is irrelevant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' in fact for any n the span using n instead of 0 will be turn out to be isomorphic to derA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But, the span must comprise a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The interpretation of the term M of Example 2 in thin spans shall have head groupoid that with four objects [0 · tt, 1 · tt] ⊸ tt , [0 · ff, 1 · ff] ⊸ tt , [0 · tt, 1 · ff] ⊸ ff , [0 · ff, 1 · tt] ⊸ tt , morphisms reduced to identities, and display map the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The use of specific copy indices allows one to observe which occurrence of x evaluates to tt or ff, hence associating distinct points to the two evaluations leading to ff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3) Bipullbacks of groupoids: If composition-by-pullback is to allow us to compose horizontally weak morphisms, we must ensure that every composition pullback is also a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It is useful to understand a bit better the shape of bipullbacks in Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A first useful fact is that condition (b) of Definition 2 (uniqueness up to iso) automatically holds in the case of Gpd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' furthermore, we can characterise those pullbacks that are also bipullbacks (see Appendix B): Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A pullback square in Gpd, of the form P S T B l r f g is a bipullback if and only if it satisfies the following property: for all s ∈ S, t ∈ T and θ ∈ B(fs, gt), there is ϕ ∈ S(s, s′) and ψ ∈ T (t′, t) such that fs′ = gt′ and θ = fψ ◦ gϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let us comment on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We regard triples of the form s ∈ S , θ ∈ B(fs, gt) , t ∈ T as pairs of states (s, t) that match up to symmetry – we call this a reindexing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The lemma above says that given a reindexing problem, we can always find s′ symmetric to s and t′ symmetric to t matching on the nose, in a way compatible with θ – called a solution to the reindexing problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, the lemma above may be reformulated to say that a pullback is a bipullback iff all its reindexing problems have a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We show a concrete example of this reindexing process: Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Take B = Fam(o) ⊸ Fam(o), with objects [i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , in] ⊸ [j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , jk] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Take S the sub-groupoid of B with objects [i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , in] ⊸ [i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , in] and morphisms all θ ⊸ θ for θ in Fam(o);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and T the full sub-groupoid of B with objects [j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' , jn] ⊸ [0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The pullback of S → B ← T is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For instance, θ ∈ B([2] ⊸ [2], [1] ⊸ [0]) is a reindexing problem that may be solved by first applying ϕ ∈ S([2] ⊸ [2], [0] ⊸ [0]) in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We are reduced to finding morphisms in S and T w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' θ′ ∈ B([0] ⊸ [0], [1] ⊸ [0]) Now, applying ψ ∈ T ([0] ⊸ [0], [1] ⊸ [0]) in T , we have ϕ ∈ S([2] ⊸ [2], [0] ⊸ [0]) , ψ ∈ T ([0] ⊸ [0], [1] ⊸ [0]) a solution to the reindexing problem, as in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' That the pullback of two prestrategies forms a bipullback is not a property of either: in this example neither strategy is a fibration as in [23], and solving the reindexing problem requires reindexing in both groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So it is a property emerging from the harmonious interaction between two pre- strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In an appropriate game semantics setting [21], one can prove that under reasonable assumptions, such interactive reindexing always succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' However, this is a gradual process progressing over time – which we do not have access to here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Orthogonality and Uniform Groupoids 1) Definition: In the literature on models of linear logic, there is a technique for choreographing models where one only composes pairs of morphisms satisfying a given interactive property: biorthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The first step is to specify the desired interactive property via an orthogonality relation: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Take (S, ∂S) and (T, ∂T ) prestrategies on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We say they are uniformly orthogonal, written S ⊥ T , iff the pullback of the cospan S → B ← T is also a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If S ⊆ PreStrat(B), then its uniform orthogonal is set to: S⊥ = {T ∈ PreStrat(B) | ∀S ∈ S, S ⊥ T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As usual with orthogonality, this automatically entails a number of properties: for all S ⊆ PreStrat(B), we have S ⊆ S⊥⊥, and S⊥ = S⊥⊥⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We are particularly interested in sets of the form S⊥, which are invariant under biorthogonal: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A uniform groupoid is a pair (A, UA) where A is a groupoid and UA ⊆ PreStrat(A) is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' U⊥⊥ A = UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We often refer to a uniform groupoid (A, UA) just with A when it is clear from the context that it is a uniform groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) Constructions: The uniform groupoid 1 is the terminal groupoid equipped with U1 = PreStrat(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If A and B are uniform groupoids, their tensor A⊗ B is the groupoid A× B equipped with the set UA⊗B = (UA ⊗ UB)⊥⊥, writing UA ⊗ UB = {(S × T, ∂S × ∂T ) | S ∈ UA, T ∈ UB} with ∂S × ∂T : S × T → A × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The dual A⊥ of A is (A, UA⊥) with UA⊥ = U⊥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The par of A and B has UA`B = (U⊥ A ⊗ U⊥ B)⊥ yielding the De Morgan duality (A⊗ B)⊥ = A⊥ `B⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' From this we derive the linear arrow A ⊸ B = A⊥ ` B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A uniform prestrategy on uniform groupoid A is simply any S ∈ UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If A, B are uniform groupoids, then a uniform prestrategy from A to B is a uniform prestrategy on A ⊸ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3) Uniform composition: We claim that whenever compos- ing S ∈ UA⊸B with T ∈ UB⊸C, we have the orthogonality (S, ∂S B) ⊥ (T, ∂T B) so that the composition pullback is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If S is a prestrategy on A and T is a prestrategy from A to B, we write T @S from the prestrategy on B obtained by T ⊙ S S T A B called the application of T to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This lets us state: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider (A, UA) and (B, UB) uniform groupoids, and T a prestrategy from A to B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' consider fur- thermore a class S ⊆ UA s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A, idA) ∈ S and UA = S⊥⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then T ∈ UA⊸B iff the following two conditions hold: (1) for all S ∈ S, T @S ∈ UB, (2) (T, ∂T A) ∈ U⊥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Unfolding the definitions, one encounters a few dia- gram chasing lemmas on pullbacks that are also bipullbacks – themselves proved via Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The apparent asymmetry is intriguing: by definition A⊥ ` B = A⊥`B⊥⊥, so that T ∈ UA⊸B iff the span B ← T → A denoted by T ⋆ obtained by reversing the two legs, is in UB⊥⊸A⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A similar phenomenon appears in the orthogonal- ity used by Ehrhard for his extensional collapse [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Now, observe that (A, idA) ∈ UA always – not the identity span, but the identity functor regarded as a prestrategy on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Indeed, if S ∈ U⊥ A, then the pullback of A → A ← S is clearly a bipullback, so (A, idA) ∈ U⊥⊥ A = UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But now this lets us instantiate Proposition 1 with S = UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then given S ∈ UA⊸B, the application S@(A, idA) is (up to iso) the right leg (S, ∂S B), which must by (1) be in UB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Likewise, if T ∈ UB⊸C, the left leg (T, ∂T B) is in U⊥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Hence, (S, ∂S B) ⊥ (T, ∂T B) and thus the composition pullback of S and T is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 1 has more consequences, all obtained in the particular case where S = UA: we saw above that (A, idA) ∈ UA, but the same argument goes to show (A, idA) ∈ U⊥ A as well – so the identity span satisfies condition (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since it also satisfies (1), we have (A ← A → A) ∈ UA⊸A as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Likewise, if A ← S → B and B ← T → C are uniform prestrategies, then it follows fairly easily that the composition A ← T ⊙ S → C is indeed in UA⊸C (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 4) Horizontal composition of 2-cells: We have an identity uniform prestrategy in UA⊸A, and a well-defined composition of S ∈ UA⊸B and T ∈ UB⊸C such that the composition pullback is always a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So given weak morphisms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F A⇓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='⇓F B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='GB⇓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='⇓GC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='by the bipullback property of T ′ ⊙ S′ there are a functor H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='and natural isos α and β such that we have the equality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⊙ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='(FB)−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='====⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='GB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⊙ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ ⊙ S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='α=⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='β=⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='altogether yielding a weak morphism as in the diagram: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⊙ S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ ⊙ S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='⇓F A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='⇓α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='⇓β−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='⇓GC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' However, H, α, β are not unique: though Lemma 1 guar- antees the existence of solutions to all reindexing problems, those may not be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We only know by condition (b) of Definition 2 that different choices of H, α, β yield isomorphic weak morphisms of uniform prestrategies, by which we mean isomorphic morphisms of the 2-category Unif(A): Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider A a uniform groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The 2-category Unif(A) has: objects UA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' uniform pre- strategies on A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' morphisms from S to T the weak morphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' pairs (F : S → T, φ : ∂S ⇒ ∂T F);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2-cells from (F, φ) to (G, ψ) the natural transformations µ : F ⇒ G such that: S T A G ψ=⇒ = S T A G F ⇑µ φ=⇒ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, although bipullbacks guarantee the existence of a fitting weak morphism for horizontal composition, there is a priori no canonical choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One could pick a choice of horizontal composition, but there is no reason why an arbitrary choice would satisfy the coherence conditions for a bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thin Spans of Groupoids In fact, if formulated in the adequate way, the reindexing problems that arise from the interpretation of programming languages do have a unique solution – as in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But to prove that, we shall need to add more structure to uniform groupoids, starting with polarized sub-groupoids: 1) Polarized sub-groupoids: Consider the groupoid Fam(o) ⊸ Fam(o) of Example 6, interpreting the formula !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='o ⊸ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='o of intuitionis- tic linear logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Here, the two occurrences of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' are intuitively very different: on the left-hand side, as in Example 4 the program performs the copying – in game semantics the copy index would be carried by a Player move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In contrast, for the right hand side exponential, the environment does the copying – in game semantics, the copy index would be carried by an Opponent move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This assigns a polarity to certain symmetries, very clear in game semantics: those reindexing copy indices only for exponentials in covariant position (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' contravariant position) are negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We enrich the groupoids interpreting types to keep track of these special symmetries: Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A polarized groupoid is a groupoid A with two sub-groupoids A− and A+, with the same objects as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It would be natural to require additional conditions for this structure (in particular, see conditions (a) and (b) in Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We omit them here, as they shall hold automat- ically once we introduce the more complete notion of a thin groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If θ ∈ A−(a1, a2), we write θ : a1 ∼=− A a2 and likewise for positive symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Usual constructions on groupoids extend to polarized groupoids componentwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The dual of (A, A−, A+) is defined as (A, A+, A−), exchanging the two sub-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Finally, we set (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A)− = Fam(A−) and (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A)+ = Famid(A+), which has morphisms from (ai)i∈I to (bj)j∈J those (π, (θi)i∈I) such that I = J and π = idI – thus we see indeed that Player cannot reindex copy indices from the outer !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' in !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A, as it appears in covariant position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) Thinness: Solutions to reindexing problems may be computed interactively as in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Intuitively, the uniqueness of the solution relies on the fact that at each stage, there is a unique choice of reindexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is captured by the definition of thin below, imported from thin concurrent games: Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider A a polarized groupoid, and S a prestrategy on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We say that S is thin iff for all ϕ : s ∼=S s′, if ∂Sϕ is positive then s = s′ and ϕ = ids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Intuitively, this captures that positive copy indices are se- lected deterministically from negative ones – so a non-trivial symmetry ϕ : s ∼=S s′ cannot display to a purely positive symmetry on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is in contrast with the saturated case, where spans must be able to reach all positive symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We show how thinness addresses uniqueness for the reso- lution of reindexing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Call a solution to a reindexing problem ϕ, ψ as in Lemma 1 positive if writing ∂Sϕ = ϕA ⊢ ϕB and ∂T ψ = ψB ⊢ ψC, we have ϕA ⊢ ψC positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider A, B, C polarized uniform groupoids, S ∈ UA⊸B and T ∈ UB⊸C s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⊙ S ∈ UA⊸C is thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, any reindexing problem in the composition pullback of S and T has at most one positive solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider a reindexing problem s ∈ S, t ∈ T, θ : ∂S Bs ∼=B ∂T Bt with solutions ϕ1 : s ∼=S s′ 1 and ψ1 : t′ 1 ∼=T t with ∂S Bs′ 1 = ∂T Bt′ 1 and ∂T Bψ1 ◦ ∂S Bϕ1 = θ, and ϕ2 : s ∼=S s′ 2 and ψ2 : t′ 2 ∼=T t with ∂S Bs′ 2 = ∂T Bt′ 2 and ∂T Bψ2 ◦ ∂S Bϕ1 = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, ∂S(ϕ2 ◦ ϕ−1 1 ) = ∂T (ψ2 ◦ ψ−1 1 ), so that we have Ω = (ϕ2 ◦ ϕ−1 1 , ψ2 ◦ ψ−1 1 ) : (s′ 1, t′ 1) ∼=T ⊙S (s′ 2, t′ 2) , whose display to A ⊢ C is positive since ϕ1, ψ1 and ϕ2, ψ2 are positive solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Hence, by thin, Ω is an identity map which entails ϕ1 = ϕ2 and ψ1 = ψ2 as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, thinness allows us to find canonical solutions to reindexing problems by insisting on finding positive solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' However, this relies on thinness not of S and T , but of T ⊙S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Again in thin concurrent games, this follows by induction on the causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In the absence of a handle on causality, we must as for uniformity treat the fact that T ⊙ S is thin as an interactive property, again handled by biorthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thin Spans 1) The thin orthogonality: We observe that for A a polar- ized groupoid, a prestrategy S on A is thin iff the pullback P S A+ A id+ A (3) is discrete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' all the morphisms in P are identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We shall base our orthogonality on this observation, and set: Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For A a polarized uniform groupoid, S ∈ UA, and T ∈ U⊥ A, we say S and T are thinly orthogonal, written S ⊥⊥ T iff the pullback T ⊙ S is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Note that this is already assuming that S and T are uniformly orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If S ⊆ UA, then its thin orthogonal is S⊥⊥ = {T ∈ U⊥ A | ∀S ∈ S, S ⊥⊥ T } , and as before we have S ⊆ S⊥⊥⊥⊥ (note that this typechecks only because U⊥⊥ A = UA) and S⊥⊥ = S⊥⊥⊥⊥⊥⊥ for all S ⊆ UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) Thin groupoids: As before, we are interested in sets of uniform prestrategies closed under bi-thin-orthogonal: Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A thin groupoid is a polarized uniform groupoid with a set TA ⊆ UA of strategies s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T⊥⊥⊥⊥ A = TA, and such that (A−, idA) ∈ TA and (A+, idA) ∈ T⊥⊥ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If S ∈ TA then S is automatically thin in the sense of Definition 8: as (A+, idA) ∈ T⊥⊥ A the pullback (3) is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This also entails properties of the polarized symmetries: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider A a thin groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then we have: (a) if θ : a ∼=− A a′ and θ : a ∼=+ A a′, then a = a′ and θ = ida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (b) if θ : a ∼=A a′, then there are unique a′′ along with θ− : a ∼=− A a′′ and θ+ : a′′ ∼=+ A a′ such that θ = θ+ ◦ θ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For (a), this follows from A− ⊥⊥ A+, as then the pullback of the cospan A− ֒→ A ←֓ A+ is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For (b), A− ∈ TA ⊆ UA and A+ ∈ T⊥⊥ A ⊆ U⊥ A, we also have A− ⊥ A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Hence, the pullback of the cospan A− ֒→ A ←֓ A+ is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But then any θ : a ∼=A a′ forms a reindexing problem, whose solution is exactly the seeked reindexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Uniqueness follows immediately from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, we get from the definition of thin groupoids some of the expected properties of the polarized sub-groupoids: if a symmetry is both positive and negative then it must be an identity, and any symmetry can be obtained by first “reindex- ing Opponent moves”, then ”reindexing Player moves”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3) Further structure: Constructions on uniform groupoids extend to thin groupoids in the expected way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The thin groupoid 1 has T1 = PreStrat(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If A and B are thin groupoids, their tensor is the uniform groupoid A ⊗ B ex- tended with TA⊗B = (TA ⊗ TB)⊥⊥⊥⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The dual of A has TA⊥ = T⊥⊥ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The par of A and B has TA`B = (T⊥⊥ A ⊗T⊥⊥ B)⊥⊥, and the linear arrow is A ⊸ B = A⊥ ` B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' To establish the compositional properties of strategies, we rely on the following analogue of Proposition 1: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider T ∈ UA⊸B for A, B thin groupoids, along with a class S ⊆ TA such that S⊥⊥⊥⊥ = TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, T ∈ TA⊸B iff T @S ∈ TB for all S ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This follows from diagram chasing lemmas on situations where the pullbacks are discrete, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Interest- ingly, this is also equivalent to T ⋆@S ∈ T⊥⊥ A for all S ∈ T⊥⊥ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It is a direct consequence of Proposition 2 that the identity span on A is in TA⊸A for any thin groupoid A, and that if S ∈ TA⊸B and T ∈ TB⊸C then T ⊙ S ∈ TA⊸C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Together with Lemma 2, we have thus identified a compositional situation where the composition pullback of spans is a bipullback, and where all arising reindexing problems have a unique solution if one insists on this solution being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 4) Positive weak morphisms: Insisting on positive solutions amounts to relating strategies not via arbitrary weak mor- phisms, but with positive weak morphisms: Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider A a thin groupoid, S, T ∈ TA, and (F, φ) a weak morphism from S to T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' F : S → T and φ : ∂S ⇒ ∂T ◦ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, (F, φ) is positive if φ is positive, that is, if ∀s ∈ S, φs : ∂Ss ∼=+ A ∂T F(s) is a positive symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Intuitively, comparing strategies with positive morphisms amounts to relating them only via maps that do not reindex Opponent moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This has the effect of making everything stricter, and cutting the higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' More precisely: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let A be a thin groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider PreThin(A) the sub-2-category of Unif(A) with objects TA, and Thin(A) where additionally morphisms are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, Thin(A) is locally discrete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' all 2-cells are identi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, PreThin(A) and Thin(A) are biequivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The first is a direct consequence of thinness: if µ : (F, φ) ⇒ (G, ψ) : S → T for φ and ψ positive, then by defi- nition of 2-cells of Unif(A), for all s ∈ S, µs ∈ T (Fs, Gs) is such that ψs = ∂T µs ◦ φs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ∂T µs = ψs ◦ φ−1 s positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, µs is an identity morphism by thinness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For the biequivalence, the crux is that if (F, φ) : S → T is a weak morphism, then there is a unique (F+, φ+) : S → T positive isomorphic to (F, φ), and a unique 2-cell µ between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Uniqueness follows from thinness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For existence, note that if s ∈ S and θ : ∂Ss ∼=A a, then there exist unique ϕ : s ∼=S s′ and θ+ : ∂Ss ∼=+ A a such that θ = θ+ ◦ ∂Sϕ – this exploits thinness, and the reindexing problem from the fact that the pullback of the cospan S ֒→ A ←֓ A+ is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We obtain (F+, φ+) by applying this lemma pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This proposition illustrates the situation well: thanks to the thin biorthogonality, the 2-category PreThin(A) is repre- sented up to biequivalence as a mere category Thin(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The higher dimensional structure simply vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 5) The bicategory Thin: With this in place, we may finally define the components of our bicategory Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Its objects are thin groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Its morphisms from A to B are strategies from A to B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' elements of TA⊸B – recall that they are (S, ∂S : S → A × B), in particular spans from A to B A ∂S A ←−− S ∂S B −−→ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The identities are identity spans, and composition is via the pullback (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If S and T are strategies from A to B, the 2- cells from S to T are the positive morphisms (F, φ) : S → T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As φ : ∂S ⇒ ∂T ◦ F is a family of positive morphisms on A⊥ ` B with underlying plain groupoid A × B, it may be equivalently presented as pair of F A : ∂S A ⇒ ∂T A ◦F and F B : ∂S B ⇒ ∂T B ◦ F, as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For horizontal composition of positive morphisms, we first proceed as in Section III-B4 and obtain a connected groupoid of (non necessarily positive) horizontal compositions – which must all have the same image through the biequivalence of Proposition 3, providing our unique positive horizontal composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Altogether, we have: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Those components form Thin, a bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' See details in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Next, we develop the further structure of Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' CARTESIAN CLOSED STRUCTURE To construct a cartesian closed bicategory, we intend to follow [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We first turn the construction Fam – thereafter denoted by !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' – into a pseudocomonad, and then equip the Kleisli bicategory Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' with the cartesian closed structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The Pseudocomonad We first develop the action of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' on objects of Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 1) The bang of thin groupoids: First, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' is defined on uniform groupoids via U!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A = (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='UA)⊥⊥, where we have used !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='UA = {(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S) | S ∈ UA} using the functorial action !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S → !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For thin groupoids, the positive and negative symmetries of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A were defined in Section III-C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The thin structure is set as T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A = (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='TA)⊥⊥⊥⊥ – it is a direct verification that this is a thin groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) The bang of strategies: If S ∈ TA⊸B, we have ∂S = ⟨∂S A, ∂S B⟩ for ∂S A : S → A and ∂S B : S → B – its bang is !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S A ←−− !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S B −−→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B packaged as (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S, ⟨!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S A, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S B⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' That this is in T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A⊸!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B relies on: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider A, B thin groupoids, and T a prestrategy from !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, the following two properties hold: (1) T ∈ U!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A⊸B iff (T, ∂T !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) ∈ U⊥ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A and for all S ∈ UA, T @!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ∈ UB, (2) T ∈ T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A⊸B iff for all S ∈ TA, T @!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ∈ TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is an immediate application of Propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since U!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A = (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='UA)⊥⊥ and T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A = (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='TA)⊥⊥⊥⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' From this lemma, it is a rather direct verification that for any S ∈ TA⊸B, we have !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ∈ T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A⊸!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3) A pseudofunctor: Since !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' is a functor, it preserves the identity span on the nose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' preserves pullbacks, for S ∈ TA⊸B and T ∈ TB⊸C, the universal property gives us mS,T : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (T ⊙ S) ∼= !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S a strong invertible 2-cell in Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As expected, this 2-cell is natural in S and T (with respect to positive morphisms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Altogether, we obtain a pseudofunctor !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Thin → Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' See Appendix F for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 4) A pseudomonad on groupoids: In fact we first turn !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' into a pseudomonad on Gpd, from which its pseudocomonad structure on Thin shall be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We noted earlier that we have a functor Fam : Gpd → Gpd – in fact, it is extended to a 2-endofunctor on the 2-category of small groupoids, noted !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Gpd → Gpd , defined on a 2-cell α : F ⇒ G : A → B as the natural transformation !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='α : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G with components all pairs (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='α)(Ai)i∈I = (idI, (αAi)i∈I) ∈ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B((FAi)i∈I, (GAi)i∈I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A µA η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ηA id!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A αA ⇒ µA βA ⇐ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Unit natural isomorphisms !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µA µA γA ⇒ µA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Associativity natural isomorphism To turn this into a pseudomonad, we must adjoin a multipli- cation and a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The components of the unit are the functors ηA : A → !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A a �→ (a){0} = [0 · a] with the obvious functorial action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The intuition is that the unit transports a single resource usage from A to !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A, arriving at a singleton family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In doing so, it must select a copy index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Any natural number will do – the rest of the paper does not depend on this choice – but for definiteness and compatibility with the traditional convention from AJM games, we pick 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For the multiplication µA : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A → !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A, we must flatten a family of families into a family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For this purpose, we fix an injective function ⟨−, −⟩ : N2 → N – again, the results of this paper do not depend on that choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given I ⊆f N and a family (Ji)i∈I where Ji ⊆f N for all i ∈ I, let us write Σi∈IJi = {⟨i, j⟩ | i ∈ I, j ∈ Ji} , which is by definition still a finite subset of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then we set µA : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A → !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ((ai,j)j∈Ji)i∈I �→ (ai,j)⟨i,j⟩∈Σi∈IJi for any groupoid A, along with the obvious functorial action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Altogether this yields η : idGpd ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and µ : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', two (strict 2-) natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The monad laws, if they were to hold on the nose, would mean that ⟨0, i⟩ = ⟨i, 0⟩ = i and ⟨⟨i, j⟩, k⟩ = ⟨i, ⟨j, k⟩⟩ for all i, j, k ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and it is clear that no injection satisfying those laws exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Nevertheless, for every groupoid A the coherence laws for a monad hold up to natural isomorphisms: we have αA, βA and γA as indicated in Figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For instance, for any (aj)j∈J ∈ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A: (αA)(aj)j∈J : (aj)j∈J ∼=!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A (aj)⟨0,j⟩∈Σi∈{0}J reindexing along the bijection J ≃ Σi∈{0}J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The other components act similarly – note that they are all negative symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The associated families (αA)A∈Gpd, (βA)A∈Gpd and (γA)A∈Gpd satisfy the conditions for modifications, and the additional coherence laws for a pseudomonad: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The 2-functor !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Gpd → Gpd along with the components above yield a pseudomonad on Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 5) Lifting functors to spans: We shall turn !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' into a pseudo- comonad on Thin by lifting the components above to spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In general, if F : B → A is a functor, then there is a span ˇF A F ←− B idB −−→ B , called the lifting of F – but we need sufficient conditions on F for this construction to yield morphisms in Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For that purpose, if A and B are thin groupoids, we say that a functor F : A → B is a renaming iff the following conditions hold: (1) for all θ : a ∼=A a′, if θ is positive then so is Fθ, (2) for all (T, ∂T ) ∈ U⊥ B, (T, F ◦ ∂T ) ∈ U⊥ A, (3) for all (T, ∂T ) ∈ T⊥⊥ B , (T, F ◦ ∂T ) ∈ T⊥⊥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Clearly, renamings compose – we consider the 2-category Ren whose objects are thin groupoids, whose morphisms are renamings, and whose 2-cells are negative natural trans- formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As expected, lifting renamings yields thin spans (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Lifting can be extended to 2-cells: if α : F ⇒ G : A → B is a negative natural transformation, then ˇα is the positive morphism described by the diagram: B A B B F idB idB α⇒ G idB , noting that this is positive as negative α is in contravariant position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Altogether, we get (see details in Appendix E): Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' There is a pseudofunctor ˇ− : Renop → Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Here, Renop is Ren with the morphisms reversed, but the 2-cells unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It can be checked that for any thin groupoid A, the functors ηA : A → !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A and µA : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A → !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A are renamings, in particular for every thin groupoid A we get ˇ ηA ∈ Thin(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A, A) ˇ µA ∈ Thin(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A, !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) the main components to turn !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' into a pseudocomonad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Unlike in Gpd, the families ˇη and ˇµ are not strict 2-natural transfor- mations but only pseudonatural transformations, with 2-cells ηS : ˇηB ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⇒ S ⊙ ˇηA µS : ˇµB ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⊙ ˇµA , positive isomorphisms obtained for S ∈ Thin(A, B) from the universal property of pullbacks, via the observation that η : idGpd ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and µ : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' are cartesian natural transormations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It may be checked that ηS and µS are natural in S and satisfy the coherence conditions of pseudonatural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Finally, the modifications α, β, γ involved in the pseudomonad structure of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' on Gpd lift to the modifica- tions required for the pseudocomonad structure of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' on Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We have a pseudocomonad !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' on Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' See details in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We move on to studying the Kleisli bicategory Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' whose horizontal composition, denoted ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', is defined as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Cartesian Closed Structure 1) Finite products: First, we show that Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' has finite products, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' is a fp-bicategory in the sense of Fiore and Saville [18] – unlike them, we work with binary products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' a) Terminal object: Write ⊤ for the empty groupoid, made a thin groupoid with U⊤ = T⊤ = {id∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For any thin groupoid A, Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A, ⊤) has exactly one element – the empty groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' has a (strict) terminal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' b) Binary product: If A and B are thin groupoids, then the with A & B has underlying groupoid A + B the disjoint union, with (A+B)− = A−+B− and (A+B)+ = A++B+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We adjoin UA&B = (UA + UB)⊥⊥ and TA&B = (TA + TB)⊥⊥⊥⊥, where as usual, UA + UB comprises the set of all (S +T, ∂S +∂T) for (S, ∂S) ∈ UA and (T, ∂T) ∈ UB, using the functorial action of + (and likewise for TA + TB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' c) Pairing and projections: The projections are simply set as L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' = ̌ (ηA+B ◦ ¯l) ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A & B, A) and R!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' = ̌ (ηA+B ◦ ¯r) ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A & B, B) for ¯l : A → A + B and ¯r : B → A + B the obvious coprojections/renamings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The pairing of S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A) and T ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, B) is (S + T, ∂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ : S + T → !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ, ∂A&B : S + T → A + B) with ∂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ the co-pairing and ∂A&B = ∂S A + ∂T B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We have: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For any thin groupoids Γ, A and B, there is Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A & B) ⊥ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A) × Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, B) (L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='−,R!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='−) ⟨−,−⟩ an adjoint equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A) and T ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, B) there are ωA S,T : L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='⟨S, T ⟩ ∼= S ωB S,T : R!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='⟨S, T ⟩ ∼= T positive isos, and for U ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A & B) there is ¯ωU : U ∼= ⟨L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='U, R!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩ a positive iso, defined in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Those are all natural in S, T, U, and satisfy the required triangle identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' See Appendix H for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Altogether, this estab- lishes that Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' is a fp-bicategory in the sense of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) Cartesian closure: If A and B are thin groupoids, then we set A ⇒ B = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ` B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Before we describe the additional components, we must observe the Seely equivalence: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ⊗ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A & B) sA,B ¯sA,B where sA,B sends (ai)i∈I, (bj)j∈J to (ck)k∈I⊲⊳J, with I⊲⊳J = ̟(I ⊔ J) for some chosen bijection ̟ = [̟l, ̟r] between N ⊔ N and N, and where c̟l(i) = ai and c̟r(j) = bj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and ¯sA,B sends (ck)k∈K to (ai)i∈I, (bj)j∈J where I ⊆ K is the subset of those i ∈ K such that ci = ai ∈ A, and likewise for bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Both functors are renamings, and the isomorphisms witnessing the equivalence are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Via the Seely equivalence, we first define the evaluation as the span with basic groupoid !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A× B, with left leg the functor !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B → (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A → !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A sA,B −−−→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((A ⇒ B) & A) and right leg the projection !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This yields a thin span evA,B ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((A ⇒ B) & A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Now, we need Λ(−) : Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) → Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B) the currying functor: given S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ&A, B), its currying Λ(S) is simply S, with display map post-composed with !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) × B ¯sΓ,A ≃ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) × B ∼= !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' With this data in place, we may finally prove: Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For any groupoids Γ, A, B, there is Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B) ⊥ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) evA,B⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−&A) Λ(−) an adjoint equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One can first show the existence of adjoint equivalence between the currying operation Λ(−), and a symmetric uncur- rying operation ¯Λ(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The unit and counit of this adjunction can be derived from the ones of the Seely (adjoint) equiva- lence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One can then prove that ¯Λ(−) is in fact isomorphic to evA,B ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−&A) in order to get the wanted equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' See Appendix I for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Altogether, we have: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We have Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', a cartesian closed bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This entails that we can interpret types of the simply-typed λ-calculus as thin groupoids, morphisms as thin spans and rewrites between terms as certain positive isomorphisms [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' CONCLUSION This paper focuses on the construction of Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', leaving for later its application to semantics of λ-calculi and program- ming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We believe this opens multiple perspectives for further research: firstly, we may explore the obtained interpretation of the λ-calculus, which syntactically should correspond to the sequence typing system of Vial [38] and to the non-uniform λ-calculus of Melli`es [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We should explore links with other models of the literature, notably with the weighted relational model recasting ideas from [37], and with generalized species of structures and template games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Another related direction consists in accommodating another feature of template games, the mechanism to capture scheduling and synchronization [34], into thin spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In more semantic directions, we believe that with respect to generalized species of structures, the fact that operations on thin spans involve no quotient may be helpful in two ways: (1) individuals may be ordered concretely, and the model should support continuous reasoning allowing one to deal easily with infinite computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and (2) adding “typed” weights coming from an SMCC as in [24] should be a lot simpler, since those weigths no longer have to themselves be saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ACKNOWLEDGMENT Work supported by the ANR projects DyVerSe (ANR- 19-CE48-0010-01) and PPS (ANR-19-CE48-0014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and by the Labex MiLyon (ANR-10-LABX-0070) of Universit´e de Lyon, within the program “Investissements d’Avenir” (ANR- 11-IDEX-0007), operated by the French National Research Agency (ANR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Girard, “Linear logic,” Theor.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [39] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Ehrhard, “The scott model of linear logic is the extensional collapse of its relational model,” Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 424, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 20–45, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [40] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Hyland, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Power, “Pseudo-distributive laws,” Elec- tronic Notes in Theoretical Computer Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 83, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 227–245, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' APPENDIX In the following, we will type spans A u ←− S v−→ B between groupoids or thin groupoids A and B as S : A ⇸ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Bipullbacks In order to complete the equational definition of bipull- backs of Definition 2, it is useful to consider the intensional definition first: given a 2-category C with invertible 2-cells (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', a (2, 1)-category) and a cospan S u−→ B v←− T in C, a bipullback is a pseudocone (P, l, r, µ) as in Figure 1 such that, for every X ∈ C, the precomposition of the pseudocone (P, l, r, µ) by morphisms X → P induces an equivalence of categories between C(X, P) and the category of pseudocones over the cospan S u−→ B v←− T and of vertex X and pseudocone morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The essential surjectiveness of this precomposition corresponds exactly to the condition (a) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Its full faithfulness can be expressed as the following condition: (b’) given a 2-cell equality X P S T B h l r u µ=⇒ v = X P S T B l◦h r◦h h′ l r β=⇒ α=⇒ u µ=⇒ v for some h, h′ : X → P and 2-cells α: l ◦ h ⇒ l ◦ h′ and β : r ◦ h′ ⇒ r ◦ h, there is a unique θ: h ⇒ h′ such that α = lθ and β = rθ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It is not too difficult to show that the latter property is equivalent to the one asserting that, given two decompositions of a pseudocone ν X S T B l′ r′ u ν=⇒ v = X P S T B l′ r′ h l r β=⇒ α=⇒ u µ=⇒ v = X P S T B l′ r′ h′ l r β′ ==⇒ α′ ==⇒ u µ=⇒ v there exists a unique θ: h ⇒ h′ such that lθ = α′ ◦ α−1 and rθ = β′−1 ◦ β, or equivalently α′ = (lθ) ◦ α and β′ = β ◦ (rθ−1), which is the complete form of the condition (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Pullbacks in Gpd It happens that pullbacks in Gpd are well-behaved w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t 2-cells: Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A pullback of a cospan S u−→ B v←− T in Gpd is a strict 2-pullback, that is, also admits a universal factorization property w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' morphisms of cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let I be the groupoid consisting of a walking isomor- phism u between two objects 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given two functors F and G between two groupoids C and D, a 2-cell α: F ⇒ G: C → D ∈ Gpd is then exactly the data of a functor H : I × C → D such that H(0, −) = F and H(1, −) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Using this correspondence, the property that a pullback (P, l, r) over the cospan is a 2-pullback easily reduces to the one that (P, l, r) is a 1-pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Note that a pullback is a cone, which is in particular a pseudocone with identity as inner 2-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One might then ask when a pullback is a bipullback, in which case we have the following characterization: Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let (P, l, r) be a pullback over a cospan S u−→ B v←− T in Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then the pseudocone induced by (P, l, r) is a bipullback if and only if it satisfies the condition (a) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Proposition 8, (P, l, r) is a 2-pullback, so that it satisfies a universal property w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' cone morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This con- dition is in fact exactly (b’) which is equivalent to (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, only (a) is left to check for (P, l, r) to be a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We can then refine the previous proposition into a “point- wise” characterization in the form of already stated Lemma 1 for which we now provide a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The implication is immediate, since the data of an isomorphism θ: f(s) → g(t) is equivalent to the one of a pseudocone on S B T f g of vertex the terminal groupoid, whose factorization in the form of the condition of Proposition 9 is exactly the wanted property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We now show the converse property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So let Z S T B h k f θ=⇒ g be a pseudocone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By hypothesis, for every z ∈ Z, there exist sz ∈ S, φz : h(z) → sz, tz ∈ T , ψz : tz → k(z) such that f(sz) = g(tz) and θz = g(ψz) ◦ f(φz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The collection of isomorphisms (φz)z∈Z induces a functor h′ defined by h′(z) = sz for z ∈ Z, and h′(w) = φz′ ◦ h(w) ◦ φ−1 z for w: z → z′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Similarly, we get a functor k′ defined by h′(z) = tz for z ∈ Z, and k′(w) = ψ−1 z′ ◦ k(w) ◦ ψz for w: z → z′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given w: z → z′ ∈ Z, we check that f ◦ h′(w) and g ◦ k′(w) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since there are only isomorphisms involved, it is enough to check that the equality holds when in the context g(ψz′) ◦ (−) ◦ f(φz): g(ψz′) ◦ f(h′(w)) ◦ f(φz) = g(ψz′) ◦ f(h′(w) ◦ φz) = g(ψz′) ◦ f(φz′ ◦ h(w)) = g(ψz′) ◦ f(φz′) ◦ f(h(w)) = θz′ ◦ f(h(w)) = g(k(w)) ◦ θz = g(k(w)) ◦ g(ψz) ◦ f(φz) = g(ψz′) ◦ g(k′(w)) ◦ f(φz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, (Z, h′, k′) is a cone on S B T f g , so there exists m: Z → P which factors h′ and k′ through l and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The collections (φz)z∈Z and (ψz)z∈Z defines natural iso- morphisms φ: h ⇒ l ◦m and ψ: r ◦m ⇒ k which satisfy (gψ) ◦ (fφ) = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Hence, the condition of Proposition 9 is satisfied and (P, l, r) is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We also have the following criterion for rectangles of bipullbacks: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given a rectangle made of two squares which are pullbacks in Gpd as in L M R A B C ⌜ πL M πL A ⌜ πM R πM B h f g , the following hold: (i) if the whole rectangle is a bipullback, then the left square is too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (ii) if the left and right square are bipullback, then the whole rectangle is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We first prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For this purpose, we use Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So let a ∈ A, m ∈ M and θ: f(a) → πM B (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We have that g(θ) is a morphism from g◦f(a) to g◦πM B (m) = h ◦ πM R (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since the outer rectangle is assumed to be a bipullback, there exist a′ ∈ A, r′ ∈ R, uA: a → a′ ∈ A, vR : r′ → πM R (m) such that g(θ) = h(vR) ◦ (g ◦ f)(uA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, we have g(θ ◦ (f(uA))−1) = h(vR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since the right square is a pullback, there exists a unique wM such that πM B (wM) = θ ◦ (f(uA))−1 and πM R (wM) = vR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, by the right pullback again, the target of wM is m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' its source is some m′ such that πM B (m′) = f(a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, we have that θ = πM B (wM) ◦ f(uA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We can conclude with Proposition 9 that the left pullback is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We now prove (ii) using Proposition 9 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So let a ∈ A, r ∈ R and θ: (g ◦ f)(a) → h(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since the right square is a bipullback, there exist b′ ∈ B, r′ ∈ R, uB : f(a) → b′ and vR : r′ → r such that θ = h(vR)◦g(uB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since b′ and r′ have the same projection in C through g and h respectively, there exists m′ ∈ M such that πM B (m′) = b′ and πM R (m′) = r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, we have uB : f(a) → πM B (m′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since the left square is a bipullback, there exist a′′ ∈ A, m′′ ∈ M, ˜uA : a → a′′, ˜vM : m′′ → m′ such that uB = πM B (˜vM) ◦ f(˜uA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We thus have θ = h(vR) ◦ g(uB) = h(vR) ◦ g(πM B (˜vM) ◦ f(˜uA)) = h(vR) ◦ g(πM B (˜vM)) ◦ g(f(˜uA)) = h(vR) ◦ h(πM R (˜vM)) ◦ g(f(˜uA)) = h(vR ◦ πM R (˜vM)) ◦ (g ◦ f)(˜uA) which is precisely the factorization required by Proposition 9 to conclude that the whole rectangle is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Uniformity and thinness Several arguments concerning uniformity requires some sort of diagram chasing relative to bipullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' An important lemma for this is the following: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Consider the diagram in Gpd S P Q L M R A B lS P rS Q 2 lP L rP M 1 l Q M rQ R 3 f L A f M A f M B f R B where the square 1, 2 and 3 are pullbacks, and derive from it the following diagram using the product structure: S M L × R A × B r P M ◦ lS P (lP L ◦ lS P ,rQ R ◦ r S Q) 4 (f M A ,f M B ) f L A×f R B Then 4 is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, the following hold: (i) if 1 and the rectangle made of 2 and 3 are bipullbacks, then 4 is a bipullback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (ii) if 3 and the rectangle made of 1 and 2 are bipullbacks, then 4 is a bipullback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (iii) if 4 is a bipullback, then the rectangle made of 1 and 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2 and 3) is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The fact that 4 is a pullback is an easy consequence of the fact that 1,2,3 are pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One can then use Lemma 1 without too much trouble on the different bipullback hypotheses in order to deduce that the wanted pullbacks are bipullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' With the above tool, we can now prove Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We first prove the first implication, and start by showing (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So let (S, ∂S A) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given (U, ∂U B) ∈ U⊥ B, we must show that T @S ⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By hypothesis, we have that T ⊥ S × U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', the pullback of ∂T A×B and ∂S A × ∂U B is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, we conclude by Lemma 6(iii) that the pullback of ∂T @S B and ∂U B is a bipullback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', T @S ⊥ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Hence, T @S ∈ UB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We now show (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since idB is an isofibration, we have that the pullback of ∂T B and idB is a bipullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, given U ∈ UA, by Lemma 6, we have that ∂T A ⊥ ∂U A if and only if ∂T A×B ⊥ ∂U A × idB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But the latter holds, since T ∈ UA⊸B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Hence, ∂T A ∈ U⊥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We now show the converse implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So assume that T satisfies (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' First note that, since (A, idA) ∈ UA, we have that ∂T B ∈ U⊥ B by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given V ∈ UB, we must show that, for every U ∈ UA, we have T ⊥ U × V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since we have ∂T B ⊥ ∂V B, by Lemma 6, this is equivalent to have U ⊥ T ⋆@V for every U ∈ UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By hypothesis, it is equivalent to only check the previous condition for U ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Lemma 6 again, it is equivalent to check that T ⊥ U × V for every U ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since U ⊥ ∂T A, by Lemma 6 again, it is equivalent to check that T @U ⊥ V , but the latter holds by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, T ∈ UA⊸B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Using Proposition 1, we can prove the compatibility of uniformity with composition: Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given uniform groupoids (A, UA) and (B, UB) and prestrategies S ∈ UA⊸B and T ∈ UB⊸C, we have T ⊙ S ∈ UA⊸C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Recall that the composition of the two spans S and T is formed as in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We show the uniformity of T ⊙S using Proposition 1 with UA taken as generating class of UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The fact that (1) is satisfied for T ⊙ S is immediate from its validity for both S and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The fact that (2) holds, that is, that ∂S A ◦ l ∈ U⊥ A, is a consequence of the fact that ∂T B ∈ U⊥ B by (2) on T , and the dual of Proposition 1 for S, asserting in particular that S⋆ maps elements of U⊥ B to U⊥ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We handle thinness similarly, and start by proving Proposition 2: Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Assume that T ∈ TA⊸B and let S ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Proposition 1, we already have T @S ∈ UA⊸B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Next, we use the fact that TB = T⊥⊥⊥⊥ B to show that T @S ∈ TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So let U ∈ T⊥⊥ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Lemma 6, we have T @S ⊥⊥U iff T ⊥⊥S ×U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But the latter holds since T ∈ TA⊸B and S ∈ S ⊆ TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, T @S ∈ TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Conversely, assume that T @S ∈ TB for every S ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' First observe that TA⊸B = (TA ⊗ T⊥⊥ B)⊥⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So we must show that, for every S ∈ TA and U ∈ T⊥⊥ B , T ⊥⊥ S × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Lemma 6, for a given U, the latter is equivalent to S ⊥⊥ T ⋆@U for every S ∈ TA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', T ⋆@U ∈ T⊥⊥ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But since T⊥⊥ A = S⊥⊥, for a given U, it is equivalent to S ⊥⊥ T ⋆@U for every S ∈ S, itself equivalent to T ⊥⊥S×U for every S ∈ S, and finally equivalent to T @S ⊥⊥ U for every S ∈ S, which amounts to our initial assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Using Proposition 2, we can then prove the compatibility of thinness with composition: Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids A and B and strategies S ∈ UA⊸B and T ∈ UB⊸C, we have T ⊙ S ∈ TA⊸C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The proof is similar to (in fact simpler than) the one of Proposition 10 and follows from the criterion given by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Details about the bicategory Thin We have the following convenient characterization of 0-composition of 2-cells of Thin: Proposition 12 (Paved Characterization of Composition (PCC)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' strategies R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' R′ : A ⇸ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' S′ : B ⇸ C and weak morphisms F : R ⇒ R′ and G: S ⇒ S′ of Thin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' if there exist a functor H : S ⊙ R → ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ⊙ R′ and two natural transformations Hl and Hr as in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⊙ R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ⊙ R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Hl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⊙ R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ⊙ R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Hr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='such that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⊙ R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ⊙ R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Hl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⊙ R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ⊙ R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Hr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='GB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='and such that the natural transformations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='HA ˆ= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⊙ R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ⊙ R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Hl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='HC ˆ= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ⊙ R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ⊙ R′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Hr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='GC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='==⇒ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='are positive over A⊥ and C respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' we have that H ˆ= (H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' HC) is a positive morphism S ⊙ R ⇒ S′ ⊙ R′ of Thin and that H = G ⊙ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The fact that H is a 2-cell S⊙R ⇒ S′⊙R′ of Thin is immediate by the polarity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The equality of 2-cells given by the hypothesis can be rewritten as S ⊙ R R S S′ ⊙ R′ R′ = S′ B H F Hl ==⇒ G (Hr)−1 =====⇒ ∂R′ B ∂S′ B = S ⊙ R R = S R′ S′ B F ∂R B G ∂S B ∂R′ B ∂S′ B (F B)−1 =====⇒ GB ==⇒ so that Hl and Hr provide a factorization of the pseudocone on the right, and define an object of the groupoid of com- positions mentioned in Section III-D5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The actual horizontal composition in Thin is then obtained by applying the biequiv- alence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But since ⟨HA, HC⟩ is already a positive natural transformation on A ⊸ C, this biequivalence does nothing on this object and H = G ⊙ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids A, B, C, we have a functor (−) ⊙ (−): Thin(B, C) × Thin(A, B) → Thin(A, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By the definition we took for the composition of the 2-cells of Thin, we already have that (−) ⊙ (−) respects the sources and targets of weak morphisms, so that we are left to verify functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given R ∈ Thin(A, B) and S ∈ Thin(B, C), a solution in H, Hl and Hr for the equation S ⊙ R S ⊙ R R R B B H l Hl ==⇒ l idR ∂R B = ∂R B = S ⊙ R S ⊙ R S S B B H r Hr ==⇒ r idS ∂S B = ∂S B is given by H = idR⊙S, Hl = idl and Hr = idr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, since identities are member of A− and C+, the polarity condition of Proposition 12 is satisfied so that idS ⊙ idR = (idS⊙R, id∂R A◦l, id∂S C◦r) = idS⊙R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Now, given four positive morphisms organized as R F−→ R′ F ′ −→ R′′ ∈ Thin(A, B) , S G −→ S′ G′ −→ S′′ ∈ Thin(B, C) , the procedure to compute F ⊙ G gives us H, Hl and Hr s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' S ⊙ R S′ ⊙ R′ R R′ B B l H Hl ==⇒ l F ∂R B F B ==⇒ ∂R′ B = S ⊙ R S′ ⊙ R′ S S′ B B r H Hr ==⇒ r G ∂S B GB ==⇒ ∂S′ B and with ∂R′ A Hl and ∂S′ C Hr respectively negative on A and positive on C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' similarly, the procedure to compute F ′ ⊙ G′ gives us H′, H′l and H′r such that S′ ⊙ R′ S′′ ⊙ R′′ R′ R′′ B B l H′ H′l ==⇒ l ∂R′ B F ′ F ′B ===⇒ ∂R′′ B = S′ ⊙ R′ S′′ ⊙ R′′ S′ S′′ B B r H′ H′r ===⇒ r ∂S′ B G′ G′B ===⇒ ∂S′′ B and with ∂R′′ A H′l and ∂S′′ C H′r respectively negative on A and positive on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' On the one hand, we thus have that (G′ ⊙ F ′) ◦ (G ⊙ F) is the span morphism K = (K, Kl, Kr) with K = H′H and Kl = S ⊙ R S′ ⊙ R′ S′′ ⊙ R′′ R R′ R′′ A A A l H Hl ==⇒ l H′ H′l ==⇒ l F ∂R A F A ==⇒ ∂R′ A F ′ F ′A ===⇒ ∂R′′ A Kr = S ⊙ R S′ ⊙ R′ S′′ ⊙ R′′ S S′ S′′ C C C r H Hr ==⇒ r H′ H′r ===⇒ r G ∂S C GC ==⇒ ∂S′ C G′ G′C ===⇒ ∂S′′ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' On the other hand, we have S ⊙ R S′ ⊙ R′ S′′ ⊙ R′′ R R′ R′′ B B B l H Hl ==⇒ l H′ H′l ==⇒ l F ∂R B F B ==⇒ ∂R′ B F ′ F ′B ===⇒ ∂R′′ B = S ⊙ R S′ ⊙ R′ S′′ ⊙ R′′ S S′ S′′ B B B r H Hr ==⇒ r H′ H′r ===⇒ r G ∂S B GB ==⇒ ∂S′ B G′ G′B ===⇒ ∂S′′ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, the two natural transformations obtained as the horizontal pasting of Hl and H′l along l and the horizontal pasting of Hr and H′r along r satisfy the polarity condition of the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Hence, by considering again the diagrammatic definition of K, the PCC tells us that K is also (G′ ◦ G)⊙ (F ′ ◦ F), which concludes functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We now show the unitality of the horizontal composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given a thin groupoid A, we write ccA for the identity span A idA ←−− A idA −−→.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We have Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given a thin groupoid A, we have ccA ∈ TA⊸A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is an easy consequence of Proposition 1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We also write ccA for the corresponding functor 1 → Thin(A, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given an additional thin groupoid B, there is a transformation R between the functors Thin(A, B) ∼ −→ Thin(A, B) × 1 → · · · id×ccA −−−−−→ Thin(A, B) × Thin(A, A) (−)⊙(−) −−−−−→ Thin(A, B) and Thin(A, B) id −→ Thin(A, B) whose component at S ∈ Thin(A, B) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Recall that the span S ⊙ ccA is defined by the pullback S ⊙ ccA A S A A B l r idA idA ∂S A ∂S B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, RS ˆ= r is an isomorphism (as the pullback of an isomorphism) which moreover induces a strong isomorphism of strategies RS : S ⊙ ccA ⇒ S ∈ Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' R ˆ= (RS)S∈Thin is a natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let A, B, S, S′ : A ⇸ B and F : S → S′ in Thin(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We first picture the two compositions T ˆ= ccA ⊙S and T ′ ˆ= ccA ⊙S′ on the diagram T A S A A B A S′ T ′ lT rT idA idA ∂S A ∂S B idA idA ∂S′ A ∂S′ B lT ′ rT ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We now compute the composition F ⊙ idccA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By the PCC, it is the morphism ˜F : S ⊙ ccA ⇒ S′ ⊙ ccA defined by ˜F = T rT −−→ S F−→ S′ (rT ′)−1 −−−−−→ T ′ and ˜F A = T S S′ T ′ A A A A A A A A lT r T = ∂S A F F A ==⇒ ∂S′ A (r T ′ )−1 = lT ′ idA = idA = idA = idA ˜F B = T S S′ T ′ S S′ B B rT rT = F (rT ′ )−1 rT ′ ∂S B F F B ==⇒ ∂S′ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The naturality of R is then expressed by the equation RS′ ◦(F ⊙ idccA) = F ◦ RS that we now check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We first have RS′ ◦(F ⊙ idccA) = T rT −−→ S F−→ S′ = F ◦ RS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, (RS′ ◦(F ⊙ idccA))A = T S S′ A A A A A A lT rT = ∂S A F F A ==⇒ ∂S′ A idA = idA = idA = (F ◦ RS)A and (RS′ ◦(F ⊙ idccA))B = T S S′ S S′ B B r T r T = F ∂S B F F B ==⇒ ∂S′ B = (F ◦ RS)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Which concludes the proof that R defines a natural iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Similarly, there is a transformation L between Thin(A, B) ∼ −→ 1 × Thin(A, B) → · · · ccB ×id −−−−−→ Thin(B, B) × Thin(A, B) (−)⊙(−) −−−−−→ Thin(A, B) and Thin(A, B) id −→ Thin(A, B) whose component at a strategy S ∈ Thin(A, B) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Recall that the span ccB ⊙S is defined by the pullback ccB ⊙S S B A B B l r ∂S B ∂S A idB idB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Then, LS ˆ= l is an isomorphism (as the pullback of an isomorphism) which moreover induces a strong isomorphism of thin spans LS : ccB ⊙S ⇒ S ∈ Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As before, we have Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' L ˆ= (LS)S∈Thin is a natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' there is a transformation A: ((−) ⊙ (−)) ⊙ (−) ⇒ (−) ⊙ ((−) ⊙ (−)) : Thin(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' D) × Thin(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' C) × Thin(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' B) → Thin(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' D) whose component at S ∈ Thin(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ∈ Thin(B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' C) and U ∈ Thin(C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' D) is given by a strong morphism AS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='U : (U ⊙ T ) ⊙ S → U ⊙ (T ⊙ S) defined as expected between the two compositions using the different pullbacks involved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' as in (U ⊙ T ) ⊙ S U ⊙ T S T U A B C D S T U T ⊙ S U ⊙ (T ⊙ S) r(U⊙T )⊙S l(U⊙T )⊙S AS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='U lU⊙T rU⊙T lT ⊙S rT ⊙S lU⊙(T ⊙S) rU⊙(T ⊙S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' An inverse for AS,T,U is defined symmetrically, so that A is an isomorphic transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The transformation A is a natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let F : S ⇒ S′ : A ⇸ B, G: T ⇒ T ′: B ⇸ C and H : U ⇒ U ′ : C ⇸ D be weak morphisms in Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We compute (U ⊙ T ) ⊙ S as usual but moreover factor the projection l (U⊙T )⊙S canonically through T ⊙ S by a unique morphism ˜l (U⊙T )⊙S so that we get a diagram (U ⊙ T ) ⊙ S T ⊙ S U ⊙ T S T U A B C D ˜l (U⊙T )⊙S and we get a similar diagram for (U ⊙ T ) ⊙ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Symmetri- cally, the projection r U⊙(T ⊙S) can be factored canonically through U ⊙ T by a morphism ˜r U⊙(T ⊙S), and the projection r U′⊙(T ′⊙S′) through U ′ ⊙ T ′ by a morphism ˜r U′⊙(T ′⊙S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Note that l U⊙(T ⊙S) ◦ AS,T,U = ˜l (U⊙T )⊙S and other similar equalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By computing K ˆ= G ⊙ F, we get natural transformations ˜KA and ˜KC such that KA = T ⊙ S T ′ ⊙ S′ S S′ A A K lT ⊙S ˜ KA ==⇒ lT ′⊙S′ ∂S A F F A ==⇒ ∂S′ A and KC = T ⊙ S T ′ ⊙ S′ T T ′ C C K rT ⊙S ˜ KC ==⇒ rT ′⊙S′ ∂T C G GC ==⇒ ∂T ′ C which satisfy moreover that T ⊙ S T ′ ⊙ S′ S S′ B B K lT ⊙S ˜ KA ==⇒ lT ′⊙S′ ∂S B F F B ==⇒ ∂S′ B = T ⊙ S T ′ ⊙ S′ T T ′ B B K r T ⊙S ˜ KC ==⇒ r T ′⊙S′ ∂T B G GB ==⇒ ∂T ′ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Similarly, by computing L ˆ= H ⊙ G, we get natural transfor- mations ˜LB and ˜LD such that LB = U ⊙ T U ′ ⊙ T ′ T T ′ B B L lU⊙T ˜LB ==⇒ lU′⊙T ′ ∂T B G GB ==⇒ ∂T ′ B and LD = U ⊙ T U ′ ⊙ T ′ U U ′ D D L rU⊙T ˜LD ==⇒ rU′⊙T ′ ∂U D H HD ===⇒ ∂U′ D which satisfy moreover that U ⊙ T U ′ ⊙ T ′ T T ′ C C L lU⊙T ˜LB ==⇒ lU′⊙T ′ ∂T C G GC ==⇒ ∂T ′ C = U ⊙ T U ′ ⊙ T ′ U U ′ C C L r U⊙T ˜LD ==⇒ r U′⊙T ′ ∂U C H HC ==⇒ ∂U′ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since (U ′ ⊙ T ′) ⊙ S′ T ′ ⊙ S′ U ′ ⊙ T ′ T ′ r (U′⊙T ′)⊙S′ ˜l (U′⊙T ′)⊙S′ r T ′⊙S′ lU′⊙T ′ is a bipullback by Lemma 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and that the natural transforma- tions ˜KC and ˜LB define a pseudocone of vertex (U ⊙ T ) ⊙S on the associated cospan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' we get M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ˜ M A and ˜ M D such that (U ⊙ T ) ⊙ S (U ′ ⊙ T ′) ⊙ S′ T ⊙ S T ′ ⊙ S′ T T ′ M ˜l (U⊙T )⊙S ˜ MA ===⇒ ˜l (U′⊙T ′)⊙S′ K rT ⊙S ˜ KC ==⇒ rT ′⊙S′ G = (U ⊙ T ) ⊙ S (U ′ ⊙ T ′) ⊙ S′ U ⊙ T U ′ ⊙ T ′ T T ′ M r(U⊙T )⊙S ˜ MD ===⇒ r(U′⊙T ′)⊙S′ L rT ⊙S ˜LB ==⇒ rT ′⊙S′ G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We thus get a weak morphism M = (M, M A, M D) between (U ⊙ T ) ⊙ S and (U ′ ⊙ T ′) ⊙ S′ in Span, where M A = (U ⊙ T ) ⊙ S (U ′ ⊙ T ′) ⊙ S′ T ⊙ S T ′ ⊙ S′ S S′ A A M ˜l (U⊙T )⊙S ˜ MA ===⇒ ˜l (U′⊙T ′)⊙S′ K lT ⊙S ˜ KA ==⇒ lT ′⊙S′ ∂S A F F A ==⇒ ∂S′ A and M D = (U ⊙ T ) ⊙ S (U ′ ⊙ T ′) ⊙ S′ U ⊙ T U ′ ⊙ T ′ U U ′ D D M r(U⊙T )⊙S ˜ MD ===⇒ r(U′⊙T ′)⊙S′ L rU⊙T ˜LD ==⇒ rU′⊙T ′ ∂U D H HD ===⇒ ∂U′ D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Using the biequivalence of Proposition 3, we can suppose that ˜ M A and ˜ M B where chosen so that M A and M D are respectively negative on A and positive on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By the PCC, we can then verify directly that M = (H ⊙ G) ⊙ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Now consider the positive weak morphism ¯ M = AS′,T ′,U′ ◦M ◦ A−1 S,T,U: we have that ¯ M A and ¯ M D are as on Figure 5 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By using the PCC to characterize the composition of G⊙F with H, we have that H⊙(G⊙F) = ¯ M, so that AS′,T ′,U′ ◦((H ⊙ G) ⊙ F) = (H ⊙ (G ⊙ F)) ◦ AS,T,U which was the wanted naturality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We can now prove Theorem 2: Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Lemmas 10 to 11, the 0-composition is naturally left unital, right unital and associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, the coherence conditions on the natural isomorphisms, re- quired by the definition of bicategories, directly follow from their pullback definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Renamings Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given F : A → B ∈ Ren, ˇF ∈ Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We first prove that ˇF ∈ UB⊸A and we use the dual version Proposition 1 for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We already have that idA ∈ UA since it is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We are left to show that ˇF ⋆@S ∈ UB⊥ for every S ∈ UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Up to isomorphism of domain, ˇF ⋆@S is the composition F ◦ ∂S and, by hypothesis on F, the latter is in UB⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So ˇF ∈ UB⊸A by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We are left to show that ˇF ∈ TB⊸A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But it follows from Proposition 2 by the same arguments as for uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given F : A → B and G: B → C in Ren, there is mF,G : ˇ (GF) ⇒ ˇF ⊙ ˇG a strong morphism of Thin, defined by the universal property of the pullback as mF,G = ⟨F, idA⟩: A → ˇF ⊙ ˇG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let A, B, C be thin groupoids, and φ: F ⇒ F ′ : A → B and ψ: G ⇒ G′ : B → C be two 2-cells of Ren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The composition ˇφ ⊙ ˇψ: ˇF ⊙ ˇG ⇒ ˇF ′ ⊙ ˇG′ is given by (H, χC, χA) where χC and χA are respectively ˇF ⊙ ˇG ˇF ′ ⊙ ˇG′ B B C C H l ¯φ=⇒ l G ψ=⇒ G′ and ˇF ⊙ ˇG ˇF ′ ⊙ ˇG′ A A A A H r = r idA = idA with ¯φ = ˇF ⊙ ˇG A A ˇF ′ ⊙ ˇG′ B B B B r l = F φ=⇒ r−1 F ′ = l and H as on the top of ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is a consequence of the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids A, B, C, the 2-cells mF,G for F : A → B and G: B → C in Thin define a natural iso m of type ˇ ((−)(2) ◦ (−)(1)) ⇒ ˇ (−)(1) ⊙ ˇ (−)(2) : Ren(A, B) × Ren(B, C) → Thin(C, A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let φ: F ⇒ F ′ : A → B and ψ: G ⇒ G′ : B → C be two 2-cells of Ren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We must show that ˇ (GF) ˇ (G′F ′) ˇF ⊙ ˇG ˇF ′ ⊙ ˇG′ ˇ (ψφ) mF,G = mF ′,G′ ˇφ⊙ ˇ ψ ¯ M A = U ⊙ (T ⊙ S) (U ⊙ T ) ⊙ S (U ′ ⊙ T ′) ⊙ S′ U ′ ⊙ (T ′ ⊙ S′) T ⊙ S T ⊙ S T ′ ⊙ S′ T ′ ⊙ S′ S S S′ S′ A A A A lU⊙(T ⊙S) A−1 S,T,U = M ˜l (U⊙T )⊙S ˜ MA ===⇒ ˜l (U′⊙T ′)⊙S′ AS′,T ′,U′ = lU′⊙(T ′⊙S′) lT ⊙S = K lT ⊙S ˜ KA ==⇒ lT ′⊙S′ = lT ′⊙S′ ∂S A = ∂S A F F A ==⇒ ∂S′ A = ∂S′ A Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The natural transformation ¯ MA ¯ M D = U ⊙ (T ⊙ S) (U ⊙ T ) ⊙ S (U ′ ⊙ T ′) ⊙ S′ U ′ ⊙ (T ′ ⊙ S′) U ⊙ T U ⊙ T U ′ ⊙ T ′ U ′ ⊙ T ′ U U U ′ U ′ D D D D ˜rU⊙(T ⊙S) A−1 S,T,U = M r(U⊙T )⊙S ˜ MD ===⇒ r(U′⊙T ′)⊙S′ AS′,T ′,U′ = ˜rU′⊙(T ′⊙S′) rU⊙T = L rU⊙T ˜LD ==⇒ rU′⊙T ′ = rU′⊙T ′ ∂U D = ∂U D H HD ===⇒ ∂U′ D = ∂U′ D Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The natural transformation ¯ MD in Thin(C, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But this equation can easily be deduced from Lemma 12, whose statement implies that ˇφ ⊙ ˇψ = mF ′,G′ ◦ ˇ (ψφ) ◦ m−1 F,G We can now finish the proof of Proposition 5: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For every A and B, ˇ (−) can easily be seen to define a functor ˇ (−)A,B : Ren(A, B) → Thin(B, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We are just left to show that the usual coherence conditions for pseudofunctors are satisfied by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But the required coherence conditions follow directly from the universal property of the pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' functor and its structure We now finish the definition of the pseudofunctor !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Thin → Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' First, while we described weak morphisms between two spans S, S′ : A ⇸ B as triples (F, F A, F B), often identifying F with the whole triple, we will in the following often refer to the first element of the triple F by F, for disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We now start by proving the naturality of the coherence m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let F = (F , F A, F B): S ⇒ S′ : A ⇸ B and G = (G, GB, GC): T ⇒ T ′: B ⇸ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let χS and χT be two 2-cells given by the definition of horizontal composition so that G ⊙ F is given by the two 2-cells T ⊙ S T ′ ⊙ S′ S S′ A A G⊙F l χS ==⇒ l F ∂S A F A ==⇒ ∂S′ A idA and T ⊙ S T ′ ⊙ S′ T T ′ C C G⊙F r χT ==⇒ r G ∂T C GC ==⇒ ∂T ′ C idC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The composition !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F is then given by the two 2-cells !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (T ⊙ S) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (T ′ ⊙ S′) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A m−1 S,T l = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (G⊙F ) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' l !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='χS ==⇒ mS′,T ′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' l = l id!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S A = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F A ===⇒ id!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S′ A = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂S′ A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='idA !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='idA !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='idA and !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (T ⊙ S) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (T ′ ⊙ S′) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='C m−1 S,T r = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (G⊙F ) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' r !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='χT ==⇒ mS′,T ′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' r = r id!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T C = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T C !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='GC ===⇒ id!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ′ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T ′ C = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='∂T ′ C !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='idC !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='idC !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='idC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We use the PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The respective positivity and nega- tivity of the proposed 2-cells follow from the positivity of the vertical composition of χS and F A and the negativity of the vertical composition of χT and GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We are left to show the top row of the two proposed 2-cells satisfy the equality required by the PCC, but it follows from that satisfied by χS and χT by the functoriality of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Gpd → Gpd on 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We can now conclude naturality: Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The morphisms mA,B,C S,T define a natural iso mA,B,C : !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((−) ⊙ (−)) ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−) ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−) : Thin(A, B) × Thin(B, C) → Thin(A, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let F : S ⇒ S′ : A ⇸ B and G: T ⇒ T ′: B ⇸ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We must show that mA,B,C S′,T ′ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (G ⊙ F) = (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F) ◦ mA,B,C S,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' But it directly follows from Lemma 13, whose conclusion states in particular that mA,B,C S′,T ′ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (G ⊙ F) ◦ (mA,B,C S,T )−1 = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='G ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We can thus conclude that !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' is a pseudofunctor: Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The functor !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Gpd → Gpd induces a pseudofunctor !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Thin → Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Lemma 14, we have an adequate natural isomor- phism expressing the functoriality of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' on Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The coher- ence laws for pseudofunctors can be directly verified by the universal properties of the pullbacks involved in the horizontal compositions appearing in these laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' pseudocomonad We are going to derive the !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' pseudocomonad on Thin from the !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' pseudomonad on Gpd through functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Before using this functoriality argument, we need to describe what are the (higher) categories we are going to apply it to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The domain (bi)category will be the one of endofunctors on Gpd with properties similar to the ones of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Gpd → Gpd, while the codomain (bi)category will be the one of endopseudofunctors on Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We shall first discuss how to relate some functors on Gpd to pseudofunctors on Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given a functor H: Gpd → Gpd, there is a canonical uniform groupoid HA = (HA, UHA) associated to any uniform groupoid A, where UHA = {HS | S ∈ UA}⊥⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' If H preserves pullbacks, and pullbacks which are bipullbacks, then given uniform groupoids A and B, and S ∈ UA⊸B, we have HS ∈ UHA⊸HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is a direct consequence of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We call bicartesian functors the functors H which satisfy the hypothesis of the above property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A ±-functor is a tuple (H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' H+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ι) with H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' H+ being functors Gpd → Gpd where H and H+ are bicartesian and preserve functors (between groupoids) that are bijective on objects (of the groupoids),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and such that H+ preserves discrete groupoids,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and ι: H+ ⇒ H being a natural transformation which is pointwise (that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' such that each ιX is) monomorphic and surjective on objects (of the groupoids),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' satisfying more- over that it is bicartesian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' meaning that its naturality squares are both pullbacks and bipullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Intuitively, the definition of ±-functor is an abstraction of the case of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Gpd → Gpd, from which we derive a functor Thin → Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In the case of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', given a thin groupoid A, a positive sub-groupoid (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A)+ is defined from a construction which is not derivable from the definition of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' and the data of A and A+, so that we have to take it into account in our definition of ±-functor, in the form of a functor H+ and a natural transformation ι: H+ ⇒ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We should a priori also require similar data for the negative side, but it so happens that, in the case of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A)− = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A−), so that it is in fact not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In order to show that !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' induces a pseudocomonad on Thin, the ±-functors we will consider will only be iterated compositions of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='. Given a ±-functor (H, H+, ι) and a thin groupoid A, there is a canonical thin groupoid HA whose underlying uniform groupoid is defined as earlier, whose class of thin prestrategies is THA = {HS | S ∈ TA}⊥⊥⊥⊥, and whose negative and positive sub-groupoids are (HA)− = HA− and (HA)+ = H+A+ with embeddings given by the compositions HA− H(id− A) −−−−−→ HA and H+A+ H+(id+ A) −−−−−→ H+A ιA −→ HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By the conditions of ±-functors, they are elements of THA and T⊥⊥ HA as required (exercise to the reader).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given a ±-functor (H, H+, ι) and thin groupoids A and B, and S ∈ TA⊸B, HS ∈ THA⊸HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is an easy consequence of the hypotheses on a ±-functor and Propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given a ±-functor (H, H+, ι) and a thin groupoid A, H preserves negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' positive) 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given a negative 2-cell φ: F ⇒ F ′ : X → A, writing X(0) for the discrete groupoid with the same object as X, we have a commutative diagram X(0) X A− A e φ− ==⇒ ¯ F ¯ F ′ φ=⇒ F F ′ id− A for some 2-cell φ− : ¯F ⇒ ¯F ′, where the top arrow e is the canonical embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By hypothesis on H, the image of e by H is bijective on objects of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, by functoriality, we have H(id− A) ◦ H(φ−) = H(φ) ◦ H(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, all the components of the natural transformation H(φ) are in the image of H(id− A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, it is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Now, given a positive 2-cell φ: F ⇒ F ′ : X → A, we have a similar commutative diagram X(0) X A+ A e φ+ ==⇒ ¯ F ¯ F ′ φ=⇒ F F ′ id+ A for some 2-cell φ+ : ¯F ⇒ ¯F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We then have the commutative diagram H+X(0) H+X HX H+A+ H+A HA H+(e) H+(φ+) =====⇒ H+( ¯ F ) H+( ¯ F ′) ιX H+(φ) ====⇒ H+(F ) H+(F ′) H(φ) ===⇒ H(F ) H(F ′) H+(id+ A) ιA where the top arrow is bijective on objects by assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, all the components of H(φ) are in the image of ιA ◦ H+(id+ A), so that H(φ) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given two ±-functors H = (H, H+, ι) and K = (K, K+, κ), a ±-transformation is a pair (α, α+) of bi- cartesian natural transformations where α: H ⇒ K and α+ : H+ ⇒ K+ are such that κ ◦ α+ = α ◦ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Now, a ±-modification between two such ±-transformations (α, α+) and (β, β+) is the data of a modification m: α ⇛ β in the 3-category of 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given a thin modification m: α ⇛ β : H ⇒ K and a thin groupoid A, the 2-cell mA : αA ⇒ βA : HA → KA is negative as a 2-cell on KA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since m is a modification, we have that mA◦H(id− A) = K(id− A) ◦ mA−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Since H(id− A) is bijective on objects by hypotheses on H and id− A, we have that all the components of mA are in the image of K(id− A), so that they are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ±-functors, ±-transformations and ±-modifications can be equipped with the evident operations in order to form a strict 3-category ±-Funct with one object (which, morally, is Gpd, the domain and codomain of each ±-functor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The important point to note here is that the data of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', η, µ, α, β and γ induces the expected way a pseudomonad in ±-Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We shall now describe an operation ˇ (−) relating ±-Funct and Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' First, a ±-functor (H, H+, ι) induces an endofunc- tor ˇH: Thin → Thin mapping a thin groupoid A to the thin groupoid HA defined as earlier, and mapping spans and their weak morphisms to their images by H, which is well-defined by Propositions 17 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Now, given a ±-transformation (α, α+): (H, H+, ι) ⇒ (K, K+, κ), we define a pseudonatural transformation ˇα: ˇK ⇒ ˇH whose component at a thin groupoid A is ˇ (αA), that is, the image of αA : HA → KA by the pseudofunctor ˇ (−) : Renop → Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We can then extend the pseudofunctor ˇ− : Renop → Thin to some sort of 3-dimensional functor ˇ (−) : ±-Functco → Bicat sending the unique 0-cell to Thin and the higher cells in the hom-bicategory Bicat(Thin, Thin) (here, ±-Functco denotes ±-Funct with 2-cells reversed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' While this is probably a trifunctor, it would be very tiresome to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Instead, we will only rely on the simpler proposition stating that Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Considering ±-Funct as a strict 2-category by forgetting the dimension 0, ˇ (−) induces a pseudofunctor ˇ (−) : ±-Functco → Bicat(Thin, Thin) between bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By checking the axioms of pseudofunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We can now briefly describe a proof of Theorem 3: Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The most satisfying proof of this state- ment would rely on the fact that ˇ (−) : ±-Functco → Bicat(Thin, Thin) is a trifunctor and that a trifunctor sends any pseudocomonad to a pseudocomonad, but we do not know a proof for the latter fact (though it is probably true) and deem a full proof of the former tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Instead, we can rely on the weaker Proposition 20 to prove the required coherences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Following [40], we are required to prove the equations of modifications of Figure 7 are verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The idea is to relate each of these equations to the equations satisfied by the pseudomonad !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' : Gpd → Gpd, and this is done through paving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For example, we use the following pavings for the two first modifications of the left hand-side of the first equation of Figure 7: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ) ⊙ ˇµ ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ (̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ˇµ) (̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ)) ⊙ ˇµ ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ̌ (µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ˇµ ̌ ((µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ◦ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ)) ̌ (µ ◦ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ))) = = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) ⇛ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ) ⊙ ˇµ ⇛ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) ⊙ ˇµ ⇛ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ) ⊙ ˇµ ⇛ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ) ⊙ ˇµ ⇛ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) ⇛ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ) ⇛ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⊙ ˇµ = (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ ⇛ (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ = (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⊙ ˇµ = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) ⇛ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ) ⇛ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⊙ ˇµ ex ≡⇛ (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ) ⊙ ˇµ ⇛ ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) ⇛ ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ) ⇛ (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⊙ ˇµ and !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇη!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) ⇛ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇη!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ (ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ) ⇛ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇη!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⊙ ˇµ = (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇη ⊙ ˇµ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ ⇛ cc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ = cc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ˇµ ⇛ ˇµ = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇη!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) ⇛ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇη!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ) ⊙ ˇµ ⇛ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (ˇη!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ ˇµ) ⊙ ˇµ ⇛ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' cc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ˇµ = cc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙ˇµ ⇛ ˇµ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The two required equations for !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' to be a pseudocomonad and (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ) ⊙ ˇµ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ ⊙ ˇµ) ⊙ ˇµ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ˇµ) ⊙ ˇµ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ˇµ) ⊙ ˇµ (̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ)) ⊙ ˇµ ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ˇµ ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ)) ⊙ ˇµ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (̌ µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ˇµ ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ)) ⊙ ˇµ ̌ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⊙ ˇµ ̌ (µ ◦ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ))) ̌ (µ ◦ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ)) = = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The other elementary modifications of Figure 7 are paved similarly, so that the first equation of Figure 7 is reduced to the equation ̌ (µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⇛ ̌ (µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⇛ ̌ (µ ◦ µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⇛ ̌ (µ ◦ µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ◦ µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') = ̌ (µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ⇛ ̌ (µ ◦ µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) = ̌ (µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⇛ ̌ (µ ◦ µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ◦ µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') which is the image by ˇ (−) of an equation satisfied by the pseudomonad (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', η, µ) on Gpd, and the second equation of Figure 7 to the equation ̌ ((µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ⇛ ̌ ((µ ◦ µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') = ̌ (µ ◦ (µ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=')) ⇛ ̌ (µ ◦ id!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') = ˇµ = ̌ ((µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ) ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') = ̌ (µ ◦ (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='µ ◦ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=')) ⇛ ̌ (µ ◦ id!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=') = ˇµ also an image of an equation satisfied by the pseudomonad (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', η, µ) on Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So that (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=', ˇη, ˇµ) is indeed a pseudocomonad on Thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The cartesian product Given thin groupoids A, B, we write¯l A,B and ¯r A,B, simply denoted ¯l and ¯r as earlier when A, B can be deduced from the context, for the coprojections ¯l : A ֒→ A + B and ¯r : B ֒→ A + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By applying the functor ˇ (−), we get thin spans ˇ¯l: A&B ⇸ A and ˇ¯r: A & B ⇸ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids Γ, A, B, we define a functor ⟨−, −⟩Γ,A,B : Thin(Γ, A)×Thin(Γ, B) → Thin(Γ, A&B), often abbreviated ⟨−, −⟩, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given S ∈ Thin(Γ, A) and T ∈ Thin(Γ, B), we define ⟨S, T ⟩ as the span ⟨S, T ⟩ = S + T Γ A + B [∂S Γ ,∂T Γ ] ∂S A+∂T B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given S ∈ Thin(Γ, A) and T ∈ Thin(Γ, B), we have ⟨S, T ⟩ ∈ Thin(Γ, A & B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By an adequate use of Propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given morphisms F : S → S′ ∈ Thin(Γ, A) and G: T → T ′ ∈ Thin(Γ, B), ⟨F, G⟩ is defined as the morphism H with H = F + G and HΓ = S + T S′ + T ′ Γ Γ F +G [∂S Γ ,∂T Γ ] [F Γ,GΓ] =====⇒ [∂S′ Γ ,∂T ′ Γ ] and HA+B = S + T S′ + T ′ A + B A + B F +G ∂S A+∂T B F A+GB ======⇒ ∂S′ A +∂T ′ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' One immediately verifies that these two 2-cells have the ade- quate polarities, so that ⟨F, G⟩ ∈ Thin(Γ, A& B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, the functoriality of ⟨−, −⟩Γ,A,B is immediately verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given S ∈ Thin(Γ, A & B), we write SA for the span SA = SA S Γ A ∂ SA A ∂S Γ where SA is the submonoid of S whose image by ∂S A+B is in A, and where ∂SA A is the induced map SA → A from ∂S A+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We define a span SB similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given S ∈ Thin(Γ, A & B), we have SA ∈ Thin(Γ, A) and SB ∈ Thin(Γ, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As the result of the composition of two thin spans, we know that ˇ¯l ⊙ S is in Thin(Γ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It is the span ˇ¯l ⊙ S S A Γ A + B A l r ∂S Γ ∂S A+B ¯l idA which, by an isomorphism of pullbacks, is isomorphic to SA S A Γ A + B A ∂ SA A ∂S Γ ∂S A+B ¯l idA which is exactly SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, the latter is in Thin(Γ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A similar argument holds for SB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The mapping S �→ SA extends to a functor (−)A : Thin(Γ, A & B) → Thin(Γ, A) the expected way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Similarly, we obtain a functor (−)B : Thin(Γ, A & B) → Thin(Γ, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The functors (−)A and (−)B are isomorphic to the functors ˇ¯l ⊙ (−) and ˇ¯r ⊙ (−) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Using the PCC, one can show the naturality of the family of isomorphisms of thin spans ˇ¯l ⊙ S ∼= SA described in the proof of Proposition 22, obtaining an isomorphism of functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A similar argument holds for (−)B and ˇ¯l ⊙(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids Γ, A, B, there is an adjoint equivalence Thin(Γ, A & B) ⊥ Thin(Γ, A) × Thin(Γ, B) ((−)A,(−)B) ⟨−,−⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given S ∈ Thin(Γ, A & B), write ιS A : SA ֒→ S and ιS B : SB ֒→ S for the canonical inclusions of Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' There is a canonical γS : S → ⟨SA, SB⟩ defined as the inverse of SA + SB [ιS A,ιS B] −−−−→ S and it induces a strong morphism of thin span γS : S ⇒ ⟨SA, SB⟩ ∈ Thin(Γ, A&B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The naturality of γ can be shown using the PCC, so that we obtain a natural isomorphism γ : idThin(Γ,A&B) ⇒ ⟨(−)A, (−)B⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given (R, T ) ∈ Thin(Γ, A) × Thin(Γ, B), we have a canonical isomorphism δA R,T : ⟨R, T ⟩A → R defined as the pullback isomorphism between ⟨R, T ⟩A ⟨R, T ⟩ A A + B ι⟨R,T ⟩ A ⌜ ∂⟨R,T ⟩ A+B ¯l and R ⟨R, T ⟩ A A + B ¯l ∂R A ⌜ ∂⟨R,T ⟩ A+B ¯l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It extends to a morphism δA R,T : ⟨R, T ⟩A → R ∈ Thin(Γ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We define similarly a morphism δB R,T : ⟨R, T ⟩B → T ∈ Thin(Γ, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Writing δR,T = ⟨δA R,T , δB R,T ⟩, the naturality of δR,T with respect to R and T can be checked using the PCC, so that we obtain a natural isomorphism δ: (⟨(−)(1), (−)(2)⟩A, ⟨(−)(1), (−)(2)⟩B) ⇒ idThin(Γ,A)×Thin(Γ,B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We thus have an equivalence, and we verify that it is adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We check the first zigzag equation, namely (δ((−)A, (−)B)) ◦ (((−)A, (−)B)γ) = id((−)A,(−)B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (4) In order to verify the above equality, by symmetry, we just need to check its projection on Thin(Γ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So let S ∈ Thin(Γ, A & B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The component of the left-hand side of (4) at S is then SA (γS)A −−−−→ (⟨SA, SB⟩)A δA ⟨SA,SB⟩ −−−−−−→ SA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By unfolding the definition of γ and δ, we compute that SA (δA ⟨SA,SB⟩)−1 −−−−−−−−−→ (SA + SB)A ((γS)A)−1 −−−−−−→ SA ιS A −→ S is precisely ιS A, which happens to be a monomorphism, so that ((γS)A)−1 ◦ (δA ⟨SA,SB⟩)−1 = idSA, which is, up to inverses, what we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, the first zigzag equation holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We now verify the second zigzag equation, namely (⟨−, −⟩δ) ◦ (γ⟨−, −⟩) = id⟨−,−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (5) So let (R, T ) ∈ Thin(Γ, A) × Thin(Γ, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The component of the left-hand side of (5) at (R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ) is ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩ γ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩ −−−−→ ⟨⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩B⟩ ⟨δA R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='δB R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩ −−−−−−−→ ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩ By unfolding the definition of γ and δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' we compute that R ¯l−→ ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩ ⟨δA R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='δB R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩−1 −−−−−−−−−→ ⟨⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩B⟩ γ−1 ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩ −−−−→ ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩ reduces to R ¯l−→ ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩ and similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ¯r−→ ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩ ⟨δA R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='δB R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩−1 −−−−−−−−−→ ⟨⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩B⟩ γ−1 ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩ −−−−→ ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩ reduces to T ¯r−→ ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' T ⟩ so that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' since ¯l and ¯r are jointly surjective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' γ−1 ⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩ ◦ ⟨δA R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' δB R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩−1 = id⟨R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='T ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' which is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' up to inverses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' what we wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So the second zigzag holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids Γ, A, B, there is an adjoint equivalence Thin(Γ, A & B) ⊥ Thin(Γ, A) × Thin(Γ, B) (ˇ¯l⊙(−),ˇ¯r⊙(−)) ⟨−,−⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This is a consequence of Propositions 23 and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Note that, given (R, T ) ∈ Thin(Γ, A) × Thin(Γ, B), the component at (R, T ) of the counit associated to the adjoint equivalence of Proposition 25 is the composite (ˇ¯l ⊙ ⟨R, T ⟩, ˇ¯r ⊙ ⟨R, T ⟩) ∼ = −→ (⟨R, T ⟩A, ⟨R, T ⟩B) δ−→ (R, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Now, we can conclude the proof of Proposition 6: Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We have the equalities Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A& B) = Thin(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ, A & B) and Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A) × Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, B) = Thin(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ, A) × Thin(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, it is quite immediate that, up to these identifications, there is an isomorphism of functors (L!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−), R!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−)) ∼= (ˇ¯l ⊙(−), ˇ¯r ⊙ (−)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, the unit/counit pair of the adjoint equivalence of Proposition 25 can be adjusted to get a unit/counit pair witnessing that we have an adjoint equivalence as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The evaluation adjunction We give here some additional details for the proof of Proposition 7, stating the existence of an adjoint equivalence between the currying operation and the evaluation one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' This adjoint equivalence will be derived from the Seely equivalence already introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 1) Properties of the Seely equivalence: Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The family of functors sA,B for groupoids A, B form a 2-natural transformation s: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−) × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−) ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((−) + (−)): Gpd × Gpd → Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The naturality w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t 2-cells is checked by direct point- wise computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The natural transformation s is bicartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By a direct use of the point-wise characterization of pullbacks and Lemma 1 on the naturality squares of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Similarly, we have the same kind of properties for ¯s: Proposition 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The family of functors ¯sA,B for groupoids A, B form a 2-natural transformation ¯s: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((−) + (−)) ⇒ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−) × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−): Gpd × Gpd → Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The natural transformation ¯s is bicartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 2) The Seely coherence 2-cell: While the Seely isomor- phisms of 1-categorical models of linear logic are required to satisfy an equality, in our 2-categorical setting we only have the following 2-cell !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A + !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A + B) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A + B) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (A + B) s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B SeeA,B =====⇒ µA×µB !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='(¯l),!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (¯r)] sA,B µA+B We first compute the action of the two vertical morphisms on objects of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So let a = ((ai,j)j∈JA i )i∈IA ∈ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A and b = ((bi,j)j∈JB i )i∈IB ∈ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The mappings associated with the left vertical morphism are (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' b) �→(¯l((ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j)j∈JA i ))̟l(i)∈̟l(IA) ∪ (¯r((bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j)j∈JB i ))̟r(i)∈̟r(IB) �→((¯l(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j))j∈JA i )̟l(i)∈̟l(IA) ∪ ((¯r(bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j))j∈JB i )̟r(i)∈̟r(IB) �→(¯l(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j))⟨̟l(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j⟩∈� i∈̟l(IA) JA i ∪ (¯r(bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j))⟨̟r(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j⟩∈� i∈̟r(IB ) JB i and the mappings associated with the right are (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' b) �→ ((ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j)⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j⟩∈� i∈IA JA i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j)⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j⟩∈� i∈IB JB i ) �→ (¯l(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j))̟l(⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j⟩)∈̟l(� i∈IA JA i ) ∪ (¯l(bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j))̟r(⟨i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='j⟩)∈̟r(� i∈IB JB i ) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' given D ∈ Gpd and (di)i∈I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (d′ j)j∈J ∈ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='D with I ∩ J = ∅, we write (di)i∈I ∪ (d′ j)j∈J for the evident family indexed by I ∪ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We thus define (SeeA,B)a,b as the map (π, (id)k∈K) where π is the bijection K → K′ with K = � i∈̟l(IA)JA i ∪ � i∈̟r(IB)JB i and K′ = ̟l(� i∈IAJA i ) ∪ ̟r(� i∈IBJB i ) such that π maps [̟l(i), j] ∈ K to ̟l([i, j]) ∈ K′, and [̟r(i), j] ∈ K to ̟r([i, j]) ∈ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The naturality of SeeA,B with respect to morphisms (a, b) → (a′, b′) of !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='B can be readily checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So SeeA,B is indeed a 2-cell of Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We moreover verify that Proposition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The family of 2-cells SeeA,B for A, B ∈ Gpd is natural with respect to functors F : A → A′ and G: B → B′ of Gpd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In other words, See = (SeeA,B)A,B∈Gpd is a modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' A direct point-wise computation of naturality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 3) Properties of the evaluation span: Given thin groupoids A, B, recall that we introduced a span evA,B : (A ⇒ B) & A ⇸ B defined by evA,B = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) & A) B (l,r,l) r η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B,A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We have that Proposition 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We have evA,B ∈ T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((A⇒B)&A)⊸B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By an adequate use of Propositions 1 and 2, using Proposition 27, the bicartesianness of η and Lemma 5 to get the required bipullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 4) The currying operation: Recall that, given thin groupoids Γ, A, B and S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B), we defined Λ(S) as the span Λ(S) = S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B l ◦¯sΓ,A◦∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ+A) (r ◦¯sΓ,A◦∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ+A),∂S B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proposition 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids Γ, A, B and S ∈ Thin(Γ & A, B), we have Λ(S) ∈ T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ⊸(A⇒B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' While more involved than the other instances, this still relies on Propositions 1 and 2, using Proposition 29 and Lemma 5 to get the required intermediate bipullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The operation Λ(−) can be extended to weak morphisms the expected way, and it is compatible with the polarities, and we moreover have Proposition 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids Γ, A, B, the operation Λ(−): Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) → Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B) is functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 5) The uncurrying operation: We can define conversely an uncurrying operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B), we define ¯Λ(S) as the span ¯Λ(S) = S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A) B sΓ,A◦(∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) ∂S B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' As before, using similar methods, we can verify that Proposition 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids Γ, A, B and σ ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B), we have ¯Λ(σ) ∈ T!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ&A)⊸B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Moreover, we can extend this uncurrying operation to the weak morphisms the expected way, and this operation is compatible with the polarities, and we moreover have Proposition 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Given thin groupoids Γ, A, B, the operation ¯Λ(−): Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B) → Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) is functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 6) The adjoint equivalences: We have a first adjoint equiv- alence between the currying and uncurrying operations: Proposition 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' There is an adjoint equivalence Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B) ⊥ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) ¯Λ(−) Λ(−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The application of ¯Λ(−) and Λ(−) to spans essentially amounts to adequately postcompose the display maps of these spans by sΓ,A and ¯sΓ,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The unit/counit pair witnessing the ad- joint equivalence are then easily derived from a unit/counit pair (ΣΓ,A, ¯ΣΓ,A) witnessing the adjoint equivalence sΓ,A ⊣ ¯sΓ,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For example, given S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B), the component of the unit of ¯Λ(−) ⊣ Λ(−) at S is the weak morphism (idS, φ) with φ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ = S S S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ ∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ = (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) = (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) ΣΓ,A ===⇒ sΓ,A ¯sΓ,A l = l and φ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B = S S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B ∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B (φ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A,φB) ======⇒ ∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B where φ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A = S S S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A = (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) = (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) ΣΓ,A ===⇒ sΓ,A ¯sΓ,A r = r and φB = S S B B ∂S B = ∂S B where we write ∂S A and ∂S B for l ◦∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B and r ◦∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B re- spectively (and similarly for ∂S A and ∂S B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The counit is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The fact that this unit/counit pair satisfies the zigzag equations is a consequence of the fact that (ΣΓ,A, ¯ΣΓ,A) satisfies the same equations, by vertical pasting of 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' In order to get another adjoint equivalence, we prove that the uncurrying functor is isomorphic to the uncurrying-through- evaluation operation: Proposition 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The functor ev ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (− & A): Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B) → Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) is isomorphic to the uncurrying functor ¯Λ(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Let S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We compute ev⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (S&A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' It is the composition of the spans !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (S + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ + !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ+ηA) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B+A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='(¯l),!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (¯r)] µΓ+A and !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) + A) B (l,r,l) r η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B,A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We compute the inner pullback of this composition as the rectangle of pullbacks shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, up to a canonical isomorphism of pullbacks, ev ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (S & A) is ¯S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) with S as the support of ¯S and ∂ ¯S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ+A) = S ⟨S,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A⟩ −−−−→ S × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ηS×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A −−−−→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A sS,A −−−→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (S + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ+ηA) −−−−−−→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ + !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='(¯l),!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (¯r)] −−−−−−→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) µΓ+A −−−→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) and ∂ ¯S B = S ∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B −−−−→ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B r−→ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The operation S �→ ¯S can be shown to extend nat- urally to weak morphisms, so that we obtain a functor ¯ (−): Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B) → Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) which is natu- rally isomorphic to ev⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (−&A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' So we are left to show that ¯ (−) ∼= ¯Λ(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' For this purpose, for S ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ, A ⇒ B), we define an isomorphism θS = (θS, θ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ+A) S , θB S ): ¯S ⇒ ¯Λ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We take θS = idS and θB S = id∂S B, and define θ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ+A) S as the (vertically expressed) 2-cell of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We directly observe that θ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ+A) S and θB S have the adequate polarities, so that θS ∈ Thin!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ & A, B) and it is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The naturality of θS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' S can be checked diagrammatically, by pasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Thus, θ defines an isomorphism ¯ (−) ∼= ¯Λ(−), so that we have ev ⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((−) & A) ∼ = =⇒ ¯ (−) θ=⇒ ¯Λ(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' We can now conclude the proof of Proposition 7: Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Proposition 36, we have an ad- joint equivalence between the currying and uncurrying op- erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' By Proposition 37, we can replace the uncurrying operation by ev⊙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((−)&A): by adjusting the unit and counit the expected way, we keep the adjoint equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (S + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='S × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A S × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' ((!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B) × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A × B !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B+A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B)×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A sS,A ⌝ ∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ηS×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ⌝ ∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B (S,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) ⌝ s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B,A η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A×B×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A (l,r,l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The inner rectangle of the composition of the two spans S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ + !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ × !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ + A) (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) = η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ηA s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A ⇓ SeeΓ,A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' [!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='(¯l),!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (¯r)] µΓ+A (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) = η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ×!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='ηA ⇓ αΓ × βA µΓ×µA sΓ,A = (∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='Γ,∂S !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content='A) sΓ,A Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' The θ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} +page_content=' (Γ+A) S 2-cell Recall the definition of α and β from Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FKT4oBgHgl3EQfnC5b/content/2301.11860v1.pdf'} diff --git a/VNAyT4oBgHgl3EQfV_dJ/content/tmp_files/2301.00153v1.pdf.txt b/VNAyT4oBgHgl3EQfV_dJ/content/tmp_files/2301.00153v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c698cad952e454254d2f72f79d232e664b8319e1 --- /dev/null +++ b/VNAyT4oBgHgl3EQfV_dJ/content/tmp_files/2301.00153v1.pdf.txt @@ -0,0 +1,792 @@ +KNOWLEDGE-BASED DATASET FOR TRAINING +PE MALWARE DETECTION MODELS +TECHNICAL REPORT +Peter Švec +Institute of Computer Science and Mathematics +Faculty of Electrical Engineering and Information Technology +Slovak University of Technology +Ilkoviˇcova 3, Bratislava, Slovakia +peter.svec1@stuba.sk +Štefan Balogh +Institute of Computer Science and Mathematics +Faculty of Electrical Engineering and Information Technology +Slovak University of Technology +Ilkoviˇcova 3, Bratislava, Slovakia +stefan.balogh@stuba.sk +Martin Homola +Department of Applied Informatics +Faculty of Mathematics, Physics and Informatics +Comenius University +Mlynská dolina, Bratislava, Slovakia +homola@fmph.uniba.sk +Ján Kl’uka +Department of Applied Informatics +Faculty of Mathematics, Physics and Informatics +Comenius University +Mlynská dolina, Bratislava, Slovakia +kluka@fmph.uniba.sk +January 3, 2023 +ABSTRACT +Ontologies are a standard for semantic schemata in many knowledge-intensive domains of human +interest. They are now becoming increasingly important also in areas until very recently dominated +by subsymbolic representations and machine-learning-based data processing. One such area is +information security, and more specifically malware detection. We propose PE Malware Ontology +that offers a reusable semantic schema for Portable Executable (PE, Windows binary format) malware +files. The ontology was inspired by the structure of the data in the EMBER dataset and it currently +covers the data intended for static malware analysis. With this proposal, we hope to achieve: a) a +unified semantic representation for PE malware datasets that are available or will be published in +the future; (b) applicability of symbolic, neural-symbolic, or otherwise explainable approaches in +the PE Malware domain that may lead to improved interpretability of results which may now be +characterized by the terms defined in the ontology; and (c) by joint publishing of semantically treated +EMBER data, including fractional datasets, also improved reproducibility of experiments. +Keywords Ontology · Dataset · Malware · Windows · Intepretability · Explainable AI +1 +Introduction +There are currently a number of datasets that may be used to train malware detection models. Among the most popular +recent datasets we find Elastic Malware Benchmark for Empowering Researchers (EMBER) [Anderson and Roth, 2018] +(approx. 1.1 million samples) and more recently Sophos/ReversingLabs 20 million sample dataset (SoReL) [Harang +arXiv:2301.00153v1 [cs.CR] 31 Dec 2022 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +and Rudd, 2020] (approx. 20 million samples). The advantage of these datasets is that they were already analyzed +by security experts and for each sample they provide extensive feature data extracted and pre-processed in structured +textual format. In addition, SoReL provides disarmed binaries to some of the samples. Several other online data sources +are available, even some that contain genuine malicious binaries in their samples. Such binaries may be used if some +additional information about the sample needs to be extracted. +Specifically both EMBER and SoReL cover the domain of PE Malware (i.e., Windows executables and libraries). Both +are well tailored for processing by various machine learning tools; for example EMBER directly provides a script to +vectorize the data which is required by many sub-symbolic classifiers. +However, in reality the data contained in these datasets is structured (JSON files) and all textual characteristics provided +for each sample can be interpreted as meaningful properties. Such data is essentially symbolic and can be viewed as +(or, exported into) a knowledge base or a knowledge graph. Such treatment of the data would enable to capitalize on +knowledge-processing tools, or even some tools rooted in the more recent neural-symbolic AI movement, especially to +improve the interpretability of the resulting classifier models. +To give an example, approaches such as concept-learning may be used to obtain a symbolic characterization of a +malware sample in the form of a concept expression based on an ontology [Švec et al., 2021]. Knowledge-base +embedding [Bordes et al., 2013] may be used to improve effectiveness of learned models by injecting prior symbolic +knowledge. More recently, neural network activation patterns recognition and alignment with ontology concepts de +Sousa Ribeiro and Leite [2021] may be used to explain and diagnose trained neural network classifiers. +Also, EMBER and SoReL are vast, they contain millions of samples. Some of the studies that have been published +[Vinayakumar et al., 2019, Liu et al., 2020, Ghouti and Imam, 2020] had to resort to reducing the dataset of interest to a +smaller size, especially if the trained classifier is computationally demanding. However no unified methodology for +reducing the datasets has been established. +To alleviate the outlined issues, our work targets the following goals: +Unified semantic representation: To provide a unified vocabulary for relevant PE malware characteristics that can be +then used to represent any relevant data sample regardless of its source. +Interpretability of results: To ensure that the vocabulary provides a suitable nomenclature that is meaningful and +recognized by human users. Consequently prototype samples, concept descriptions, rules, or any other +characterizations expressed in the vocabulary would be understandable and explainable. +Reproducibility of experiments: Ensure that experiments may be executed on datasets of different suitable sizes +without the need to reduce the full original dataset to smaller ones again and again in each study. +To address the first two goals, we have developed a unified ontology that can provide a reusable semantic schema for PE +malware files. To the best of our knowledge, such ontology was missing so far. We have also aligned the nomenclature +used to describe actions possibly performed by PE files with the Malware Attribute Enumeration and Characterization +(MAEC) standard [MITRE Corp., 2020] which is widely accepted for malware descriptions. +Our goal was to improve interpretability of the captured data. The ontology is partly based on the EMBER dataset +structure derived from the EMBER data features, but it is not a direct ontological mapping of this or any other individual +dataset. The emphasis is placed on data features that are meaningful in malware characterization. For this reason +we included only those features that make sense from an expert’s point of view, while other less meaningful features +were left out. Some of the left-out features (e.g., file size) may even cause the trained classifiers to be prone to trial +counterattacks. Also some of the features captured in the ontology can be considered as derived in that they combine +multiple low-level EMBER data entries into a meaningful feature. Two features are based on thresholds on top of +numeric features (e.g., “high entropy”). This is useful for systems that have difficulties with handling numeric values. +But our goal was also to create a dataset that could be effectively used as a standard benchmark to help advance learning +algorithms in the malware detection domain and that would also offer unambiguous reproducibility of experimental +results. To this end, we release EMBER data (2018 version) which has been translated into RDF semantic format, +including smaller fractional datasets which have been fixed for sake of better comparison of diverse experiments that +require smaller data volume than the full EMBER. +Our ontology, including the fractional datasets, and mapping scripts is available in our GitHub repository1. +1https://github.com/orbis-security/pe-malware-ontology +2 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +2 +Data sources +In this section we describe in more detail the EMBER dataset, which we used as the foundation for our ontology, along +with the modifications we proposed. The main difference between the EMBER and SoReL datasets is that SoReL +contains more samples together with disarmed malware binaries. However, the static properties of samples that these +datasets offer are almost identical (a few are missing in SoReL). Among other things, we decided on the EMBER +dataset because it is generally more used in research (261 vs. 32 citations). +2.1 +EMBER +The dataset contains a total of 1.1 million samples and includes 400,000 malicious samples, 400,000 benign samples +and 300,000 unlabeled samples, so the dataset can also be applied to unsupervised learning algorithms. The labeled +samples from the dataset are also divided into training and testing sets (600,000 for training and 200,000 for testing). +The dataset itself is composed of a collection of JSON objects, where each object represents data statically extracted +from the PE file of one sample. A simplified example of one sample can be seen in Listing 1. The static properties +themselves are organized as follows: +General file information: This set of features is dedicated to general information about the file, such as file size, +information whether the file contains a digital signature, presence of debugging symbols, presence of a TLS +section, or the number of imported and exported functions. +Header information: These are properties found in file headers such as the target architecture for which the file was +compiled, linker version, various timestamps, etc. +Section information: This set is dedicated to individual sections in the binary file. For each section dataset contains its +name, content type (code, initialized or uninitialized data), various properties (such as read, write, and execute +rights), or the value of the entropy of the section’s content. +Imported functions: List of imported functions organized by the DLL (here it is necessary to note that if the file +imports a certain function, it does not necessarily have to be called in the code). +Exported functions: List of exported functions. These are mostly included only in cases where the PE file is a library. +In addition to the categories mentioned above, the EMBER dataset also contains other numerical properties such as +byte histogram, byte-entropy histogram, or simple statistics for the strings found in the file. +Listing 1: Static features for single binary sample. +{ +"sha256": " eb87d82ad7bdc1b753bf91858d2986063ebd8aabeb8e7e91c0c78db21982a0d6 ", +"md5": " aba129a3d1ba9d307dad05617f66d8e7 ", +"appeared": "2018 -01", +"label": 1, +"avclass": "fareit", +"histogram": [ 96506 , 8328 , 5582 , ... ], +" byteentropy ": [0, 4229 , 269, 247, ... ], +"strings": { +" numstrings": 7762 , +"avlength": 181.60641587219789 , +" printabledist ": [591 , 51, 96, 46, ... ], +" printables": 1409629 , +"entropy": 5.037064474164528 , +"paths": 0, +"urls": 9, +"registry": 0, +"MZ": 11 +}, +"general": { +"size": 2261028 , +"vsize": 1912832 , +"has_debug": 0, +"exports": 0, +"imports": 17, +" has_relocations ": 1, +" has_resources ": 1, +" has_signature ": 0, +"has_tls": 1, +"symbols": 0 +}, +"header": { +"coff":{ +"timestamp": 708992537 , +3 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +"machine": "1386", +" characteristics ": [" CHARA_32BIT_MACHINE ", " BYTES_REVERSED_LO ", " EXECUTABLE_IMAGE ", +... ] +}, +"optional": { +"subsystem": " WINDOWS_GUI", +" dll_characteristics ": [], +"magic": "PE32", +" major_image_version ": 0, +" minor_image_version ": 0, +" major_linker_version ": 2, +" minor_linker_version ": 25, +... +} +}, +"section": { +"entry": "CODE", +"sections": [ +{ +"name": "CODE", +"size": 443392 , +"entropy": 6.532932639432919 , +"vsize": 442984 , +"props": ["CNT_CODE", "MEM_EXECUTE ", "MEM_READ"] +}, +... +] +}, +"imports": { +"kernel32.dll": [" DeleteCriticalSection ", " TlsSetValue ", "Sleep", ... ], +}, +"exports": [], +" datadirectories ": [ { "name": " EXPORT_TABLE ", " virtual_address ": 0 }, ... ] +} +2.2 +Other sources +Following the success of EMBER, Sophos/ReversingLabs 20 million sample dataset (SoReL) was published by Harang +and Rudd [2020] who especially aimed at scaling up the volume of samples available. The structure of SoReL samples +is similar to EMBER and it covers the same static features with very minor differences. These once more represented in +JSON. In addition to approximately 15M JSON structured samples that are labelled and over 4M additional ones that +are unlabelled, SoReL also contains approximately 10M real binary malware samples that have been disarmed in by +modifying their header file). The ontology presented in this work may be directly applied on SoReL. +Among the most popular sources (that provide real binaries) are VirusTotal [vir, b], which, however, provides paid +services (along with benign files), VirusShare [vir, a], which currently contains approx. 50 million malicious samples or +MalShare [mal]. Known datasets that contain dynamic properties include MALREC [Severi et al., 2018], which captures +the overall activity of malware in the form of logging all sources of indeterminism in the system, such as system calls, +peripheral devices, etc. In total, it contains approximately 66,000 samples. A smaller dynamic dataset that focuses +primarily on logging API calls is Mal-API-2019 [Catak and Yazı, 2019] and contains approximately 7,000 samples. In +2022, a dataset from AVAST was released [Bosansky et al., 2022], which, unlike previous works, combines static and +dynamic features and contains approx. 50,000 malware samples (its main purpose is to classify malware into individual +families). Other well-known datasets include ClaMP [cla], which contains only Portable Executable (PE) file headers +(approx. 5,000) or MalImg [Nataraj et al., 2011], which contains malicious samples, encoded in gray scale in form of +approx. 10,000 samples from different families. The DREBIN dataset is also worth mentioning, but unlike the previous +works, it contains binary samples from the Android platform [Arp et al., 2014]. +3 +Data preprocessing +3.1 +Dataset standardization +Individual samples may import a large number of functions from various standard DLLs implementing system APIs. +They provide useful clues about the possible actions a sample can perform when running, although not all imported +functions may be called and not all of these actions are relevant from the malware-detection perspective. In order to +extract only relevant information from samples’ imports to the ontological representation in a way that is aligned with +standard best practice, we have turned to Malware Attribute Enumeration and Characterization (MAEC) [MITRE Corp., +2020]. MAEC is a community-developed structured language for describing information about malware, combining +static and dynamic features (different kinds of behavior, interactions between processes, and so on). We map the space +4 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +of imported functions to malware actions, defined in one of MAEC’s vocabularies [MITRE Corp., 2017]. This aids not +only standardization of the dataset, but also achieves an effect similar to dimensionality reduction in traditional machine +learning – irrelevant imported functions are disregarded and multiple functions with a similar effect can be mapped to a +single action. The complete mapping of API functions to actions is available in our GitHub repository. +The MAEC vocabulary describes malware actions in general, i.e., regardless of whether they are determined by static or +dynamic analysis. So, in some cases, we cannot map an imported function to a MAEC malware action. Compare, for +example, the CreateFile and HttpSendRequest API calls. We can map the first one directly to the create-file ac- +tion. However, a call to the HttpSendRequest function could be mapped to any of the send-http-method -request +actions in the vocabulary (send-http-get-request, send-http-put-request, etc.) depending on its input param- +eters, which are not available in the dataset. +In order to map more API calls, we have extended the MAEC actions vocabulary with additional actions summarized +in Table 1. The unmappable method-specific HTTP request actions have been removed in favor of more general +send-http-request. New actions pertaining to calls to cryptographic API functions, commonly used by malware, +have been added. +Moreover, we have categorized the actions into several classes, such as networking, file access, or system manipulation, +in order to aid generalization in concept learning applications of our dataset. We describe these classes in more detail in +Section 4.7. +Table 1: Extensions of the MAEC actions vocabulary +Action +Description +send-http-request +The action of sending an HTTP client request to a server +encrypt +The action of file encryption +decrypt +The action of file decryption +generate-key +The action of cryptographic key generation +3.2 +File and section features +Samples in the EMBER dataset are characterized by a number of properties of various types (cf. Lst. 1). We have +selected the most salient of them based on malware detection expert knowledge to be mapped to the ontological +representation as features of samples (PE files) or samples’ sections. There are three kinds of such features in the +ontology: +1. direct representations of boolean (actually 0/1) properties from the original dataset; +2. features obtained by pre-processing original properties not directly represented in the ontology; +3. features obtained by pre-processing properties that are also represented directly. +Features of the first kind arise from sample properties such as general.has_relocations. The original properties +mapped to these features are listed within the description of features in Sec. 4.2. +There are two features of the second kind: One of them is the presence of a non-empty Common Language Runtime +data directory (in the datadirectories property) in the sample, which is indicative of .NET binaries. The other is the +sample’s entry point being located in a non-executable section (checked by inspecting the section.entry property +and the respective member of the section.sections array). Machine-learning algorithms can thus take advantage of +these two important features even though the underlying properties are not represented in the ontology directly. +Features of the third kind, also called derived features, can be defined by OWL 2 expressions, but they are known to +experts to be important malware indicators and are thus worth naming explicitly. Moreover, assigning these features +to samples during dataset pre-processing exposes them even to tools that lack the required expressivity (cardinality +restrictions, data type restrictions, enumerated data types with long lists of literals) or enabling it imposes a significant +performance penalty. If, e.g., a classifier trained on the data set is capable of working efficiently with constructs required +to define some of the derived features, these may be considered redundant and the user of our dataset may instruct the +tool to selectively ignore them so as not to enlarge the dimensionality or search space unnecessarily. Likewise, if the +user targets a learning algorithm to learn from the derived features, e.g., for sake of efficiency, it may be indicated to +remove or to ignore the original data from which they were constructed. +Two features derived from numerical sample properties in the EMBER dataset stand out due to their important role in +malware detection and a less trivial way of determining a suitable threshold controlling the assignment of the feature to +5 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +a sample. These features are a low number of imports of the sample and a high entropy of a sample’s section. They +indicate that the sample may be packed. The threshold values for these features are inspired by pestudio [pes], a tool +commonly used by security teams for the initial assessment of malware samples. Based on pestudio’s defaults, a sample +importing less than 10 functions is considered having a low number of imports, and a section with entropy greater +than 7.0 is marked as having a high entropy. +(a) Benign entropy histogram +(b) Malware entropy histogram +(c) Benign imports histogram +(d) Malware imports histogram +Figure 1: Entropy and imports histograms for EMBER dataset +In order to verify the threshold values from the pestudio tool, we performed a basic statistical analysis for the entire +EMBER dataset. Specifically, we were interested in histograms representing the number of imports for each sample and +the entropy of each section in the dataset. The results are depicted in Fig. 1. The threshold value for section entropy +was confirmed. We can see that there are significantly more sections in malware samples with entropy higher than 7.0 +compared to the number of such sections in benign samples. As for the number of imports, the histograms in Figs. 1c +and 1d do not show any clear indicative value that separates a significant volume of malware and benign samples. In +particular, a large number of both kinds of samples import less than 10 API functions. This feature alone is thus not a +strong indicator of a sample’s maliciousness, although it may become useful in combination with other features. Hence +we decided to include it anyway and keep pestudio’s preset threshold of 10. +Another non-trivially derived feature of sections is a non-standard name. While we map section names to the ontological +representation, describing this feature in OWL 2 requires a data property restriction with a negated list of section names +usually produced by compilers and linkers. We consider it improbable that a learning algorithm would discover such a +restriction. We have thus opted to collect usual section names and to add this feature to sections when EMBER data is +transformed to the ontology. The list of usual section names is available in our GitHub repository. +4 +PE Malware Ontology +We now introduce PE Malware Ontology. While we used the structure of the EMBER as a starting point, it should be +noted that our ontology is not a direct mapping of this or any other individual dataset. Since one of the most important +goals was interpretability, we included only those features that make sense from an expert’s point of view. For this +reason, we did not use various features such as file sizes, linker versions, byte histograms, etc. In theory, there may be +6 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +has_ +section +has_ +file_ +feature +has_action +PEFile +exports_count: integer +imports_count: integer +mz_count: integer +symbols_count: integer +path_strings_count: integer +registry_strings_count: integer +url_strings_count: integer +FileFeature +SectionFeature +has_ +section_ +feature +Low +ImportsCount +Debug +Exports +has_ +section_ +flag +ExecutableFile +DynamicLink +Library +Action +ProcessHandling +Networking +AntiDebugging +Access +Management +Connect +ToURL +CheckForKernel +Debugger +AddUser +CLR +Multiple +Executable +Sections +Nonexecutable +EntryPoint +LogonAsUser +WriteExecute +Section +Nonstandard +SectionName +HighEntropy +Shareable +Executable +Writable +SendData +OnSocket +CreateProcess +CodeSection +Initialized +DataSection +Uninitialized +DataSection +Section +section_name: string +section_entropy: double +SectionFlag +Readable +WriteTo +ProcesMemory +Figure 2: Core classes of the EMBER ontology and their properties +patterns between in features and various classifiers can learn them, but from an expert’s perspective, such features are +meaningless for distinguishing between malware and benign samples. Also, the use of such features could subsequently +cause the learned classifiers being prone to trivial adversarial attacks. +Our ontology is partially depicted in Figure 2. The core classes and properties are fully depicted, but only an illustrative +sample of file features and actions is shown for brevity. The ontology contains 195 classes, 6 object properties, and +9 data properties in total. +4.1 +PE files +Each sample of a Portable Executable file [Bridge et al., 2022] in the original dataset is represented by an instance +of the PEFile class, central to our ontology. Each such instance is further classified either as a proper executable in +the ExecutableFile subclass of PEFile or as a shared library (DLLs) in the DynamicLinkLibrary subclass. The +respective subclass is determined based on the COFF header available in the EMBER dataset. +The PEFile class is the domain of 7 data properties (see Tab. 2) which provide information on the number of imports +and exports, symbols, and MZ headers (mapped from the general property of JSON descriptions of samples), as well +as the the number of path, registry, and URL strings (mapped from the strings property). +PE files structurally consist of sections, possess features relevant to malware detection, and may perform actions (based +on their imported functions) when executed. These entities are represented by instances of the respective classes +in the ontology (Section, FileFeature, Action) which are linked to a PEFile instance via the has_section, +has_file_feature, and has_action object property, respectively. +7 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +Table 2: PEFile properties +Data property +Range +exports_count +xsd:integer +imports_count +xsd:integer +mz_count +xsd:integer +path_strings_count +xsd:integer +symbols_count +xsd:integer +registry_strings_count +xsd:integer +url_strings_count +xsd:integer +Object property +Range +has_action +Action +has_file_feature +FileFeature +has_section +Section +4.2 +File features +The FileFeature class has 15 subclasses representing various qualitative features of PE files that are, based on expert +knowledge, relevant to malware detection. They are enumerated and described in Table 3. As detailed in Sect. 3.2, these +features can be sorted into three categories: directly represented features, pre-processed features, and additional derived +features that are obtained from underlying values that are also represented. +Each subclass of FileFeature has a single prototypical instance. All PEFile instances that possess a feature are linked +via the has_file_feature object property to the feature’s prototypical instance. Although this way of modeling +features may be uncommon, it allows for easily readable descriptions of possible malware in the Manchester syn- +tax, e.g., ExecutableFile has_file_feature some {multiple_executable_sections} – to be read e.g. “an +executable file with the feature of having multiple executable sections”. +Table 3: PE file features +File feature class +Description (sample property in JSON or equivalent OWL 2 expression) +Directly represented features +Debug +Contains a debug section (general.has_debug) +Relocations +Contains a relocation section (general.has_relocations) +Resources +Contains resources (fonts, images, etc.; general.has_resources) +Signature +Is digitally signed (general.has_signature) +TLS +Includes a Thread Location Storage section (possibly a secret entry point; +general.has_tls) +Pre-processed features +CLR +Contains a Common Language Runtime data directory +(used in .NET binaries) +NonexecutableEntryPoint +Entry point is not in an executable section +Derived features +Exports +Exports functions (mostly in DLLs; exports_count some +xsd:integer[> 0]) +MultipleExecutableSections +Has multiple sections with the executable flag (has_section min 2 +(has_section_flag some Executable)) +LowImportsCount +The number of imported functions is smaller than the threshold value +(imports_count some xsd:integer[< imports_threshold ]) +NonstandardMZ +Has no or more than one MZ headers (possibly contains an embedded PE +file; mz_count some xsd:integer[> 1]) +PathStrings +Contains strings defining paths (path_strings_count some +xsd:integer[> 0]) +RegistryStrings +Contains strings defining registry keys (registry_strings_count some +xsd:integer[> 0]) +Symbols +Has COFF debug symbols (deprecated in executables; symbols_count +some xsd:integer[> 0]) +URLStrings +Contains strings defining URLs (url_strings_count some +xsd:integer[> 0]) +8 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +4.3 +Sections +Instances of the Section class represent sections of PE file samples in the original dataset. A PEFile instance is +linked to all its sections via the has_section property. Sections are further classified based on the type of data +found in the given section into one of Section’s three subclasses: CodeSection, InitializedDataSection, and +UninitializedDataSection. +Section instances have data properties assigning them their names and the entropies of their content. Furthermore, +they are linked to their permission flags and features by the respective object properties shown in Table 4. +Table 4: Section properties +Data property +Range +section_entropy +xsd:double +section_name +xsd:string +Object property +Range +has_section_feature +SectionFeature +has_section_flag +SectionFlag +4.4 +Section flags +The SectionFlag class classifies the flags given to PE file sections. In the EMBER dataset, these flags are included in +the props property of each section. From the malware-detection point of view, the most interesting of them are the flags +controlling how processes executing the PE file may access (read, write, execute) and share the memory region into +which this section is mapped. These flags can be, in theory, combined arbitrarily. Only these four flags are represented +in the ontology by the Executable, Readable, Writable, and Shareable subclasses of SectionFlag. Similarly +to file features, each subclass has a prototypical instance that sections with the respective flag are linked to via the +has_section_flag object property. +4.5 +Section features +Similarly to FileFeature, SectionFeature’s subclasses represent features of the sections that, based on our expert +knowledge, are relevant for malware detection. They are listed in Table 5. All of these features are derived, and their +values are computed from the section’s data properties and flags, as detailed in Sect. 3.2. +Instances of Section are linked to the prototypical instances of the three subclasses of SectionFeature using the +has_section_feature property. +Table 5: Section features +Section feature class +Description (equivalent OWL 2 expression) +HighEntropy +The value of section’s entropy is larger than the threshold value +(section_entropy some xsd:double[> entropy_threshold ]) +NonstandardSectionName +Section’s name is not in the list of standard section names (section_name some +not { ".text", ".data", ".rsrc", ... }) +WriteExecuteSection +Section has write and execute permissions ((has_section_flag some +Writable) and (has_section_flag some Executable)) +4.6 +Annotation of derived features +As discussed in Sec. 3.2, derived features of PE files and sections may be considered redundant in some applications +and retaining them in the data set may have adverse effects. We have documented such features in the ontology with +the derived_as annotation property, in order to facilitate their automatic identification. Each derived feature class is +annotated using this property with the respective defining OWL 2 expression (see Tables 3 and 5), encoded as a string +in the Manchester syntax. +4.7 +Actions +Instances of class Action represent actions that may be taken by a process executing the code from a PE file. Actions +are classified in 139 leaf subclasses of Action (a leaf class is one with no named subclasses except itself) that are +9 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +aligned with the MAEC standard’s Malware actions vocabulary as described in Sec. 3.1. In order to aid generalization +during classifier learning, we have added 17 categories of actions listed in Table 6. Each category is represented by an +intermediate class (e.g., ProcessHandling; see Fig. 2) – a direct subclass of Action and a direct superclass of the +leaf MAEC-based action classes falling into the category (e.g., CreateProcess, WriteToProcessMemory). +Currently, each leaf action class has a prototypical instance and all PEFile instances that may perform this action are +connected to it by the has_action object property. This connection is created by mapping imported functions from the +imports property of samples in the EMBER dataset to the corresponding MAEC malware actions if possible. For +example, when generating an ontological representation of a sample importing an API function that creates a process, a +prototypical instance create-process of the CreateProcess class is linked to the sample’s PEFile instance via the +has_action property. Of course, importing such an API function does not necessarily mean that it will actually be +called (thus creating a process) when this sample is run. +Alternatively, if dynamic tracing data from running samples in a sandboxed environment are available, every action +actually performed by a sample could be represented by an individual, non-prototypical instance of Action, classified +in the appropriate leaf subclass and further characterized by additional properties, such as parameters, previous and next +action, or timestamp. Extending our ontology to cover such dynamic data may be interesting future work. However, to +the best of our knowledge, no such dataset is currently available. +Table 6: Action classes +Action class +Description +AccessManagement +Managing users on the system (adding new user, enumerating exist- +ing users, etc.) +AntiDebugging +Debugger detection techniques +Cryptography +Encrypting/decrypting files, generating keys, etc. +DirectoryHandling +Manipulating with directories (creation, deletion, etc.) +DiskManagement +Mounting/unmounting disks, enumerating existing disks +FileHandling +Manipulating with files (creation, deletion, etc.) +InterProcessCommunication +Communication between processes (named pipes) +LibraryHandling +Loading library into running processes +Networking +Various networking activities (connecting to a socket, sending DNS +requests) +ProcessHandling +Various APIs for process handling, including creation or modifying +process memory +RegistryHandling +Enumerating registry keys, writing new values, etc. +ResourceSharing +Manipulating with resources shared over network +ServiceHandling +Manipulating with systems’s services +SynchronizationPrimitivesHandling +Handling mutexes/semaphores +SystemManipulation +Various APIs for obtaining system information +ThreadHandling +Creating remote threads, enumerating running threads +WindowHandling +Window manipulation APIs (creating new windows, dialog boxes, +etc.) +5 +Fractional datasets +One of the important goals was to create datasets that would allow unambiguous reproduction of experimental results. +For this reason, we generated multiple datasets with a clear definition of which samples belong to the training set and +which belong to the testing set. Each dataset contains 50% positive samples (malware) and 50% negative samples +(benign). We generated datasets of various sizes from 1000 to 800,000 (i.e. the entire EMBER dataset excluding +unlabeled samples). The main reason was that concept learning algorithms are computationally more demanding than +some traditional machine learning algorithms, and for that reason it can be useful to have less robust datasets available. +We also generated several variants for each fractional dataset size, which were randomly selected from the entire +EMBER dataset. More details can be seen in Table 7. +There are three files for each individual dataset: +dataset_N_1000.owl: this is an ontology filled with individuals of OWL type, which contains 1000 malware/benign +samples; +10 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +Table 7: Fractional datasets properties +Name +Total samples +Positive +Negative +Variants +dataset_N_1000.owl +1000 +500 +500 +10 +dataset_N_10000.owl +10,000 +5000 +5000 +10 +dataset_N_100000.owl +100,000 +50,000 +50,000 +10 +dataset_N_800000.owl +800,000 +400,000 +400,000 +1 +dataset_N_1000_raw.json: original JSON samples from the EMBER dataset (including unused features) from +which the ontological dataset was generated; +dataset_N_1000_examples.json: list of positive and negative samples in the given dataset in JSON format. +We refrain from publishing a predefined split of the datsets into the training and testing part, as committing to one split +may lead into lucky or unlucky setting for certain learning algorithms or even to overfitting on the given predefined split. +Instead we propose to follow the k-fold cross-evaluation methodology [Refaeilzadeh et al., 2009] in which the dataset +is split in k equal parts and in each run one of them is used for testing while the rest of the data is used for training. The +results are then averaged. Fivefold and tenfold splits are commonly used. +6 +Usage in concept learning +The ontology can be used by various machine learning algorithms where the goal is to build a malware detection +model. One such example includes the family of concept learning algorithms. Which can be used to provide descriptive +characterizations of (parts of) the malware sample using therms from the ontology. Three examples of class expressions +(in standard description logics syntax [Baader et al., 2003]) learned using Parallel Class Expression Learner (PARCEL) +Tran et al. [2012] algorithm are shown below: +(1) +(∃has_file_feature.{multiple_executable_sections}) +⊓ (≥ 2has_section.(∃has_section_feature.{high_entropy}) +(2) +(∃has_action.{read-from-process-memory}) +⊓ (∃has_section.(∃has_section_feature.{write_execute_section}) +(3) +(¬DynamicLinkLibrary) ⊓ (∃has_action.{connect-to-ftp-server}) +⊓ (∃has_action.{enumerate-registry-key-values}) +We can see that expressions are highly interpretable and human readable. For instance, expression (1) could be +interpreted as follows: a binary is malicious if it has multiple sections that are executable and has at least two sections +with high value of entropy. Given that the class expressions are in fact logical formulae, they may also be used with an +ontology-based retrieval system to query for samples that satisfy each given expressions. +7 +Conclusions +The PE Ontology, described in this report, is intended as an inter-operable representation format that can be used to +publish datasets of samples of PE malware. The ontology is partly based on the EMBER dataset but does not copy +the EMBER data properties one-to-one but instead relied on expert knowledge and standard nomenclature to improve +interpretability and reusability. +We hope it may also serve as a first step to establish an industry-standard for a semantic schema specifying what data +should actually be included in these datasets – considering its future gradual extensions, if needed. +It can be readily used to extract class expressions of learned malware samples either ex post or during the learning +process when malware detection models are learned. As briefly hinted in Section 6 such class expressions can than +serve e.g. to generate explanations in human language. +The ontology, being based on EMBER, is currently suited especially for data resulting from static malware analysis. +For future work, it might be useful to extend it to support data resulting from dynamic malware analysis as well. +11 + +Knowledge-Based Dataset for Training PE Malware Detection Models +TECHNICAL REPORT +8 +Acknowledgement +This research was sponsored by Slovak Republic under grants APVV-19-0220 (ORBIS) and by the EU H2020 +programme under Contract no. 952215 (TAILOR) +References +Classification of malware PE headers. URL https://github.com/urwithajit9/ClaMP. [Online; accessed 2022- +10-08]. +MalShare. URL https://malshare.com/. [Online; accessed 2022-10-15]. +Malware initial assessment. URL https://www.winitor.com/. [Online; accessed 2022-10-08]. +VirusShare, a. URL https://virusshare.com. [Online; accessed 2022-05-15]. +VirusTotal, b. 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Springer, 2018. +Peter Švec, Štefan Balogh, and Martin Homola. Experimental evaluation of description logic concept learning algorithms +for static malware detection. In ICISSP, pages 792–799, 2021. +An C Tran, Jens Dietrich, Hans W Guesgen, and Stephen Marsland. An approach to parallel class expression learning. +In International Workshop on Rules and Rule Markup Languages for the Semantic Web, pages 302–316. Springer, +2012. +R. Vinayakumar, Mamoun Alazab, K. P. Soman, Prabaharan Poornachandran, and Sitalakshmi Venkatra- +man. +Robust intelligent malware detection using deep learning. +IEEE Access, 7:46717–46738, 2019. +doi:10.1109/ACCESS.2019.2906934. +13 + diff --git a/VNAyT4oBgHgl3EQfV_dJ/content/tmp_files/load_file.txt b/VNAyT4oBgHgl3EQfV_dJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..deb7f7422395abcab2dac1627945d572db65fd44 --- /dev/null +++ b/VNAyT4oBgHgl3EQfV_dJ/content/tmp_files/load_file.txt @@ -0,0 +1,592 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf,len=591 +page_content='KNOWLEDGE-BASED DATASET FOR TRAINING PE MALWARE DETECTION MODELS TECHNICAL REPORT Peter Švec Institute of Computer Science and Mathematics Faculty of Electrical Engineering and Information Technology Slovak University of Technology Ilkoviˇcova 3, Bratislava, Slovakia peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='svec1@stuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='sk Štefan Balogh Institute of Computer Science and Mathematics Faculty of Electrical Engineering and Information Technology Slovak University of Technology Ilkoviˇcova 3, Bratislava, Slovakia stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='balogh@stuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='sk Martin Homola Department of Applied Informatics Faculty of Mathematics, Physics and Informatics Comenius University Mlynská dolina, Bratislava, Slovakia homola@fmph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='uniba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='sk Ján Kl’uka Department of Applied Informatics Faculty of Mathematics, Physics and Informatics Comenius University Mlynská dolina, Bratislava, Slovakia kluka@fmph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='uniba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='sk January 3, 2023 ABSTRACT Ontologies are a standard for semantic schemata in many knowledge-intensive domains of human interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' They are now becoming increasingly important also in areas until very recently dominated by subsymbolic representations and machine-learning-based data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' One such area is information security, and more specifically malware detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We propose PE Malware Ontology that offers a reusable semantic schema for Portable Executable (PE, Windows binary format) malware files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The ontology was inspired by the structure of the data in the EMBER dataset and it currently covers the data intended for static malware analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' With this proposal, we hope to achieve: a) a unified semantic representation for PE malware datasets that are available or will be published in the future;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' (b) applicability of symbolic, neural-symbolic, or otherwise explainable approaches in the PE Malware domain that may lead to improved interpretability of results which may now be characterized by the terms defined in the ontology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' and (c) by joint publishing of semantically treated EMBER data, including fractional datasets, also improved reproducibility of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Keywords Ontology · Dataset · Malware · Windows · Intepretability · Explainable AI 1 Introduction There are currently a number of datasets that may be used to train malware detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Among the most popular recent datasets we find Elastic Malware Benchmark for Empowering Researchers (EMBER) [Anderson and Roth, 2018] (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='1 million samples) and more recently Sophos/ReversingLabs 20 million sample dataset (SoReL) [Harang arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='00153v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='CR] 31 Dec 2022 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT and Rudd, 2020] (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 20 million samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The advantage of these datasets is that they were already analyzed by security experts and for each sample they provide extensive feature data extracted and pre-processed in structured textual format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In addition, SoReL provides disarmed binaries to some of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Several other online data sources are available, even some that contain genuine malicious binaries in their samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Such binaries may be used if some additional information about the sample needs to be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Specifically both EMBER and SoReL cover the domain of PE Malware (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', Windows executables and libraries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Both are well tailored for processing by various machine learning tools;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' for example EMBER directly provides a script to vectorize the data which is required by many sub-symbolic classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' However, in reality the data contained in these datasets is structured (JSON files) and all textual characteristics provided for each sample can be interpreted as meaningful properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Such data is essentially symbolic and can be viewed as (or, exported into) a knowledge base or a knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Such treatment of the data would enable to capitalize on knowledge-processing tools, or even some tools rooted in the more recent neural-symbolic AI movement, especially to improve the interpretability of the resulting classifier models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' To give an example, approaches such as concept-learning may be used to obtain a symbolic characterization of a malware sample in the form of a concept expression based on an ontology [Švec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Knowledge-base embedding [Bordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2013] may be used to improve effectiveness of learned models by injecting prior symbolic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' More recently, neural network activation patterns recognition and alignment with ontology concepts de Sousa Ribeiro and Leite [2021] may be used to explain and diagnose trained neural network classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Also, EMBER and SoReL are vast, they contain millions of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Some of the studies that have been published [Vinayakumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2019, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2020, Ghouti and Imam, 2020] had to resort to reducing the dataset of interest to a smaller size, especially if the trained classifier is computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' However no unified methodology for reducing the datasets has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' To alleviate the outlined issues, our work targets the following goals: Unified semantic representation: To provide a unified vocabulary for relevant PE malware characteristics that can be then used to represent any relevant data sample regardless of its source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Interpretability of results: To ensure that the vocabulary provides a suitable nomenclature that is meaningful and recognized by human users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Consequently prototype samples, concept descriptions, rules, or any other characterizations expressed in the vocabulary would be understandable and explainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Reproducibility of experiments: Ensure that experiments may be executed on datasets of different suitable sizes without the need to reduce the full original dataset to smaller ones again and again in each study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' To address the first two goals, we have developed a unified ontology that can provide a reusable semantic schema for PE malware files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' To the best of our knowledge, such ontology was missing so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We have also aligned the nomenclature used to describe actions possibly performed by PE files with the Malware Attribute Enumeration and Characterization (MAEC) standard [MITRE Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2020] which is widely accepted for malware descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Our goal was to improve interpretability of the captured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The ontology is partly based on the EMBER dataset structure derived from the EMBER data features, but it is not a direct ontological mapping of this or any other individual dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The emphasis is placed on data features that are meaningful in malware characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' For this reason we included only those features that make sense from an expert’s point of view, while other less meaningful features were left out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Some of the left-out features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', file size) may even cause the trained classifiers to be prone to trial counterattacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Also some of the features captured in the ontology can be considered as derived in that they combine multiple low-level EMBER data entries into a meaningful feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Two features are based on thresholds on top of numeric features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', “high entropy”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' This is useful for systems that have difficulties with handling numeric values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' But our goal was also to create a dataset that could be effectively used as a standard benchmark to help advance learning algorithms in the malware detection domain and that would also offer unambiguous reproducibility of experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' To this end, we release EMBER data (2018 version) which has been translated into RDF semantic format, including smaller fractional datasets which have been fixed for sake of better comparison of diverse experiments that require smaller data volume than the full EMBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Our ontology, including the fractional datasets, and mapping scripts is available in our GitHub repository1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='com/orbis-security/pe-malware-ontology 2 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT 2 Data sources In this section we describe in more detail the EMBER dataset, which we used as the foundation for our ontology, along with the modifications we proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The main difference between the EMBER and SoReL datasets is that SoReL contains more samples together with disarmed malware binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' However, the static properties of samples that these datasets offer are almost identical (a few are missing in SoReL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Among other things, we decided on the EMBER dataset because it is generally more used in research (261 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 32 citations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='1 EMBER The dataset contains a total of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='1 million samples and includes 400,000 malicious samples, 400,000 benign samples and 300,000 unlabeled samples, so the dataset can also be applied to unsupervised learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The labeled samples from the dataset are also divided into training and testing sets (600,000 for training and 200,000 for testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The dataset itself is composed of a collection of JSON objects, where each object represents data statically extracted from the PE file of one sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' A simplified example of one sample can be seen in Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The static properties themselves are organized as follows: General file information: This set of features is dedicated to general information about the file, such as file size, information whether the file contains a digital signature, presence of debugging symbols, presence of a TLS section, or the number of imported and exported functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Header information: These are properties found in file headers such as the target architecture for which the file was compiled, linker version, various timestamps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Section information: This set is dedicated to individual sections in the binary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' For each section dataset contains its name, content type (code, initialized or uninitialized data), various properties (such as read, write, and execute rights), or the value of the entropy of the section’s content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Imported functions: List of imported functions organized by the DLL (here it is necessary to note that if the file imports a certain function, it does not necessarily have to be called in the code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Exported functions: List of exported functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' These are mostly included only in cases where the PE file is a library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In addition to the categories mentioned above, the EMBER dataset also contains other numerical properties such as byte histogram, byte-entropy histogram, or simple statistics for the strings found in the file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Listing 1: Static features for single binary sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' { "sha256": " eb87d82ad7bdc1b753bf91858d2986063ebd8aabeb8e7e91c0c78db21982a0d6 ", "md5": " aba129a3d1ba9d307dad05617f66d8e7 ", "appeared": "2018 -01", "label": 1, "avclass": "fareit", "histogram": [ 96506 , 8328 , 5582 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ], " byteentropy ": [0, 4229 , 269, 247, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ], "strings": { " numstrings": 7762 , "avlength": 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='60641587219789 , " printabledist ": [591 , 51, 96, 46, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ], " printables": 1409629 , "entropy": 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='037064474164528 ,' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' "has_debug": 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' "exports": 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' "imports": 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' " has_relocations ": 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' " has_resources ": 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' " has_signature ": 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' "has_tls": 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' "symbols": 0 },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' "header": { "coff":{ "timestamp": 708992537 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 3 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT "machine": "1386",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' " characteristics ": [" CHARA_32BIT_MACHINE ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' " BYTES_REVERSED_LO ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' " EXECUTABLE_IMAGE ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ] }, "optional": { "subsystem": " WINDOWS_GUI", " dll_characteristics ": [], "magic": "PE32", " major_image_version ": 0, " minor_image_version ": 0, " major_linker_version ": 2, " minor_linker_version ": 25, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' } }, "section": { "entry": "CODE", "sections": [ { "name": "CODE", "size": 443392 , "entropy": 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='532932639432919 , "vsize": 442984 , "props": ["CNT_CODE", "MEM_EXECUTE ", "MEM_READ"] }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ] }, "imports": { "kernel32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='dll": [" DeleteCriticalSection ", " TlsSetValue ", "Sleep", .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ], }, "exports": [], " datadirectories ": [ { "name": " EXPORT_TABLE ", " virtual_address ": 0 }, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ] } 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='2 Other sources Following the success of EMBER, Sophos/ReversingLabs 20 million sample dataset (SoReL) was published by Harang and Rudd [2020] who especially aimed at scaling up the volume of samples available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The structure of SoReL samples is similar to EMBER and it covers the same static features with very minor differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' These once more represented in JSON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In addition to approximately 15M JSON structured samples that are labelled and over 4M additional ones that are unlabelled, SoReL also contains approximately 10M real binary malware samples that have been disarmed in by modifying their header file).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The ontology presented in this work may be directly applied on SoReL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Among the most popular sources (that provide real binaries) are VirusTotal [vir, b], which, however, provides paid services (along with benign files), VirusShare [vir, a], which currently contains approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 50 million malicious samples or MalShare [mal].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Known datasets that contain dynamic properties include MALREC [Severi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2018], which captures the overall activity of malware in the form of logging all sources of indeterminism in the system, such as system calls, peripheral devices, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In total, it contains approximately 66,000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' A smaller dynamic dataset that focuses primarily on logging API calls is Mal-API-2019 [Catak and Yazı, 2019] and contains approximately 7,000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In 2022, a dataset from AVAST was released [Bosansky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2022], which, unlike previous works, combines static and dynamic features and contains approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 50,000 malware samples (its main purpose is to classify malware into individual families).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Other well-known datasets include ClaMP [cla], which contains only Portable Executable (PE) file headers (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 5,000) or MalImg [Nataraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2011], which contains malicious samples, encoded in gray scale in form of approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 10,000 samples from different families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The DREBIN dataset is also worth mentioning, but unlike the previous works, it contains binary samples from the Android platform [Arp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 3 Data preprocessing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='1 Dataset standardization Individual samples may import a large number of functions from various standard DLLs implementing system APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' They provide useful clues about the possible actions a sample can perform when running, although not all imported functions may be called and not all of these actions are relevant from the malware-detection perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In order to extract only relevant information from samples’ imports to the ontological representation in a way that is aligned with standard best practice, we have turned to Malware Attribute Enumeration and Characterization (MAEC) [MITRE Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' MAEC is a community-developed structured language for describing information about malware, combining static and dynamic features (different kinds of behavior, interactions between processes, and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We map the space 4 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT of imported functions to malware actions, defined in one of MAEC’s vocabularies [MITRE Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' This aids not only standardization of the dataset, but also achieves an effect similar to dimensionality reduction in traditional machine learning – irrelevant imported functions are disregarded and multiple functions with a similar effect can be mapped to a single action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The complete mapping of API functions to actions is available in our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The MAEC vocabulary describes malware actions in general, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', regardless of whether they are determined by static or dynamic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' So, in some cases, we cannot map an imported function to a MAEC malware action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Compare, for example, the CreateFile and HttpSendRequest API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We can map the first one directly to the create-file ac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' However, a call to the HttpSendRequest function could be mapped to any of the send-http-method -request actions in the vocabulary (send-http-get-request, send-http-put-request, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=') depending on its input param- eters, which are not available in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In order to map more API calls, we have extended the MAEC actions vocabulary with additional actions summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The unmappable method-specific HTTP request actions have been removed in favor of more general send-http-request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' New actions pertaining to calls to cryptographic API functions, commonly used by malware, have been added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Moreover, we have categorized the actions into several classes, such as networking, file access, or system manipulation, in order to aid generalization in concept learning applications of our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We describe these classes in more detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Table 1: Extensions of the MAEC actions vocabulary Action Description send-http-request The action of sending an HTTP client request to a server encrypt The action of file encryption decrypt The action of file decryption generate-key The action of cryptographic key generation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='2 File and section features Samples in the EMBER dataset are characterized by a number of properties of various types (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Lst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We have selected the most salient of them based on malware detection expert knowledge to be mapped to the ontological representation as features of samples (PE files) or samples’ sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' There are three kinds of such features in the ontology: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' direct representations of boolean (actually 0/1) properties from the original dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' features obtained by pre-processing original properties not directly represented in the ontology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' features obtained by pre-processing properties that are also represented directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Features of the first kind arise from sample properties such as general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_relocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The original properties mapped to these features are listed within the description of features in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' There are two features of the second kind: One of them is the presence of a non-empty Common Language Runtime data directory (in the datadirectories property) in the sample, which is indicative of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='NET binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The other is the sample’s entry point being located in a non-executable section (checked by inspecting the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='entry property and the respective member of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='sections array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Machine-learning algorithms can thus take advantage of these two important features even though the underlying properties are not represented in the ontology directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Features of the third kind, also called derived features, can be defined by OWL 2 expressions, but they are known to experts to be important malware indicators and are thus worth naming explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Moreover, assigning these features to samples during dataset pre-processing exposes them even to tools that lack the required expressivity (cardinality restrictions, data type restrictions, enumerated data types with long lists of literals) or enabling it imposes a significant performance penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' If, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', a classifier trained on the data set is capable of working efficiently with constructs required to define some of the derived features, these may be considered redundant and the user of our dataset may instruct the tool to selectively ignore them so as not to enlarge the dimensionality or search space unnecessarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Likewise, if the user targets a learning algorithm to learn from the derived features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', for sake of efficiency, it may be indicated to remove or to ignore the original data from which they were constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Two features derived from numerical sample properties in the EMBER dataset stand out due to their important role in malware detection and a less trivial way of determining a suitable threshold controlling the assignment of the feature to 5 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' These features are a low number of imports of the sample and a high entropy of a sample’s section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' They indicate that the sample may be packed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The threshold values for these features are inspired by pestudio [pes], a tool commonly used by security teams for the initial assessment of malware samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Based on pestudio’s defaults, a sample importing less than 10 functions is considered having a low number of imports, and a section with entropy greater than 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='0 is marked as having a high entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' (a) Benign entropy histogram (b) Malware entropy histogram (c) Benign imports histogram (d) Malware imports histogram Figure 1: Entropy and imports histograms for EMBER dataset In order to verify the threshold values from the pestudio tool, we performed a basic statistical analysis for the entire EMBER dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Specifically, we were interested in histograms representing the number of imports for each sample and the entropy of each section in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The threshold value for section entropy was confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We can see that there are significantly more sections in malware samples with entropy higher than 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='0 compared to the number of such sections in benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' As for the number of imports, the histograms in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 1c and 1d do not show any clear indicative value that separates a significant volume of malware and benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In particular, a large number of both kinds of samples import less than 10 API functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' This feature alone is thus not a strong indicator of a sample’s maliciousness, although it may become useful in combination with other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Hence we decided to include it anyway and keep pestudio’s preset threshold of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Another non-trivially derived feature of sections is a non-standard name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' While we map section names to the ontological representation, describing this feature in OWL 2 requires a data property restriction with a negated list of section names usually produced by compilers and linkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We consider it improbable that a learning algorithm would discover such a restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We have thus opted to collect usual section names and to add this feature to sections when EMBER data is transformed to the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The list of usual section names is available in our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 4 PE Malware Ontology We now introduce PE Malware Ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' While we used the structure of the EMBER as a starting point, it should be noted that our ontology is not a direct mapping of this or any other individual dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Since one of the most important goals was interpretability, we included only those features that make sense from an expert’s point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' For this reason, we did not use various features such as file sizes, linker versions, byte histograms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' there may be ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Knowledge-Based Dataset for Training PE Malware Detection Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='TECHNICAL REPORT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='section ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='file_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='PEFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='exports_count: integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='imports_count: integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='mz_count: integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='symbols_count: integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='path_strings_count: integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='registry_strings_count: integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='url_strings_count: integer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='FileFeature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='SectionFeature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='section_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='ImportsCount ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Debug ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Exports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='section_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='flag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='ExecutableFile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='DynamicLink ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Library ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Action ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='ProcessHandling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Networking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='AntiDebugging ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Access ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Management ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Connect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='ToURL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='CheckForKernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Debugger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='AddUser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='CLR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Multiple ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Executable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Sections ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Nonexecutable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='EntryPoint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='LogonAsUser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='WriteExecute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Section ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Nonstandard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='SectionName ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='HighEntropy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Shareable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Executable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Writable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='SendData ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='OnSocket ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='CreateProcess ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='CodeSection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Initialized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='DataSection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Uninitialized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='DataSection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Section ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='section_name: string ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='section_entropy: double ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='SectionFlag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Readable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='WriteTo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='ProcesMemory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='Figure 2: Core classes of the EMBER ontology and their properties ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='patterns between in features and various classifiers can learn them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' but from an expert’s perspective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' such features are meaningless for distinguishing between malware and benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Also, the use of such features could subsequently cause the learned classifiers being prone to trivial adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Our ontology is partially depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The core classes and properties are fully depicted, but only an illustrative sample of file features and actions is shown for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The ontology contains 195 classes, 6 object properties, and 9 data properties in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='1 PE files Each sample of a Portable Executable file [Bridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2022] in the original dataset is represented by an instance of the PEFile class, central to our ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Each such instance is further classified either as a proper executable in the ExecutableFile subclass of PEFile or as a shared library (DLLs) in the DynamicLinkLibrary subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The respective subclass is determined based on the COFF header available in the EMBER dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The PEFile class is the domain of 7 data properties (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 2) which provide information on the number of imports and exports, symbols, and MZ headers (mapped from the general property of JSON descriptions of samples), as well as the the number of path, registry, and URL strings (mapped from the strings property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' PE files structurally consist of sections, possess features relevant to malware detection, and may perform actions (based on their imported functions) when executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' These entities are represented by instances of the respective classes in the ontology (Section, FileFeature, Action) which are linked to a PEFile instance via the has_section, has_file_feature, and has_action object property, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 7 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT Table 2: PEFile properties Data property Range exports_count xsd:integer imports_count xsd:integer mz_count xsd:integer path_strings_count xsd:integer symbols_count xsd:integer registry_strings_count xsd:integer url_strings_count xsd:integer Object property Range has_action Action has_file_feature FileFeature has_section Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='2 File features The FileFeature class has 15 subclasses representing various qualitative features of PE files that are, based on expert knowledge, relevant to malware detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' They are enumerated and described in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' As detailed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='2, these features can be sorted into three categories: directly represented features, pre-processed features, and additional derived features that are obtained from underlying values that are also represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Each subclass of FileFeature has a single prototypical instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' All PEFile instances that possess a feature are linked via the has_file_feature object property to the feature’s prototypical instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Although this way of modeling features may be uncommon, it allows for easily readable descriptions of possible malware in the Manchester syn- tax, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', ExecutableFile has_file_feature some {multiple_executable_sections} – to be read e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' “an executable file with the feature of having multiple executable sections”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Table 3: PE file features File feature class Description (sample property in JSON or equivalent OWL 2 expression) Directly represented features Debug Contains a debug section (general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_debug) Relocations Contains a relocation section (general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_relocations) Resources Contains resources (fonts, images, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_resources) Signature Is digitally signed (general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_signature) TLS Includes a Thread Location Storage section (possibly a secret entry point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='has_tls) Pre-processed features CLR Contains a Common Language Runtime data directory (used in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='NET binaries) NonexecutableEntryPoint Entry point is not in an executable section Derived features Exports Exports functions (mostly in DLLs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' exports_count some xsd:integer[> 0]) MultipleExecutableSections Has multiple sections with the executable flag (has_section min 2 (has_section_flag some Executable)) LowImportsCount The number of imported functions is smaller than the threshold value (imports_count some xsd:integer[< imports_threshold ]) NonstandardMZ Has no or more than one MZ headers (possibly contains an embedded PE file;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' mz_count some xsd:integer[> 1]) PathStrings Contains strings defining paths (path_strings_count some xsd:integer[> 0]) RegistryStrings Contains strings defining registry keys (registry_strings_count some xsd:integer[> 0]) Symbols Has COFF debug symbols (deprecated in executables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' symbols_count some xsd:integer[> 0]) URLStrings Contains strings defining URLs (url_strings_count some xsd:integer[> 0]) 8 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='3 Sections Instances of the Section class represent sections of PE file samples in the original dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' A PEFile instance is linked to all its sections via the has_section property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Sections are further classified based on the type of data found in the given section into one of Section’s three subclasses: CodeSection, InitializedDataSection, and UninitializedDataSection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Section instances have data properties assigning them their names and the entropies of their content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Furthermore, they are linked to their permission flags and features by the respective object properties shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Table 4: Section properties Data property Range section_entropy xsd:double section_name xsd:string Object property Range has_section_feature SectionFeature has_section_flag SectionFlag 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='4 Section flags The SectionFlag class classifies the flags given to PE file sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In the EMBER dataset, these flags are included in the props property of each section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' From the malware-detection point of view, the most interesting of them are the flags controlling how processes executing the PE file may access (read, write, execute) and share the memory region into which this section is mapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' These flags can be, in theory, combined arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Only these four flags are represented in the ontology by the Executable, Readable, Writable, and Shareable subclasses of SectionFlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Similarly to file features, each subclass has a prototypical instance that sections with the respective flag are linked to via the has_section_flag object property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='5 Section features Similarly to FileFeature, SectionFeature’s subclasses represent features of the sections that, based on our expert knowledge, are relevant for malware detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' They are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' All of these features are derived, and their values are computed from the section’s data properties and flags, as detailed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Instances of Section are linked to the prototypical instances of the three subclasses of SectionFeature using the has_section_feature property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Table 5: Section features Section feature class Description (equivalent OWL 2 expression) HighEntropy The value of section’s entropy is larger than the threshold value (section_entropy some xsd:double[> entropy_threshold ]) NonstandardSectionName Section’s name is not in the list of standard section names (section_name some not { ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='text", ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='data", ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='rsrc", .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' }) WriteExecuteSection Section has write and execute permissions ((has_section_flag some Writable) and (has_section_flag some Executable)) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='6 Annotation of derived features As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='2, derived features of PE files and sections may be considered redundant in some applications and retaining them in the data set may have adverse effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We have documented such features in the ontology with the derived_as annotation property, in order to facilitate their automatic identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Each derived feature class is annotated using this property with the respective defining OWL 2 expression (see Tables 3 and 5), encoded as a string in the Manchester syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='7 Actions Instances of class Action represent actions that may be taken by a process executing the code from a PE file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Actions are classified in 139 leaf subclasses of Action (a leaf class is one with no named subclasses except itself) that are 9 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT aligned with the MAEC standard’s Malware actions vocabulary as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' In order to aid generalization during classifier learning, we have added 17 categories of actions listed in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Each category is represented by an intermediate class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', ProcessHandling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 2) – a direct subclass of Action and a direct superclass of the leaf MAEC-based action classes falling into the category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', CreateProcess, WriteToProcessMemory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Currently, each leaf action class has a prototypical instance and all PEFile instances that may perform this action are connected to it by the has_action object property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' This connection is created by mapping imported functions from the imports property of samples in the EMBER dataset to the corresponding MAEC malware actions if possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' For example, when generating an ontological representation of a sample importing an API function that creates a process, a prototypical instance create-process of the CreateProcess class is linked to the sample’s PEFile instance via the has_action property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Of course, importing such an API function does not necessarily mean that it will actually be called (thus creating a process) when this sample is run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Alternatively, if dynamic tracing data from running samples in a sandboxed environment are available, every action actually performed by a sample could be represented by an individual, non-prototypical instance of Action, classified in the appropriate leaf subclass and further characterized by additional properties, such as parameters, previous and next action, or timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Extending our ontology to cover such dynamic data may be interesting future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' However, to the best of our knowledge, no such dataset is currently available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Table 6: Action classes Action class Description AccessManagement Managing users on the system (adding new user, enumerating exist- ing users, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=') AntiDebugging Debugger detection techniques Cryptography Encrypting/decrypting files, generating keys, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' DirectoryHandling Manipulating with directories (creation, deletion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=') DiskManagement Mounting/unmounting disks, enumerating existing disks FileHandling Manipulating with files (creation, deletion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=') InterProcessCommunication Communication between processes (named pipes) LibraryHandling Loading library into running processes Networking Various networking activities (connecting to a socket, sending DNS requests) ProcessHandling Various APIs for process handling, including creation or modifying process memory RegistryHandling Enumerating registry keys, writing new values, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' ResourceSharing Manipulating with resources shared over network ServiceHandling Manipulating with systems’s services SynchronizationPrimitivesHandling Handling mutexes/semaphores SystemManipulation Various APIs for obtaining system information ThreadHandling Creating remote threads, enumerating running threads WindowHandling Window manipulation APIs (creating new windows, dialog boxes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=') 5 Fractional datasets One of the important goals was to create datasets that would allow unambiguous reproduction of experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' For this reason, we generated multiple datasets with a clear definition of which samples belong to the training set and which belong to the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Each dataset contains 50% positive samples (malware) and 50% negative samples (benign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We generated datasets of various sizes from 1000 to 800,000 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' the entire EMBER dataset excluding unlabeled samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The main reason was that concept learning algorithms are computationally more demanding than some traditional machine learning algorithms, and for that reason it can be useful to have less robust datasets available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We also generated several variants for each fractional dataset size, which were randomly selected from the entire EMBER dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' More details can be seen in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' There are three files for each individual dataset: dataset_N_1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='owl: this is an ontology filled with individuals of OWL type, which contains 1000 malware/benign samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 10 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT Table 7: Fractional datasets properties Name Total samples Positive Negative Variants dataset_N_1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='owl 1000 500 500 10 dataset_N_10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='owl 10,000 5000 5000 10 dataset_N_100000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='owl 100,000 50,000 50,000 10 dataset_N_800000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='owl 800,000 400,000 400,000 1 dataset_N_1000_raw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='json: original JSON samples from the EMBER dataset (including unused features) from which the ontological dataset was generated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' dataset_N_1000_examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='json: list of positive and negative samples in the given dataset in JSON format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We refrain from publishing a predefined split of the datsets into the training and testing part, as committing to one split may lead into lucky or unlucky setting for certain learning algorithms or even to overfitting on the given predefined split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Instead we propose to follow the k-fold cross-evaluation methodology [Refaeilzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2009] in which the dataset is split in k equal parts and in each run one of them is used for testing while the rest of the data is used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The results are then averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Fivefold and tenfold splits are commonly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 6 Usage in concept learning The ontology can be used by various machine learning algorithms where the goal is to build a malware detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' One such example includes the family of concept learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Which can be used to provide descriptive characterizations of (parts of) the malware sample using therms from the ontology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Three examples of class expressions (in standard description logics syntax [Baader et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 2003]) learned using Parallel Class Expression Learner (PARCEL) Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' [2012] algorithm are shown below: (1) (∃has_file_feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' {multiple_executable_sections}) ⊓ (≥ 2has_section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='(∃has_section_feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' {high_entropy}) (2) (∃has_action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' {read-from-process-memory}) ⊓ (∃has_section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='(∃has_section_feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' {write_execute_section}) (3) (¬DynamicLinkLibrary) ⊓ (∃has_action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' {connect-to-ftp-server}) ⊓ (∃has_action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' {enumerate-registry-key-values}) We can see that expressions are highly interpretable and human readable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' For instance, expression (1) could be interpreted as follows: a binary is malicious if it has multiple sections that are executable and has at least two sections with high value of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Given that the class expressions are in fact logical formulae, they may also be used with an ontology-based retrieval system to query for samples that satisfy each given expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 7 Conclusions The PE Ontology, described in this report, is intended as an inter-operable representation format that can be used to publish datasets of samples of PE malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The ontology is partly based on the EMBER dataset but does not copy the EMBER data properties one-to-one but instead relied on expert knowledge and standard nomenclature to improve interpretability and reusability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' We hope it may also serve as a first step to establish an industry-standard for a semantic schema specifying what data should actually be included in these datasets – considering its future gradual extensions, if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' It can be readily used to extract class expressions of learned malware samples either ex post or during the learning process when malware detection models are learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' As briefly hinted in Section 6 such class expressions can than serve e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' to generate explanations in human language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' The ontology, being based on EMBER, is currently suited especially for data resulting from static malware analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' For future work, it might be useful to extend it to support data resulting from dynamic malware analysis as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 11 Knowledge-Based Dataset for Training PE Malware Detection Models TECHNICAL REPORT 8 Acknowledgement This research was sponsored by Slovak Republic under grants APVV-19-0220 (ORBIS) and by the EU H2020 programme under Contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' 952215 (TAILOR) References Classification of malware PE headers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' URL https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='com/urwithajit9/ClaMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' accessed 2022- 10-08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' MalShare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' URL https://malshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' accessed 2022-10-15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Malware initial assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' URL https://www.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' Secur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=', 89, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='cose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNAyT4oBgHgl3EQfV_dJ/content/2301.00153v1.pdf'} +page_content='2019.' metadata={'source': 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b/ZtFIT4oBgHgl3EQfkyt3/content/tmp_files/2301.11302v1.pdf.txt @@ -0,0 +1,2292 @@ +Minimax estimation of discontinuous optimal transport +maps: The semi-discrete case +Aram-Alexandre Pooladian1,∗, Vincent Divol3,∗, Jonathan Niles-Weed1,2 +1Center for Data Science, New York University +2Courant Institute of Mathematical Sciences, New York University +3Universit´e Paris Dauphine - PSL +aram-alexandre.pooladian@nyu.edu,vincent.divol@psl.eu, jnw@cims.nyu.edu +January 27, 2023 +Abstract +We consider the problem of estimating the optimal transport map between two +probability distributions, P and Q in Rd, on the basis of i.i.d. samples. All existing +statistical analyses of this problem require the assumption that the transport map +is Lipschitz, a strong requirement that, in particular, excludes any examples where +the transport map is discontinuous. As a first step towards developing estimation +procedures for discontinuous maps, we consider the important special case where the +data distribution Q is a discrete measure supported on a finite number of points in Rd. +We study a computationally efficient estimator initially proposed by [PNW21], based on +entropic optimal transport, and show in the semi-discrete setting that it converges at the +minimax-optimal rate n−1/2, independent of dimension. Other standard map estimation +techniques both lack finite-sample guarantees in this setting and provably suffer from +the curse of dimensionality. We confirm these results in numerical experiments, and +provide experiments for other settings, not covered by our theory, which indicate that +the entropic estimator is a promising methodology for other discontinuous transport +map estimation problems. +1 +Introduction +The theory of optimal transport (OT) defines a natural geometry on the space of probability +measures [San15, Vil09] and has become ubiquitous in modern data-driven tasks. In this area, +optimal transport maps are a central object of study: suppose P and Q are two probability +distributions with finite second moments, with P having a density with respect to the +Lebesegue measure on Rd. Then, Brenier’s theorem (see Section 2.1) states that there exists +*Pooladian and Divol contributed equally to this work. +1 +arXiv:2301.11302v1 [math.ST] 26 Jan 2023 + +a convex function ϕ0 whose gradient defines a unique optimal transport map between P and +Q. This map is optimal in the sense that it minimizes the following objective function: +∇ϕ0 := argmin +T∈T (P,Q) +� +1 +2∥x − T(x)∥2 dP(x) , +(1) +where T (P, Q) := {T : Rd → Rd | X ∼ P, T(X) ∼ Q} is the set of transport maps between +P and Q. The optimal value of the objective function in Equation (1) is called the (squared) +2-Wasserstein distance, written explicitly as +S0(P, Q) = +� +1 +2∥x − ∇ϕ0(x)∥2 dP(x) , +though a more general formulation is available (see Section 2.1). Computing or approximating +S0(P, Q) as well as ∇ϕ0 has found use in several academic communities, such as economics +[CCG16, CGHH17, TGR21, GX21], computational biology [BSG+21, BKC22, LBG+22, +SST+19, MSF+21, DSSS22, YDV+20], and computer vision [SPKS16, SDGP+15, FCVP17], +among many others. +Practitioners seldom have access to P or Q, but instead have access to i.i.d. samples +X1, . . . , Xn ∼ P and Y1, . . . , Yn ∼ Q. On the basis of these samples, practitioners face both +computational and statistical challenges when estimating ∇ϕ0. From a theoretical perspective, +the statistical task of estimating optimal transport maps has attracted much interest in the +last few years [HR21, MVB+21, MBNWW21, DGS21, PNW21, DNWP22, GS22]. +The first finite-sample analysis of this problem was performed by [HR21], who proposed +an estimator for ∇ϕ0 under the assumption that ϕ0 is s + 1-times continuously differentiable, +for s > 1. They showed that a wavelet-based estimator ˆϕW satisfies +E∥∇ ˆϕW − ∇ϕ0∥2 +L2(P) ≲ n− +2s +2s+d−2 log2(n) , +and that this rate is minimax optimal up to logarithmic factors. Their analysis requires that +P and Q have bounded densities with compact support Ω ⊆ Rd, and that ϕ0 be both strongly +convex and smooth. Implementing the estimator ˆϕW is computationally challenging even in +moderate dimensions, and is practically infeasible for d > 3. Follow up work has proposed +alternative estimators which improve upon ˆϕW either in computational efficiency or in the +generality in which they apply. Though these subsequent works go significantly beyond the +setting considered by [HR21], none has eliminated the crucial assumption that ϕ0 is smooth, +i.e., that the transport map ∇ϕ0 is Lipschitz. +We highlight two estimators proposed in this line of work that are particularly practical. +[MBNWW21] study the 1-Nearest Neighbor estimator ˆT1NN. This estimator is obtained by +solving the empirical optimal transport problem between the samples, which is then extended +to a function defined on Rd using a projection scheme; see Section 4 for more details. Given +n samples from the source and target measures in Rd, ˆT1NN has a runtime of O(n3) via the +Hungarian Algorithm [see PC19, Chapter 3], and, for d ≥ 5, achieves the rate +E∥ ˆT1NN − ∇ϕ0∥2 +L2(P) ≲ n− 2 +d +(2) +whenever the optimal transport map ϕ0 is smooth and strongly convex, and under mild +regularity conditions on P. In another work, [PNW21] conducted a statistical analysis of +2 + +an estimator originally proposed by [SDF+18] based on entropic optimal transport. The +efficiency of Sinkhorn’s algorithm for large-scale problems [Cut13, PC19] makes this estimator +attractive from a computational perspective, and [PNW21] also give statistical guarantees, +though these fall short of being minimax-optimal. +Despite this progress, none of the aforementioned results can be applied in situations +where ∇ϕ0 is not Lipschitz. And in practice, even requiring the continuity of the transport +map can be far too stringent. It is indeed too much to hope for that an underlying data +distribution (e.g. over the space of images) has one single connected component; this is +supported by recent work that stipulates that the underlying data distribution is the union of +disjoint manifolds of varying intrinsic dimension [BCR+22]. In such a setting, the transport +map ∇ϕ0 will not be continuous, demonstrating the need of considering the problem of the +statistical estimation of discontinuous transport maps to get closer to real-world situations. +As a first step, we choose to focus on the case where the target distribution Q = �J +j=1 qjδyj +is discrete while the source measure P has full support, often called the semi-discrete setting +in the optimal transport literature. In this setting, the optimal transport map ∇ϕ0 is constant +over regions known as Laguerre cells (each cell corresponding to a different atom of the +discrete measure), while displaying discontinuities on their boundaries (see Section 2.1.1 for +more details). Figure 1 provides such an example. Semi-discrete optimal transport therefore +provides a natural class of discontinuous transport maps. +Figure 1: An illustration of a semi-discrete optimal transport map. The support of P, the +whole rectangle, is partitioned into regions, each of which is transported to one of the atoms +of the discrete target measure Q. The resulting map is discontinuous at the boundaries of +each cell. +We focus on this setting for two reasons. First, it has garnered a lot of attention in recent +years, in both computational and theoretical circles [see, e.g., MSS21, ANWS22, CAN22], +due in particular to its connection with the quantization problem [GL07]. Second, the +semi-discrete setting is intriguing from a statistical perspective: existing results show that +statistical estimation problems involving semi-discrete optimal transport can escape the curse +of dimensionality [FHN+19, dBL19, dBGSL22, HSM22]. For example, [HSM22, Theorem +3 + +O +O3.2] show that if Pn and Qn are empirical measures consisting of i.i.d. samples from P and Q, +then the semi-discrete assumption implies +E|S0(P, Q) − S0(Pn, Qn)| ≲ n−1/2 . +These results offer the tantalizing possibility that semi-discrete transport maps can be +estimated at the rate n−1/2, in sharp contrast to the dimension-dependent rates obtained +in bounds such as (2). However, the optimal rates of estimation for semi-discrete transport +maps are not known, and no estimators with finite-sample convergence guarantees exist. +Main Contributions +We show that the computationally efficient estimator ˆTε based on entropically regularized +optimal transport, originally studied in [SDF+18, PNW21], provably estimates discontinuous +semi-discrete optimal transport maps at the optimal rate. More precisely, our contributions +are the following: +1. For Q discrete and P with full support on a compact, convex set, we show that ˆTε +achieves the following dimension-independent convergence rate to the optimal transport +map (see Theorem 3.1) +E∥ ˆTε − ∇ϕ0∥2 +L2(P) ≲ n−1/2 , +(3) +when the regularization parameter ε ≍ n−1/2. We further show (Proposition 4.1) that +this rate is minimax optimal. +2. As a by-product of our analysis, we give new parametric rates of convergence to the +entropic Brenier map Tε, a result which improves exponentially on prior work in the +dependence on ε (see Theorem 3.5 and Remark 3.6). +3. Our proof technique requires several new results, including a novel stability bound for +the entropic Brenier maps (Proposition 3.7), and a new stability result for the entropic +dual Brenier potentials in the semi-discrete case (Proposition 3.9). +4. We show that, unlike ˆTε, the 1-Nearest-Neighbor estimator is provably suboptimal in +the semi-discrete setting (see Proposition 4.2) by exhibiting a discrete measure Q such +that the risk suffers from the curse of dimensionality: +E∥ ˆT1NN − ∇ϕ0∥2 +L2(P) ≳ n−1/d . +5. In Section 4, we verify our theoretical findings on synthetic experiments. We also show +by simulation that the entropic estimator appears to perform well even outside the +semi-discrete setting, suggesting it as a promising choice for estimating other types of +discontinuous maps. +4 + +2 +Background on optimal transport +2.1 +Optimal transport +We define P(Ω) to be the space of probability measures whose support lies in a compact +subset Ω ⊆ Rd. If a probability measure P has a density with respect to the Lebesgue +measure on Rd with support Ω ⊆ Rd, then we write P ∈ Pac(Ω). +For two probability measures P, Q ∈ P(Ω), we define the (squared) 2-Wasserstein distance +to be [Kan42] +S0(P, Q) := +min +π∈Γ(P,Q) +�� +1 +2∥x − y∥2 dπ(x, y) , +(4) +where π ∈ Γ(P, Q) ⊆ P(Ω × Ω) such that for any event A, +π(A × Ω) = P(A) , +π(Ω × A) = Q(A) . +We call Γ(P, Q) the set of couplings between P and Q. In this work, we focus on the +squared-Euclidean cost but Equation (4) is well-defined for convex, lower-semicontinuous +costs; see [Vil09, San15] for more information on optimal transport under general costs. +Equation (4) is a convex optimization problem on the space of joint measures, and a +minimizer, denoted π0, always exists; we call π0 an optimal plan from P to Q. Moreover, +Equation (4) possesses the following dual formulation, +S0(P, Q) = 1 +2M2(P) + 1 +2M2(Q) − +inf +(ϕ,ψ)∈Φ +� +ϕ dP + +� +ψ dQ +(5) +where M2(P) := +� +∥x∥2 dP(x) (similarly for M2(Q)) and the functions (ϕ, ψ) ∈ Φ ⊆ L1(P) × +L1(Q) satisfy +⟨x, y⟩ ≤ ϕ(x) + ψ(y) for all x, y ∈ Ω . +As with the primal formulation, the infimum in Equation (5) is attained at functions (ϕ0, ψ0). +These minimizers are called (optimal) Brenier potentials. In particular, at optimality, we +have that these Brenier potentials are convex conjugates of one another, i.e. the Legendre +transform of one of the potentials gives the other: +ϕ∗ +0(y) := sup +x {⟨x, y⟩ − ϕ0(x)} = ψ0(y) , +(6) +and vice-versa. +Apart from these two formulations of optimal transport under the squared-Euclidean cost, +there exists a third, known as the Monge problem: +T0 := argmin +T∈T (P,Q) +� +1 +2∥x − T(x)∥2 dP(x) , +(7) +where T (P, Q) is the set of admissible transport maps, i.e. for X ∼ P, T(X) ∼ Q. This +optimization problem is non-convex in T, and a solution is not always guaranteed to exist for +arbitrary P and Q. +The following theorem unifies these three formulations of optimal transport under the +squared-Euclidean cost: +5 + +Theorem 2.1 (Brenier’s theorem; Bre91). Let P ∈ Pac(Ω) and let Q ∈ P(Ω), then +1. the solution to Equation (7) exists and is of the form T0 = ∇ϕ0, where ϕ0 solves +Equation (5). +2. π0 is also uniquely defined as +dπ0(x, y) = dP(x)δ{∇ϕ0(x)}(y) . +When we want to place emphasis on the underlying measures, we will write ϕ0 = ϕP→Q +0 +, +ψ0 = ψP→Q +0 +and T0 = T P→Q +0 +. +2.1.1 +OT in the semi-discrete case +In optimal transport, the semi-discrete setting refers to the case where P has as density +with respect to the Lebesgue measure on Rd, and Q is a discrete measure supported on +points. The following theorem characterizes the optimal transport map in this situation, +which exhibits a particular structure compared to the general results in the previous section. +Let [J] = {1, . . . , J}. +Proposition 2.2 (AHA98). If P ∈ Pac(Ω) and Q is a discrete measure supported on the +points y1, . . . , yJ, then the optimal transport map ∇ϕ0 is given by +∇ϕ0(x) := argmax +j∈[J] +{⟨x, yj⟩ − ψ0(yj)} , +(8) +where ψ0 is the dual to ϕ0 in the sense of Equation (6). +Here, the optimal dual Brenier potential ψ0 can be identified with a vector in RJ, defined +by the number of atoms, and the optimal Brenier potential is consequently given by +ϕ0 := max +j∈[J]{⟨x, yj⟩ − ψ0(yj)} . +Although ϕ0 is not differentiable, only subdifferentiable, we still use the gradient notation as +∇ϕ0 is well-defined P-almost everywhere. +The map ∇ϕ0 partitions the space into J convex polytopes Lj := ∇ϕ−1 +0 ({yj}) called +Laguerre cells; recall Figure 1. From this definition, it is clear that for a given x ∈ Lj, +x �→ ∇ϕ0(x) = yj is the optimal transport mapping. The difficulty in finding this map lies in +determining the cells Lj, or equivalently the dual variables ψ0(yj). +2.2 +Entropic optimal transport +Entropic regularization was introduced to both optimal transport and machine learning +communities in the seminal paper by [Cut13], allowing approximate optimal transport +distances to be computed at unprecedented speeds. Entropic optimal transport (EOT) is +defined as the following regularized version of Equation (4): for ε > 0 +Sε(P, Q) := +min +π∈Γ(P,Q) +�� +1 +2∥x − y∥2 dπ(x, y) + εKL(π∥P ⊗ Q) , +(9) +6 + +where KL(µ∥ν) = +� +log dµ +dν dµ when µ ∈ P(Ω) is absolutely continuous with respect to +ν ∈ P(Ω). This speedup is due to the elegant connection of (9) to Sinkhorn’s algorithm; we +refer the interested reader to [PC19, Chapter 4] for more information. The computational +tractability of Sε compared to S0 when dealing with many samples lends itself to being a central +object of study in its own right [see, e.g., GCB+19, MNW19, CRL+20, RS22, GSLNW22]. +Equation (9) admits the following dual formulation, which is now an unconstrained +optimization problem [Gen19, MG20] +Sε(P, Q) = 1 +2M2(P) + 1 +2M2(Q) − inf +ϕ,ψ +� � +ϕ dP + +� +ψ dQ ++ ε +�� +(e(⟨x,y⟩−ϕ(x)−ψ(y))/ε − 1) dP(x) dQ(y) +� +, +(10) +where (ϕ, ψ) ∈ L1(P) × L1(Q). When P and Q have finite second moments, Equation (9) +admits a unique minimizer, πε and we have the existence of minimizers to Equation (10), +which we denote as (ϕε, ψε). We call πε the entropic optimal plan and (ϕε, ψε) are called +entropic Brenier potentials. The following optimality relation further relates these primal +and dual solutions [Csi75]: +dπε(x, y) := e(⟨x,y⟩−ϕε(x)−ψε(y))/ε dP(x) dQ(y) . +As a consequence, the following relationship holds at optimality: +Sε(P, Q) = 1 +2M2(P) + 1 +2M2(Q) − +� +ϕε dP − +� +ψε dQ , +and, moreover, we can define versions of ϕε and ψε such that the following relationships hold +[see MNW19, NW22] over all x ∈ Rd and y ∈ Rd, respectively: +ϕε(x) = ε log +� +e(⟨x,y⟩−ψε(y))/ε dQ(y) , +(11) +ψε(y) = ε log +� +e(⟨x,y⟩−ϕε(x))/ε dP(x) , +(12) +which are smoothed version of the Legendre transform, see Appendix A for details. In what +follows, we always assume that we have selected ϕε and ψε so that these identities hold. +2.2.1 +Entropic Brenier Map +If (X, Y ) ∼ πε, we may define the conditional probability πx +ε of Y given that X = x, with +density +dπx +ε +dQ (y) ∝ exp ((⟨x, y⟩ − ψε(y))/ε) . +(13) +The barycentric projection of the optimal entropic coupling πε, or entropic Brenier map, +is a central object of study in several works e.g. [GKRS22, PNW21, dBGSLNW22, RS22], +defined as +Tε(x) = +� +y dπx +ε(y) = ∇ϕε(x) , +(14) +7 + +where πx +ε is as in Equation (13). Note that this quantity is well defined for all x ∈ Rd as long +as the source and target measures have compact support; in particular, it applies to both +discrete and continuous measures. The second equality follows from Equation (11) and the +dominated convergence theorem. As in the unregularized case, we will write ϕε = ϕP→Q +ε +, +ψε = ψP→Q +ε +and Tε = T P→Q +ε +when we want to emphasize on the dependency with respect to +the underlying measures. +This particular barycentric projection was proposed as a tool for large-scale optimal +transport by [SDF+18], but analyzed statistically for the first time by [PNW21] as an +estimator for the optimal transport map. We mention some of their results to highlight the +differences with our new results for the semi-discrete setting in Section 3. First, they prove +the following approximation result for Tε. +Proposition 2.3 (PNW21, Corollary 1). Let P, Q be compactly supported absolutely contin- +uous measures on a compact set Ω ⊆ Rd with densities p and q, that are bounded away from +0 and ∞. Assume that ϕ0 is smooth and strongly convex, and that ϕ∗ +0 is at least C3. Then, +∥Tε − ∇ϕ0∥2 +L2(P) ≲ ε2 . +(15) +Their main statistical result is the following theorem: +Proposition 2.4 (PNW21, Theorem 3). Suppose the same assumptions as Proposition 2.3, +and let Pn and Qn denote the empirical measures of P and Q constructed from i.i.d. samples. +Let ˆTε = T Pn→Qn +ε +denote the entropic Brenier map from Pn to Qn and let T0 = ∇ϕ0 be the +optimal transport map from P to Q. Then, if ε ≍ n− +1 +d′+3 +E∥ ˆTε − T0∥2 +L2(P) ≲ n− +3 +2(d′+3) log(n) , +(16) +where d′ = 2⌈d/2⌉. +Note that in particular the the rate of convergence of the entropic estimator critically +depends on the ambient dimension d in the continuous-to-continuous case. +2.2.2 +Related work +Characterizing the convergence of entropic objects (e.g. potentials, cost, plans) to their +unregularized counterparts in the ε → 0 regime has been a topic of several works in recent years. +Convergence of the costs Sε to S0 with precise rates was investigated in [Pal19, CRL+20, CT21]. +The works [CDPS17, L´eo12, BGN22, GNB22] study the convergence of the minimizers πε to +π0 under varying assumptions. Convergence of the potentials in a very general setting was +established in [NW22], though without a rate of convergence in ε. In the semi-discrete case, +this gap was closed in [ANWS22] followed closely by [Del22], which gave non-asymptotic +rates. The Sinkhorn Divergence, a non-negative, symmetric version of Sε, was introduced +in [GPC18], was statistically analysed in [GKRS22] and also in [GSLNW22, dBGSLNW22], +and was connected to the entropic Brenier map in [PCNW22]. The recent preprint by [RS22] +proved parametetric rates of estimation between the empirical entropic Brenier map and its +population counterpart, though with an exponentially poor dependence on the regularization +parameter (see Remark 3.6). Using covariance inequalities, the entropic Brenier potentials +8 + +were used give a new proof of Caffarelli’s contraction theorem; see [CP22]; this approach was +recently generalized in [Con22]. Entropic optimal transport has also come into contact with +the area of deep generative modelling through the following works [FGOP20, DBTHD21], +among others. +3 +Statistical performance of the entropic estimator in +the semi-discrete setting +Let Pn and Qn be the empirical measures associated with two n-samples from P and Q. We +make the following regularity assumptions on P, already introduced by [Del22]. +(A) The measure P has a compact convex support Ω ⊆ B(0; R), with a density p satisfying +0 < pmin ≤ p ≤ pmax < ∞ for positive constants pmin, pmax and R. +For example, P can be the uniform distribution over Ω, or a truncated Gaussian distribution. +Furthermore, we will need the following assumption on Q. +(B) The discrete probability measure Q = �J +j=1 qjδyj is such that qj ≥ qmin > 0 and +yj ∈ B(0; R) for all j ∈ [J]. +The goal of this section is to prove the following theorem: +Theorem 3.1. Let P satisfy (A) and let Q satisfy (B). Let ˆTε = T Pn→Qn +ε +. Then, for +ε ≍ n−1/2 and n large enough, +E∥ ˆTε − T0∥2 +L2(P) ≲ n−1/2 . +(17) +Let Tε = T P→Q +ε +denote the entropic Brenier map associated to P and Q. Our proof relies +on the following bias-variance decomposition: +E∥ ˆTε − T0∥2 +L2(P) ≲ E∥ ˆTε − Tε∥2 +L2(P)+∥Tε − T0∥2 +L2(P). +Following the next two results (Theorem 3.2 and Theorem 3.5) and the preceding decomposi- +tion, the proof of Theorem 3.1 is merely a balancing act in the regularization parameter ε. +Theorem 3.2. Let P satisfy (A) and let Q satisfy (B). Then, for ε small enough, +∥Tε − T0∥2 +L2(P) ≲ ε . +(18) +The proof of Theorem 3.2 relies on the following qualitative picture: if a point x belongs +to some Laguerre cell Lj, and is far away from the boundary of Lj, then the entropic optimal +plan πε will send almost all of its mass towards the point yj = T0(x), sending an exponentially +small amount of mass to the other points yj. Such a picture is correct as long as x is at +distance at least ε from the boundary of the Laguerre cell Lj, incurring a total error of order +ε. A rigorous proof of Theorem 3.2 can be found in Appendix B. +Note that this rate is slower than the rate appearing in Proposition 2.3 in the continuous- +to-continuous case. The following example shows that the dependency in ε is optimal in +Theorem 3.2, indicating that the presence of discontinuities necessarily affects the approxima- +tion properties of the entropic Brenier map. +9 + +Example 3.3. Let P be a probability measure on R having a symmetric bounded density p +continuous at 0, and let Q = 1 +2(δ−1 + δ1). Following [ANWS22, Section 3], one can check that +the entropic Brenier map in this setting is the following scaled sigmoidal function +Tε(x) = tanh(2x/ε) , +whereas the optimal transport map T0(x) = sign(x). Then, performing a computation +∥Tε − T0∥2 +L2(P) = 2 +� ∞ +0 +(1 − tanh(2x/ε))2p(x) dx += ε +� ∞ +0 +(1 − tanh(u))2p(uε/2) du += εp(0)(log(4) − 1) + o(ε) , +where in the last step we invoked the dominated convergence theorem, and computed the +limiting integral. +Remark 3.4. Assumption (A) can be relaxed for Theorem 3.2 to hold. More precisely, it can +be replaced by Assumptions 2.2 and 2.9 of [ANWS22], that hold for unbounded measures +such as the normal distribution. +Finally, we present the sample-complexity result: +Theorem 3.5. Let P satisfy (A) and let Q satisfy (B). Then, for 0 < ε ≤ 1 such that +log(1/ε) ≲ n/ log(n) +E∥ ˆTε − Tε∥2 +L2(P) ≲ ε−1n−1 . +(19) +Remark 3.6. In [RS22], the authors show that if P and Q are merely compactly supported +with supp(P), supp(Q) ⊆ B(0; R), then +E∥ ˆTε − Tε∥2 +L2(P) ≲ ecR2/εε−1n−1 , +(20) +where c > 0 is some absolute positive constant. Thus, under the additional structural +assumptions of the semi-discrete formulation, we are able to significantly improve the rate of +convergence between the empirical and population entropic Brenier maps. +The proof of Theorem 3.5 relies on a novel stability result, reminiscent of [MBNWW21, +Theorem 6], which is of independent interest. We provide the proof in Appendix C. +Proposition 3.7. Let µ, ν, µ′, ν′ be four probability measures supported in B(0; R). Then the +entropic maps T µ→ν +ε +and T µ′→ν′ +ε +satisfy +ε +8R2∥T µ→ν +ε +− T µ′→ν′ +ε +∥2 +L2(µ) ≤ +� +(ϕµ′→ν′ +ε +− ϕµ→ν +ε +) dµ + +� +(ψµ′→ν′ +ε +− ψµ→ν +ε +) dν + εKL(ν∥ν′). +Remark 3.8. The right side of the bound in Proposition 3.7 is equal to +Sε(µ, ν) − Sε(µ′, ν′) + +� +f µ′→ν′ +ε +d(µ′ − µ) + +� +gµ′→ν′ +ε +d(ν′ − ν) + εKL(ν∥ν′) , +where f µ′→ν′ +ε += 1 +2∥ · ∥2 − ϕµ′→ν′ +ε +and gµ′→ν′ +ε += 1 +2∥ · ∥2 − ψµ′→ν′ +ε +. Proposition 3.7 is therefore the +entropic analogue of the stability bounds of [MBNWW21, Theorem 6] and [GS22, Lemma +5.1]. Unlike those results, Proposition 3.7 allows both the source and target measure to be +modified, and does not require any smoothness assumptions. +10 + +Proof sketch of Theorem 3.5 +To prove Theorem 3.5, we first consider the one-sample setting, where we assume that we only +have access to samples Y1, . . . , Yn ∼ Q, but we have full access to P. We then consider the +one-sample entropic estimator T P→Qn +ε +. We apply Proposition 3.7 with µ = µ′ := P, ν := Qn +and ν′ := Q, yielding (see Corollary C.1 for details) +ε +8R2E∥T P→Qn +ε +− Tε∥2 +L2(µ) ≤ E +� � +(ψε − ψP→Qn +ε +) d(Qn − Q) + εKL(Qn∥Q) +� +. +Let χ2(P∥Q) denote the χ2-divergence between probability measure. Young’s inequality +(see Lemma H.1) and the inequality KL(Qn∥Q) ≤ χ2(Qn∥Q) yield the following bound: +E∥T P→Qn +ε +− Tε∥2 +L2(P) ≤ 8R2 +ε +�E[VarQ(ψP→Qn +ε +− ψε)] +2 ++ E[χ2(Qn∥Q)] +2 +� ++ 8R2E[χ2(Qn∥Q)] . +To complete our proof sketch, we use a new stability result on the entropic dual Brenier +potentials, catered for the semi-discrete setting. +Proposition 3.9. Let µ be a measure that satisfies (A). Let ν, ν′ be two discrete probability +measures supported on {y1, . . . , yJ}, with ν′ ≥ λν for some λ > 0. Then, for 0 < ε ≤ 1, +Varν(ψµ→ν′ +ε +− ψµ→ν +ε +) ≤ C +λ2χ2(ν′∥ν), +(21) +where C depends on R, pmin and pmax. +Moreover, a computation provided in Lemma H.2 shows that E[χ2(Qn∥Q)] = J−1 +n , which +is enough to conclude the proof of the one-sample case, see Appendix E for details. The two- +sample setting is tackled using similar reasoning, where we ultimately prove in Appendix F +that the risk E∥ ˆTε − T P→Qn +ε +∥2 +L2(P) is upper bounded by +8R2 +ε E +� +(ϕP→Qn +ε +− ϕPn→Qn +ε +) d(Pn − P) . +Such a quantity can again be related to the estimation of the dual potentials ψP→Qn +ε +and +ψPn→Qn +ε +. Using the same reasoning as before, we expect a parametric rate of convergence for +this term as well. Merging the two results completes the proof of Theorem 3.5. We refer to +Appendix F for full details. +4 +Comparing against the 1NN estimator +4.1 +Rate optimality of the entropic Brenier map +The upper bound of Theorem 3.5 shows that our estimator achieves the n−1/2 rate. In fact, +the following simple proposition tells us that this rate is optimal in the semi-discrete case. +11 + +Proposition 4.1. Let P be the uniform distribution on [−1/2, 1/2]d and for any J ≥ 2, +let QJ denote the space of of probability measures with at most J atoms, supported on +[−1/2, 1/2]d. Define the minimax rate of estimation +Rn(QJ) = inf +ˆT +sup +Q∈QJ +EQ⊗n[∥ ˆT − T P→Q +0 +∥2 +L2(P)] . +Then, it holds that Rn(QJ) ≥ n−1/2/64. +Proof. Let e be a vector of the canonical basis of Rd, scaled by 1/2. Fix 0 < r < 1/2 and let +Q0 = 1 +2δ−e+ 1 +2δe and Q1 = ( 1 +2−r)δ−e+( 1 +2+r)δe. A computation gives ∥T P→Q0 +ε +−P→Q1 +ε +∥2 +L2(P) = r. +Therefore, by Le Cam’s lemma [see, e.g., Wai19, Chapter 15], +Rn(QJ,R) ≥ r +8(1 − dTV(Q⊗n +0 , Q⊗n +1 )). +(22) +Let dH2(Q0, Q1) denote the (squared) Hellinger distance between measures. +We have +dTV(Qn +0, Qn +1)2 ≤ dH2(Qn +0, Qn +1) ≤ ndH2(Q0, Q1). Furthermore, a computation gives +dH2(Q0, Q1) = +�� +1 +2 − r − +� +1 +2 +�2 ++ +�� +1 +2 + r − +� +1 +2 +�2 += 2 − ( +√ +1 + 2r + +√ +1 − 2r) +≤ 4r2. +We obtain the conclusion by picking r = n−1/2/4. +4.2 +The 1NN estimator is proveably suboptimal +The 1-Nearest-Neighbor estimator, henceforth denoted ˆT1NN, was proposed by [MBNWW21] +as a computational surrogate for estimating optimal transport maps in the low smoothness +regime. Written succinctly, their estimator is ˆT1NN(x) = �n +i=1 1Vi(x)Yˆπ(i), where (Vi)n +i=1 are +Voronoi regions i.e. +Vi := {x ∈ Rd : ∥x − Xi∥ ≤ ∥x − Xk∥ , ∀ k ̸= i} , +and ˆπ is the optimal transport plan between the empirical measures Pn and Qn, which amounts +to a permutation. Computing the closest Xi to a new sample x has runtime O(n log(n)), +though the complexity of this estimator is determined by computing the plan ˆπ, which takes +O(n3) time via, e.g., the Hungarian Algorithm [see PC19, Chapter 3]. +When ϕ0 is smooth and strongly convex, [MBNWW21] showed that, for d ≥ 5, +E∥ ˆT1NN − ∇ϕ0∥2 +L2(P) ≲ n−2/d . +In contrast to the rate optimality of the entropic Brenier map, we now show that ˆT1NN +is proveably suboptimal in the semi-discrete setting. Not only does it fail to recover the +minimax rate obtained by the entropic Brenier map, but its performance in fact degrades in +comparison to the smooth case. A proof appears in Appendix G. +Proposition 4.2. There exist a measure P satisfying (A) and a discrete measure Q satisfying +(B) such that for d ≥ 3 +E∥ ˆT1NN − T P→Q +0 +∥2 +L2(P) ≳ n−1/d . +12 + +4.3 +Experiments +We briefly verify our theoretical findings on synthetic experiments. To create the following +plots, we draw two sets of n i.i.d. points from P, (X1, . . . , Xn) and (X′ +1, . . . , X′ +n), and create +target points Yi = T0(X′ +i), where T0 is known to us in advance in order to generate the data. +Our estimators are computed on the data (X1, . . . , Xn) and (Y1, . . . , Yn), and we evaluate the +Mean-Squared error criterion +MSE( ˆT) = ∥ ˆT − T0∥2 +L2(P) +of a given map estimator ˆT using Monte Carlo integration, using 50000 newly sampled +points from P. We plot the means across 10 repeated trials, accompanied by their standard +deviations. +101 +102 +103 +n +10 +2 +10 +1 +MSE +T slope=-0.485 +T1NN slope=-0.174 +101 +102 +103 +n +10 +3 +10 +2 +10 +1 +MSE +T slope=-0.675 +T1NN slope=-0.081 +Figure 2: ˆTε versus ˆT1NN for: J = 2 and d = 10 (left), and J = 10 and d = 50 (right). +102 +103 +n +2 × 100 +3 × 100 +4 × 100 +6 × 100 +MSE +T slope=-0.294 +T1NN slope=-0.205 +Figure 3: ˆTε versus ˆT1NN in d = 10 for estimating the splitting map (23). +4.3.1 +Semi-discrete example +First consider P = Unif([0, 1]d) and create atoms {y1, . . . , yJ} by partitioning the points +along the first coordinate for all j ∈ [J]: +(yj)[1] = (j − 1/2) +J +, +(yj)[2] = · · · = (yj)[d] = 0.5 . +13 + +We choose uniform qj = 1/J for j ∈ [J]. In this case, it is easy to see that the optimal +transport map T0(x) is uniquely defined by the first coordinate of x1. Figure 2 illustrates +the rate-optimal performance of the entropic Brenier map, and the proveably suboptimal +performance of the 1-Nearest-Neighbor estimator. +4.3.2 +Discontinuous example +We turn our attention to a discontinuous transport map, where for x ∈ Rd, all the coordinates +are fixed except for the first one +T0(x) = 2sign(x[1]) ⊗ x[2] ⊗ · · · ⊗ x[d] . +(23) +We choose P = Unif([−1, 1]d) to exhibit a discontinuity in the data. Focusing on d = 10, we +see in Figure 3 that the entropic map estimator avoids the curse of dimensionality and enjoys +a faster convergence rate, with better constants. +5 +Conclusion +Understanding optimal transport maps in the semi-discrete case is a natural stepping-stone +to understanding the case for general discontinuous transport maps. In this work, we propose +a tractable, minimax optimal estimator of the Brenier map in the semi-discrete setting, where +the rate of estimation is dimension independent. To prove our result, we require several new +results and techniques, and, as a by-product of our analysis, give the first parametric rates of +estimation the entropic Brenier map, without exponential dependence in the regularization +parameter. Our synthetic experiments indicate that the entropic Brenier map might be useful +in estimating other variants of discontinuous transport maps, which constitutes an interesting +direction for future research. +Acknowledgements +AAP would like to thank Tudor Manole for fruitful discussions, and gratefully thanks funding +sources NSF Award 1922658, and Meta AI Research. JNW is thanks the Sloan Research +Fellowship and NSF grant DMS-2210583 +14 + +A +Reminders on semi-discrete entropic optimal trans- +port +We recall in this section some known results on entropic optimal transport that will be needed +later. Let µ, ν ∈ P(Ω), where Ω ⊂ B(0; R) is a compact set. +Lemma A.1 (GCB+19). The entropic potential (ϕµ→ν +ε +, ψµ→ν +ε +) have a bounded amplitude, in +the sense that +max +x∈Ω ϕµ→ν +ε +− min +x∈Ω ϕµ→ν +ε +≤ cR +(24) +for some absolute constant c, and similarly for ψµ→ν +ε +. +Assume now that ν = �J +j=1 νjδyj is a discrete measure. In this situation, only the values +of the dual potential ψµ→ν +ε +on the points y1, . . . , yJ are relevant. We therefore consider ψµ→ν +ε +as a vector in RJ. The potentials ϕµ→ν +ε +and ψµ→ν +ε +are dual of one another, in the sense of the +ε-Legendre transform. Given a finite measure ρ, the ε-Legendre transform of a function h +with respect to ρ is given by +Φρ +ε(h)(y) = ε log +� +e(⟨x,y⟩−h(x))/ε dρ(y). +(25) +Relations (11) and (12) express that ϕµ→ν +ε += Φν +ε(ψµ→ν +ε +) and vice-versa. In the semi-discrete +setting, it is also convenient to introduce the ε-Legendre transform with respect to the +counting measure σ on {y1, . . . , yJ}. For a vector ψ ∈ RJ, we have +Φε(ψ)(x) := Φσ +ε(ψ)(x) = ε log +� +e(⟨x,yj⟩−ψ(yj))/ε. +(26) +The Φε transform and the Φν +ε transform are linked through the relation +Φν +ε(ψ) = Φε( ˜ψ) +where +˜ψ(yj) = ψ(yj) − ε log νj, +(27) +while we call ˜ψ a shifted potential. With such a notation, the optimality condition on the +potentials can be rephrased. Let +F µ→ν +ε +: ψ ∈ RJ → +� +Φε(ψ) + +� +ψ dν . +(28) +Then, the function F µ→ν +ε +is minimized at ˜ψµ→ν +ε +. For ψ ∈ RJ and x ∈ Rd, we introduce the +probability measure supported on {y1, . . . , yJ} given by +∀i ∈ [J], +πx +ε[ψ](yi) = +e(⟨x,yi⟩−ψ(yi))/ε +�J +j=1 e(⟨x,yj⟩−ψ(yj))/ε = e(⟨x,yi⟩−Φε(ψ)(x)−ψ(yi))/ε. +(29) +A computation gives ∇F µ→ν +ε +(ψ) = +� +πx +ε[ψ] dµ(x) − ν, so that at optimality, we have +� +πx +ε[ ˜ψµ→ν +ε +] dµ(x) = ν. +(30) +In this case, πx +ε = πx +ε [ ˜ψµ→ν +ε +] is the conditional distribution of the second marginal of πε given +that the first is equal to x, as in Section 2.2.1. More generally, for any potential ψ, the first +order condition implies that ψ is equal to ˜ψ +µ→νψ +ε +, the optimal dual potential between µ an +νψ = +� +πx +ε[ψ] dµ(x). +15 + +B +Bound on the approximation error +Proof of Theorem 3.2. Let i, j ∈ [J]. We define the jth slack at x ∈ Li by +1 +2∆ij(x) = −⟨x, yj⟩ + ϕ0(x) + ψ0(yj). +(31) +As ϕ0 is the Legendre transform of ψ0, we have ∆ij(x) ≥ 0. If the cells Li and Lj have a +nonempty intersection, the set Hij(t) = {x ∈ Li : ∆ij(x) = t} represents the trace on Li of +the hyperplane spanned by the boundary between Li and Lj, shifted by t. It is stated in +[ANWS22] that for every nonnegative measurable function f : R → R, +� +Li +f(∆ij(x))p(x) dx = +1 +2∥yi − yj∥ +� ∞ +0 +f(t)hij(t) dt, +(32) +where hij(t) = +� +Hij(t) p(x) dHd−1(x) and Hd−1 is the (d − 1)-dimensional Hausdorff measure. +In particular, wij = hij(0) is the (weighted) surface of the boundary between the ith and jth +Laguerre cells (should it exist). Given x ∈ Li, let s(x) = minj̸=i +1 +2∆ij(x). When the point +x is sufficiently inside its Laguerre cell, the conditional probability πx +ε becomes extremely +concentrated around the point yi, as the next lemma shows. Note that πx +0 = δyi when x ∈ Li. +Lemma B.1. Let x ∈ Li. For ε small enough, it holds that for every j ∈ [J], |πx +ε(yj) − +πx +0(yj)| ≤ ce−s(x)/ε, where c depends on J, the distances ∥yi − yj∥ and on the quantities wij. +Such a result was already stated in [Del22, Corollary 2.2], although while requiring that +the source measure P has a H¨older continuous density. Only assumption (A) is needed here. +Proof. According to [ANWS22, Proposition 4.6], for ε small enough, +ε−1∥ ˜ψε − ψ0∥∞ ≤ C, +(33) +where ˜ψε is the shifted version of ψε (see (26)) and C depends on the distances ∥yi − yj∥ and +on the wijs. Following [Del22, Proof of Corollary 2.2] and (29), we have for j ̸= i +|πx +ε(yj) − πx +0(yj)| = πx +ε(yj) = +e(⟨x,yj⟩− ˜ψε(yj))/ε +�J +j′=1 e(⟨x,yj′⟩− ˜ψε(yj′))/ε ≤ e2C +e(⟨x,yj⟩−ψ0(yj))/ε +�J +j′=1 e(⟨x,yj′⟩−ψ0(yj′))/ε ≤ e2Ce−s(x)/ε. +A similar computation yields that |πx +ε(yi) − πx +0(yi)| = |πx +ε(yi) − 1| ≤ Je2Ce−s(x)/ε. +We can bound for any x ∈ Li, +∥Tε(x) − T0(x)∥ = ∥ +J +� +j=1 +yj(πx +ε(yj) − πx +0(yj))∥ ≤ c +J +� +j=1 +∥yj∥e−s(x)/ε. +(34) +Therefore, letting C′ denote a constant, which may depend on J, whose value may change +from line to line, we obtain +∥Tε − T0∥2 +L2(P) = +J +� +i=1 +� +Li +∥Tε(x) − T0(x)∥2 dP(x) ≤ C′ +J +� +i=1 +� +Li +J +� +j=1 +e−2s(x)/ε dP(x) +(35) +≤ C′ � +i̸=j +� +Li +e−∆ij(x)/ε dP(x) ≤ C′ � +i̸=j +1 +2∥yi − yj∥ +� ∞ +0 +e−t/εhij(t) dt , +(36) +16 + +where in the second equality, we used the definition of s(x). Assumption (A) ensures that +the functions hijs are bounded, which implies that the right-hand side in (36) is of order ε. +C +Stability of entropic transport plans +Proof of Proposition 3.7. Note that we may assume without loss of generality that ν ≪ ν′ +and that KL(ν∥ν′) < ∞, for otherwise the bound is vacuous. For notational convenience, we +omit the dependence on ε in the subscripts. +Write πµ,ν = γµ,ν(x, y)dµ(x)dν(y) for the entropic optimal plan between µ and ν, where +γµ,ν = exp +�1 +ε(⟨x, y⟩ − ϕµ→ν(x) − ψµ→ν(y)) +� +, +and analogously define +γµ′,ν′ = exp +�1 +ε(⟨x, y⟩ − ϕµ′→ν′(x) − ψµ′→ν′(y)) +� +. +Consider the measure γµ′,ν′(x, y) dµ(x) dν′(y). The first-order optimality condition for +(ϕµ′→ν′, ψµ′→ν′) implies that +� +γµ′,ν′(y) dν′(y) = 1 +∀x ∈ Ω , +(37) +so that γµ′,ν′(x, y) dν′(y) is a probability measure. Let us write dπx(y) = γµ,ν(x, y) dν(y) and +dρx(y) = γµ′,ν′(x, y) dν′(y). +We make the following observations: first, T µ→ν(x) = +� +y dπx(y) and T µ′→ν′(x) = +� +y dρx(y). Second, the support of ρx lies inside B(0; R); since any Lipschitz function f on +B(0; R) satisfies supx f(x) − infx f(x) ≤ 2R, Hoeffding’s lemma [see BLM13, Lemma 2.2] +implies that if f is Lipschitz and +� +f dρx = 0, then +� +etf dρx ≤ e2R2t2 +∀t ∈ R . +This implies [BG99, Theorem 3.1] that +W1(πx, ρx)2 ≤ 8R2KL(πx∥ρx) . +(38) +Third, Jensen’s inequality implies that for any coupling γ between πx and ρx, +� +∥y − y′∥ dγ(y, y′) ≥ +���� +� +(y − y′) dγ(y, y′) +���� = ∥T µ→ν(x) − T µ′→ν′(x)∥ , +(39) +so that in particular, ∥T µ→ν(x) − T µ′→ν′(x)∥ ≤ W1(πx, ρx). Combining these facts, we obtain +1 +8R2∥T µ→ν(x) − T µ′→ν′(x)∥2 ≤ KL(πx∥ρx) = +� +log +� γµ,ν +γµ′,ν′ (x, y) dν +dν′(y) +� +γµ,ν(x, y) dν(y) . +(40) +17 + +Integrating both sides of this equation with respect to µ yields +1 +8R2∥T µ→ν(x) − T µ′→ν′(x)∥2 +L2(µ) ≤ +� +log +� γµ,ν +γµ′,ν′ (x, y) dν +dν′(y) +� +dπµ,ν(x, y) . +(41) +Expanding the definition of γµ,ν and γµ′,ν′ and using that +� +log dν +dν′(y) dπµ,ν(x, y) = +� +log dν +dν′(y) dν(y) = KL(ν∥ν′) +yields the claim. +We now record two corollaries of this bound, which apply when either the source or the +target measures of the entropic maps agree. +Corollary C.1. For any µ, ν, ν′ supported in B(0; R), +1 +8R2∥T µ→ν +ε +− T µ→ν′ +ε +∥2 +L2(µ) ≤ ε−1 +� +(ψµ→ν′ +ε +− ψµ→ν +ε +) d(ν − ν′) + KL(ν∥ν′). +(42) +Proof. We apply Proposition 3.7 with µ = µ′, which yields (once again omitting the depen- +dency in ε) +1 +8R2∥T µ→ν +ε +−T µ→ν′ +ε +∥2 +L2(µ) ≤ ε−1 +�� +(ϕµ→ν′ − ϕµ→ν) dµ + +� +(ψµ→ν′ − ψµ→ν) dν +� ++KL(ν∥ν′) . +(43) +By definition, (ϕµ→ν′, ψµ→ν′) minimizes the expression +� +ϕ dµ + +� +ψ dν′ + ε +�� +e(⟨x,y⟩−ϕ(x)−ψ(y))/ε dµ(x) dν′(y) − ε , +so, recalling that +�� +e(⟨x,y⟩−ϕµ→ν′(x)−ψµ→ν′(y))/ε dµ(x) dν′(y) = 1, we have in particular +� +ϕµ→ν′ dµ + +� +ψµ→ν′ dν′ ≤ +� +ϕµ→ν dµ + +� +ψµ→ν dν′ + ε +�� +e(⟨x,y⟩−ϕµ→ν(x)−ψµ→ν(y))/ε dµ(x) dν′(y) − ε += +� +ϕµ→ν dµ + +� +ψµ→ν dν′ , +where we have used that the first-order optimality condition for (ϕµ→ν, ψµ→ν) implies that +�� +e(⟨x,y⟩−ϕµ→ν(x)−ψµ→ν(y))/ε dµ(x) dν′(y) = 1 as well (see (11)). This implies +� +(ϕµ→ν′ − ϕµ→ν) dµ ≤ − +� +(ψµ→ν′ − ψµ→ν) dν′ . +(44) +Applying this inequality to (43) yields +1 +8R2∥T µ→ν +ε +− T µ→ν′ +ε +∥2 +L2(µ) ≤ ε−1 +� +(ψµ→ν′ − ψµ→ν) d(ν − ν′) + KL(ν∥ν′). +18 + +Corollary C.2. For any µ, µ′, ν supported in B(0; R), +1 +8R2∥T µ→ν +ε +− T µ′→ν +ε +∥2 +L2(µ) ≤ ε−1 +� +(ϕµ′→ν +ε +− ϕµ→ν +ε +) d(µ − µ′) . +(45) +Proof. We apply Proposition 3.7 with ν = ν′, yielding (dropping the dependency on ε) +1 +8R2∥T µ→ν − T µ′→ν∥2 +L2(µ) ≤ ε−1 +�� +(ϕµ′→ν − ϕµ→ν) dµ + +� +(ψµ′→ν − ψµ→ν) dν +� +. +(46) +An argument analogous to the one used in the proof of Corollary C.1 gives the inequality +� +ϕµ′→ν dµ′ + +� +ψµ′→ν dν ≤ +� +ϕµ→ν dµ′ + +� +ψµ→ν dν , +(47) +or, equivalently, +� +(ψµ′→ν − ψµ→ν) dν ≤ − +� +(ϕµ′→ν − ϕµ→ν) dµ′ , +(48) +and combining this inequality with (46) proves the claim. +D +Strong convexity of the entropic semi-dual problem +Proposition D.1 (Strong convexity of F µ→ν +ε +). Let ν = �J +j=1 νjδyj be a measure supported +on {y1, . . . , yJ} ⊆ B(0; R) and let µ supported on a compact convex set Ω ⊆ B(0; R) with +a density p satisfying pmin ≤ p ≤ pmax for some pmax ≥ pmin > 0. For ψ ∈ RJ, define +νψ = +� +πx +ε(ψ) dµ(x) and assume that νψ ≥ λν for some 0 < λ ≤ 1. Then, we have for ε > 0 +F µ→ν +ε +(ψ) − min +ψ F µ→ν +ε +≥ Cλ · Varν(ψ − ψµ→ν +ε +), +(49) +where C = +� +e2R2 pmax +pmin + ε +�−1 pmin +pmax. +Proof. As µ and ε are fixed, we will simply write ψν instead of ψµ→ν +ε +, and write similarly +Fν = F µ→ν +ε +. Recall the definition (26) of the shifted potential ˜ψν(yj) = ψν(yj) − ε log νj. +According to [Del22, Theorem 3.2], the functional Fν is minimized at the vector ˜ψν, with +∀v ∈ RJ, +Varν(v) ≤ +� +e2R2 pmax +pmin ++ ε +� +v⊤∇2Fν( ˜ψν)v. +(50) +For t ∈ [0, 1], let ψt = ˜ψν + t(ψ − ˜ψν) and let νt = +� +πx +ε(ψt) dµ(x). The potential ψt is the +(shifted) entropic Brenier potential between µ and νt, so that it minimizes the functional Fνt +(see Appendix A). Also, note that ∇2Fν does not depend on ν, so that +v⊤∇2Fν(ψt)v = v⊤∇2Fνt(ψt)v ≥ +� +e2R2 pmax +pmin ++ ε +�−1 +Varνt(v). +(51) +Let v = ψ − ψµ→ν +ε +. A Taylor expansion of Fν gives +Fν(ψ) − Fν( ˜ψν) = +� 1 +0 +v⊤∇2Fν(ψt)v dt ≥ +� +e2R2 pmax +pmin ++ ε +�−1 � 1 +0 +Varνt(v) dt. +(52) +19 + +Lemma D.2. Write νt = �J +j=1 νt,jδyj. +Then, for all t ∈ [0, 1] and j ∈ [J], we have +νt,j ≥ pmin +pmaxν1−t +0,j νt +1,j. +This lemma is enough to conclude the proof. Indeed, ν1 = νψ ≥ λν, so that it implies +that Varνt(v) ≥ pmin +pmaxλVarν(v). +Proof of Lemma D.2. According to [Del22, Proof of Proposition 4.1], +Φε(ψt)(tx + (1 − t)y) ≤ tΦε( ˜ψµ→ν +ε +)(x) + (1 − t)Φε(ψ)(y). +(53) +Therefore, if we let ht(x) = e(⟨x,yj⟩−ψt(yj)−Φε(ψt)(x))/ε, then we have ht(tx + (1 − t)y) ≥ +h0(x)th1(y)1−t. By the Pr´ekopa-Leindler inequality, +νt,j = +� +ht(x) dµ(x) ≥ pmin +� +X +ht(x) dx +≥ pmin +�� +X +h0(x) dx +�t �� +X +h1(x) dx +�1−t +≥ pmin +pmax +ν1−t +0,j νt +1,j. +Proof of Proposition 3.9. As in the previous proof, we drop the ε and µ dependency in our +notation. Write νk = �J +j=1 νk,jδyj for k = 0, 1, and define as before the shifted potentials +˜ψνk(yj) = ψν1(yj) − ε log νk,j. Let θ > 0 be a parameter to fix. According to Proposition D.1, +Lemma H.1, and using the inequality Fν1( ˜ψν1) ≤ Fν1( ˜ψν0), we have +CλVarν0( ˜ψν1 − ˜ψν0) ≤ Fν0( ˜ψν1) − Fν0( ˜ψν0) +≤ Fν0( ˜ψν1) − Fν1( ˜ψν1) + Fν1( ˜ψν0) − Fν0( ˜ψν0) += +� +( ˜ψν1 − ˜ψν0)( dν0 − dν1) +≤ θ +2Varν0( ˜ψν1 − ˜ψν0) + 1 +2θχ2(ν1∥ν0). +We pick θ = Cλ to conclude that +Varν0( ˜ψν1 − ˜ψν0) ≤ +1 +(Cλ)2χ2(ν1∥ν0). +(54) +Therefore, using the inequality | log(a/b)| ≤ |a − b|/ min{a, b} for a, b > 0, +Varν0(ψ1 − ψ0) ≤ 2Varν0( ˜ψ1 − ˜ψ0) + 2 +J +� +j=1 +ν0,j +� +log +�ν1,j +ν0,j +��2 +≤ +2 +(Cλ)2χ2(ν1∥ν0) + 2 +J +� +j=1 +ν0,j +� +ν1,j − ν0,j +min{ν0,j, ν1,j} +�2 +≤ +2 +(Cλ)2χ2(ν1∥ν0) + 2 +λ2 +J +� +j=1 +1 +ν0,j +(ν1,j − ν0,j)2 ≤ +� +2 +(Cλ)2 + 2 +λ2 +� +χ2(ν1∥ν0). +20 + +E +Control of the fluctuations in the one-sample case +Lemma E.1 (Sample complexity in the one-sample case). Assume that P satisfy (A) and +that Q satisfy (B). Then, it holds that E∥T P→Qn +ε +− Tε∥2 +L2(P) ≲ ε−1n−1. +Proof. To ease notation, we write Tε,n = T P→Qn +ε +and ψε,n = ψP→Qn +ε +. As explained in Section 3, +the stability result Proposition 3.7 implies that +E∥Tε,n − Tε∥2 +L2(P) ≤ 8R2 +ε +�E[VarQ(ψε,n − ψε)] +2 ++ E[χ2(Qn∥Q)] +2 +� ++ 8R2E[χ2(Qn∥Q)] . +(55) +Write Q = �J +j=1 qjδyj and Qn = �J +j=1 ˆqjδyj, and introduce the event E = {∀j ∈ [J], ˆqj ≥ +qj/2}. If E is satisfied, we have Qn ≥ Q/2, so that Proposition 3.9 yields +VarQ(ψε,n − ψε) ≤ Cχ2(Qn∥Q). +(56) +If E is not satisfied, we use the fact that the entropic potentials have a bounded amplitude +(see Lemma A.1), to obtain that +VarQ(ψε,n − ψε) ≤ C′. +(57) +Lemma E.2. Let E be the event that Qn ≥ Q/2. Then P(Ec) ≤ Je−cqminn for some c > 0. +Proof. By [Ver18, Exercise 2.3.2], we have P(Ec) ≤ �J +j=1 P(ˆqj < qj/2) ≤ Je−cqminn for some +c > 0. +We obtain +E∥ ˆTε,n − Tε∥2 +L2(P) ≲ R2 +ε E[χ2(Qn∥Q)] + R2 +ε Je−cqminn ≲ ε−1n−1 +(58) +by Lemma H.2. +F +Control of the fluctuations in the two-sample case +The goal of this section is to prove Theorem 3.5. We will actually prove a more general +result, and show that for any discrete measure ν = �J +j=1 νjδyj supported on {y1, . . . , yJ} +with νj ≥ νmin > 0 for all j ∈ [J], we have for log(1/ε) ≲ n/ log(n), +E∥T Pn→ν +ε +− T P→ν +ε +∥2 +L2(P) ≲ ε−1n−1. +(59) +Theorem 3.5 follows from (59) by conditioning on Qn. Let E be the event that Qn ≥ Q/2. +Then, by Lemma E.2, +E∥ ˆTε − T P→Qn +ε +∥2 +L2(P) ≤ E +� +E[∥ ˆTε − T P→Qn +ε +∥2 +L2(P)|Qn]1{E} +� ++ R2P(Ec) +≤ Cε−1n−1 + R2Je−cqminn ≲ ε−1n−1. +We obtain Theorem 3.5 by combining this bound with Lemma E.1. +21 + +To prove (59), we first use Corollary C.2 which yields +E∥T Pn→ν +ε +− T P→ν +ε +∥2 +L2(P) ≤ 8R2ε−1E +� +(ϕPn→ν +ε +− ϕP→ν +ε +) d(Pn − P) += 8R2ε−1E +� +(Φε( ˜ψPn→ν +ε +) − Φε( ˜ψP→ν +ε +)) d(Pn − P), +(60) +where we recall that for a potential ψ, the shifted potential ˜ψ is given by ˜ψj = ψj − ε log νj. +The remainder of the proof consists in bounding this integral by using localization arguments +and standard bounds on suprema of empirical processes. Our first goal is to show that the +potential ψPn→ν +ε +is close to to the potential ψP→ν +ε +for the ∞-norm. It will be convenient to +work with the “L∞-variance” +Var∞(ψ) = inf +c∈R max +j∈[J] |ψ(yj) − c|2 = +�max ψ − min ψ +2 +�2 +. +(61) +As the measure ν is lower bounded, it holds that +Varν(ψ) ≥ νminVar∞(ψ). +(62) +Lemma F.1 (Supremum of ε-Legendre transforms). Let ψ0 be a fixed potential and let τ > 0. +Then, for all j ∈ [J], +E +� +sup +Var∞(ψ−ψ0)≤τ 2 +���� +� +(πx +ε(ψ)j − πx +ε(ψ0)j) d(P − Pn)(x) +���� +� +≤ C +� +J max{log(τ/ε), 1} +n +(63) +E +� +sup +Var∞(ψ−ψ0)≤τ 2 +���� +� +(Φε(ψ)(x) − Φε(ψ0))(x) d(P − Pn)(x) +���� +� +≤ Cτ +� +J +n +(64) +for some absolute constant C. +Proof. Let us prove the first inequality. +The functional πx +ε is invariant by translation: +πx +ε(ψ + c) = πx +ε(ψ) for all c ∈ R. This implies that +sup +Var∞(ψ−ψ0)≤τ 2 +���� +� +(Φε(ψ)(x) − Φε(ψ0))(x) d(P − Pn)(x) +���� += +sup +∥ψ−ψ0∥∞≤τ +���� +� +(Φε(ψ)(x) − Φε(ψ0))(x) d(P − Pn)(x) +���� . +For a metric space (A, d) and u > 0, we let N(u, A, d) be the covering number of A at scale +u, that is the smallest number of balls of radius u needed to cover A. Let B be the L∞-ball +of radius τ in RJ, centered at ψ0, and let ∥ · ∥∞ denote the ∞-norm. For 0 < u ≤ τ, we have +log N(u, B, ∥ · ∥∞) ≤ J log(τ/u). As the function ψ �→ πx +ε (ψ)j is ε−1-Lipschitz continuous for +every x ∈ Rd, we have for 0 < u ≤ τ/ε, +log N(u, {x �→ πx +ε(ψ)j : ψ ∈ B}, ∥ · ∥∞) ≤ J log(τ/(uε)) . +22 + +Remarking furthermore that 0 ≤ πx +ε(ψ)j ≤ 1 (so that the class of functions {x �→ πx +ε(ψ)j : +ψ ∈ B} admits the constant function 1 as an envelope function), we obtain the following +control using Lemma H.3: +E +� +sup +∥ψ−ψ0∥∞≤τ +���� +� +(πx +ε(ψ)j − πx +ε(ψ0)j)( dP − dPn)(x) +���� +� +≤ c0 +√n +� c1 +0 +� +J log 2N(u, {x �→ πx +ε(ψ)j : ψ ∈ B}, ∥ · ∥∞) du +≤ +� +c2J max{log(τ/ε), 1} +n +, +where c0, c1 and c2 are absolute constants, and the last line follows from arguing on whether +c1 < τ/ε or not. +The second inequality follows from the same argument, using that the function ψ �→ Φε(ψ) +is 1-Lipschitz continuous. Indeed, the functional Φε satisfies Φε(ψ + c) = Φε(ψ) + c for all +c ∈ R. Then the set {ψ : Var∞(ψ − ψ0) ≤ τ 2} is equal to the set {ψ + c : ψ ∈ B, c ∈ R}. +As +� +c d(P − Pn) = 0, we can therefore once again restrict the supremum to vectors ψ ∈ B. +Furthermore, an envelope function of the class {Φε(ψ) − Φε(ψ0) : ψ ∈ B} is the constant +function equal to τ. Therefore, by Lemma H.3, we obtain +E +� +sup +∥ψ−ψ0∥∞≤τ +���� +� +(Φε(ψ) − Φε(ψ0))( dP − dPn) +���� +� +≤ c0 +√n +� c1τ +0 +� +J log 2N(u, {Φε(ψ) : ψ ∈ B}, ∥ · ∥∞) du +≤ +� +c3Jτ +n +. +Proposition F.2. Assume that P satisfies (A) and let ν = �J +j=1 νjδyj be a measure supported +on {y1, . . . , yJ} ⊂ B(0; R), with νj ≥ qmin for all j ∈ [J]. Then, for all 0 < ε ≤ 1 with +log(1/ε) ≲ n/ log(n), it holds that +EVar∞( ˜ψPn→ν +ε +− ˜ψP→ν +ε +) ≲ n−1. +(65) +Proof. To alleviate notation, we will write ψn = ψPn→ν +ε +and ψ0 = ψP→ν +ε +. Similarly, we write +Fn = F Pn→ν +ε +and F0 = F P→ν +ε +. Let νn = +� +πx +ε (ψPn→ν +ε +) dP(x). Under the event E = {νn ≥ ν/2}, +we have according to Proposition D.1 and the fact that ˜ψn minimizes Fn, +CνminVar∞( ˜ψn − ˜ψ0) ≤ CVarν( ˜ψn − ˜ψ0) ≤ F0( ˜ψn) − F0( ˜ψ0) +≤ F0( ˜ψn) − Fn( ˜ψn) + Fn( ˜ψ0) − F0( ˜ψ0) += +� +(Φε( ˜ψn) − Φε( ˜ψ0)) d(P − Pn) +(66) +Let us bound P(Ec). As ˜ψn is the minimum of Fn, we have ν = +� +πx +ε( ˜ψn)j dPn(x) (see +Appendix A). Therefore, we may write νn,j = +� +πx +ε( ˜ψn)j dPn(x) + +� +πx +ε( ˜ψn)j d(P − Pn)(x) = +23 + +νj + Zj, where +Zj = +� +πx +ε( ˜ψn)j d(P − Pn)(x) = +� +(πx +ε( ˜ψn)j − πx +ε( ˜ψ0)j) d(P − Pn)(x). +Note that Var∞( ˜ψn − ˜ψ0) ≲ R2 (see Lemma A.1), so that by Lemma F.1 and Lemma H.3, +P(Ec) ≤ +J +� +j=1 +P(|Zj| > νj/2) ≤ J exp +� +−c +√nqmin +( +� +J log(1/ε) + log n +� +≲ n−1, +(67) +under the condition log(1/ε) ≲ n/ log(n). +For k ≥ 0, let ak = 2k/√n and fix some p > 2. Let +Ba = +sup +Var∞(ψ− ˜ψ0)≤a2 +���� +� +(Φε(ψ) − Φε( ˜ψ0)) d(P − Pn) +���� . +Assume that E is satisfied and that Var∞( ˜ψ0 − ˜ψn) ∈ [a2, b2]. Then, according to (66), it +holds that Bb ≥ ca2. Using Markov’s inequality, Lemma F.1 and Lemma H.3, we bound +EVar∞( ˜ψn − ˜ψ0) ≤ a2 +0 + +� +k≥0 +P(Var∞( ˜ψn − ˜ψ0) ∈ [a2 +k, a2 +k+1] and E)a2 +k+1 + CP(Ec) +≲ n−1 + +� +k≥0 +P +� +Bak+1 ≥ ca2 +k +� +a2 +k+1+ ≲ n−1 + +� +k≥0 +E[Bp +ak+1] +a2p +k +a2 +k+1+ +≲ n−1 + +� +k≥0 +(2k/n)p +(4k/n)p +4k+1 +n ++ P(Ec) ≲ n−1 + +� +k≥0 +22k−pk +n +≲ n−1. +Proposition F.3. Under the same assumptions than Proposition F.2, it holds that +E∥T Pn→ν +ε +− T P→ν +ε +∥2 +∞ ≲ ε−1n−1. +(68) +Proof. Let Z = Var∞( ˜ψn − ˜ψ0). Let once again ak = 2k/√n for k ≥ 1, with a0 = 0. Fix some +p > 2, with q = +p +p−1. For a > 0, let Da = supVar∞(ψ− ˜ψ0)≤a2 +��� +� +(Φε(ψ) − Φε( ˜ψ0)) d(P − Pn) +���. +By H¨older inequality and Markov inequality, we obtain, +E +� +(Φε( ˜ψn) − Φε( ˜ψ0)) d(P − Pn) +≤ +� +k≥0 +E +� +1{Z ∈ [a2 +k, a2 +k+1]} +sup +Var∞(ψ− ˜ψ0)≤a2 +k+1 +� +(Φε(ψ) − Φε( ˜ψ0)) d(P − Pn) +� +≤ E[Da1] + +� +k≥1 +� +P(Z ≥ a2 +k) +�1/q E +� +Dp +ak+1 +�1/p +≲ n−1 + +� +k≥0 +�E[Z] +a2 +k +�1/q 2k +n ≲ +� +k≥0 +2k(1−2/q) +n +≲ n−1, +where we use Proposition F.2, Lemma F.1 and Lemma H.3 at the last line. Equation (60) +then gives the conclusion. +24 + +G +A lower bound for the performance of the 1NN es- +timator +In this section, we prove Proposition 4.2. We let P be the Lebesgue measure on Ω = [0, 1]d, +and let y0 = (0, 1/2, . . . , 1/2) and y1 = (1, 1/2, . . . , 1/2). We denote by Pn an empirical +measure consisting of i.i.d. samples from Pn. As in Appendix F, we work in a general setting +of a generic discrete target measure ν, which may either be fixed or may be a random measure +independent of Pn. We let ν = � +j=0,1 νjδyj for ν0, ν1 ≥ 1 +4; this latter condition will hold with +overwhelming probability if ν is an empirical measure Qn corresponding to n i.i.d. samples +from Q = 1 +2δy0 + 1 +2δy1. Following [MBNWW21], we define the one-nearest neighbor estimator +ˆT1NN in this general context by +ˆT1NN(x) = +n +� +i=1 +� +j=0,1 +1Vi(x)(nˆπ(Xi, yj)) , +where ˆπ is the empirical optimal coupling between Pn and ν. +We first examine the structure of the Brenier map T0 = ∇ϕ0. The considerations in +Section 2.1.1 imply that +T0(x) = +� +y0 +⟨e1, x⟩ ≤ ν0 +y1 +⟨e1, x⟩ > ν0 , +where e1 is the first elementary basis vector. The potential ϕ0 is not differentiable on the +separating hyperplane ⟨e1, x⟩ = ν0, which has measure 0 under P, but we may arbitrarily +assign points on this hyperplane to y0. +Similar arguments imply that the empirical transport plan ˆπ between Pn and ν has the +following property: there exists a (random) threshold τ ∈ (0, 1) such that +ˆπ(x, y0) = +� +1 +⟨e1, x⟩ < τ +0 +⟨e1, x⟩ > τ . +The set ⟨e1, x⟩ = τ may not have measure 0 under Pn, and ˆπ(x, y0) may take values strictly +between 0 and 1 on this set. +The following lemma shows that τ is close to ν0 with high probability. +Lemma G.1. For any t ≥ 0, +P {τ ≥ ν0 + t} ≤ e−2nt2 . +Proof. If τ ≥ ν0 + t, this implies that Pn({x : ⟨e1, x⟩ < ν0 + t}) ≤ ν0. On the other hand, +nPn({x : ⟨e1, x⟩ < ν0 + t} is a Bin(n, ν0 + t) random variable. The result then follows from +Hoeffding’s inequality [BLM13, Theorem 2.8]. +Let us write H for the halfspace {x : ⟨e1, x⟩ ≤ ν0}, and ˆH for the halfspace {x : ⟨e1, x⟩ ≤ τ}. +Let x be any point in Ω such that x ∈ H. We are interested in the event that there exists an +element Xi ∈ {X1, . . . , Xn} such that a) x ∈ Vi and b) Xi ∈ ˆHc. Call this event E(x). On +this event, ˆT1NN(x) = y1 and T0(x) = y0, so ∥ ˆT1NN(x) − T0(x)∥2 = 1. +25 + +We therefore obtain +E∥ ˆT1NN − T0∥2 +L2(P) = E +� +∥ ˆT1NN(x) − T0(x)∥2 dP(x) +≥ E +� +H +∥ ˆT1NN(x) − T0(x)∥21{E(x)} dP(x) +≳ E +� +H +1{E(x)} dP(x) += +� +H +P {E(x)} dP(x) , +where the final equality follows from the Fubin–Tonelli theorem. +We now lower bound the probability of E(x). Let us write At for the event that τ < ν0 +t, +for t > 0 to be specified, and write Ht for the halfspace {x : ⟨e1, x⟩ ≤ ν0 + t}. Given any +x ∈ H, write ∆ = d(x, Hc +t ), and let B be a ball of radius 2∆ around x, intersected with Ω. +Denote by F(x) the event that there are no samples in V = B ∩ Ht but there is at least +one point in B ∩ Hc +t . Then F(x) ∩ At ⊆ E(x), since on F(x) the nearest neighbor to x must +be a sample in Hc +t , and on At we have Hc +t ⊆ ˆHc. +Lemma G.2. +P {F(x) ∩ At} ≥ (1 − vol(V ))n − (1 − vol(B))n − e−2nt2 . +Proof. We first compute P {F(x)}. +The probability that there are no samples in V is +(1 − vol(V ))n, and this event may be written as the disjoint union of F(x) and the event that +all of B is empty. The latter event has probability (1 − vol(B))n. Therefore +(1 − vol(V ))n = P {F(x)} + (1 − vol(B))n . +Since P {Ac +t} ≤ e−2nt2, the claim follows. +We need the following lemma. +Lemma G.3. Assume that ∆ > 0 and that d(x, ∂Ω) ≥ 2∆. There exist positive constants +cd,0 < 1 and cd,1 such that +vol(V ) ≤ cd,0 vol(B) +(69) +and +vol(B) ≥ cd,1∆d +(70) +Proof. This is immediate from a scaling argument: since d(x, ∂Ω) ≥ 2∆, the set B is a +Euclidean ball of radius 2∆, and the set V is a Euclidean ball of radius 2∆ minus a spherical +dome cut off by a hyperlane at distance ∆ from the center. When ∆ = 1, it is clear that the +claimed inequalities hold, and the general case is obtained by dilation. +We assume in what follows that d(x, ∂Ω) ≥ 2∆. The inequalities (1 + x)n ≥ 1 + nx and +ex ≤ 1 + x + x2, valid for all x ∈ [−1, 0] and n ≥ 1, imply that for any δ > 0 there exists a +constant cd,δ > 0 such that if ∆ ≤ cd,δn−1/d, then we will have +(1 − vol(V ))n ≥ 1 − ncd,0 vol(B) +(71) +(1 − vol(B))n ≤ e−n vol(B) ≤ 1 − (1 − δ)n vol(B) +(72) +26 + +Choosing δ sufficiently small, we obtain the existence of a small cd,3 > 0 such that if +∆ ≤ cd,3n−1/d, then +(1 − vol(V ))n − (1 − vol(B))n ≥ Cdn∆d . +Define ∆n = cd,4n−1/d. Putting it all together, consider the set +S = {x ∈ H ∩ Ω : ∆n/2 ≤ d(x, Hc +t ) ≤ ∆n, d(x, ∂Ω) ≥ 2∆n} . +The above considerations imply that P {E(x)} ≥ Cdn(∆n/2)d − e−2nt2 ≥ C′ +d − e−2nt2 for all +x ∈ S. Choosing t to be a sufficiently large constant multiple of n−1/2, we obtain +� +H +P {E(x)} dP(x) ≥ +� +S +P {E(x)} dP(x) ≳d vol(S) . +Since t ≍ n−1/2, we will have that t ≪ ∆n for n sufficiently large (as d ≥ 3). Therefore, for n +large enough, the set S contains the set +S′ = {x ∈ Ω : ν0−∆n+t ≤ ⟨e1, x⟩ ≤ ν0−∆n/2+t, 2∆n ≤ ⟨ej, x⟩ ≤ 1−2∆n +∀j = 2, . . . , d} . +Since vol(S′) ≳d ∆n ≳ n−1/d, the claim follows. +H +Auxiliary lemmas +Lemma H.1 (Young’s inequality). Let Q0, Q1 be probability measures with Q1 ≪ Q0 and let +f be a function. Then, for θ > 0, +� +f( dQ0 − dQ1) ≤ θVarQ0(f) +2 ++ χ2(Q1∥Q0) +2θ +. +(73) +Proof. Recall Young’s inequality: for a, b ∈ R, ab ≤ +a2 +2 + b2 +2 . +As the left-hand side is +invariant by translation, we may assume without loss of generality that +� +f dQ0 = 0, so that +VarQ0(f) = +� +f 2 dQ0. We write +� +f( dQ0 − dQ1) = +� +( +√ +θf) +� +1 − dQ1 +dQ0 +� +√ +θ +dQ0 ≤ θ +2 +� +f 2 dQ0 + 1 +2θ +� � +1 − dQ1 +dQ0 +�2 +dQ0 += θVarQ0(f) +2 ++ χ2(Q1∥Q0) +2θ +. +Lemma H.2 (Expectation of empirical χ2-divergence). Let Q = �J +j=1 qjδyj be a discrete +measure supported on J atoms, and let Qn denote its empirical measure, consisting of n +i.i.d. samples. Then, +E[χ2(Qn∥Q)] = J − 1 +n +. +(74) +27 + +Proof. We can write Qn = �J +j=1 ˆqjδyj, where ˆqj is a binomial random variable with parameters +n and qj. We obtain +χ2(Qn∥Q) = +J +� +j=1 +(ˆqj − qj)2 +qj +. +Taking expectations, our bound reads +E[χ2(Qn∥Q)] = +J +� +j=1 +Var(ˆqj) +qj += +J +� +j=1 +qj(1 − qj) +nqj += J − 1 +n +. +Lemma H.3 (Control of suprema of empirical processes). Let X1, . . . , Xn be an i.i.d. sample +from some probability measure P on Rd, with Pn the associated empirical measure. Consider +F a class of functions Rd → R with ∥f∥∞ ≤ A for all f ∈ F. For u > 0, let N(u) be the +u-covering numbers of F, that is the minimal number of balls of radius u for the ∥ · ∥∞-metric +required to cover F. Then, +E +� +sup +f∈F +���� +� +f d(Pn − P) +���� +� +≤ C0 +√n +� C1A +0 +� +log 2N(u) du =: +I +√n +(75) +for two positive absolute constants C0 and C1. Furthermore, for all t > 0, +P +� +sup +f∈F +���� +� +f d(Pn − P) +���� > t +� +≤ exp +� +− +C2 +√nt +I + A log n +� +, +(76) +for some positive absolute constant C2. Eventually, for all p ≥ 2, +E +� +sup +f∈F +���� +� +f d(Pn − P) +���� +p�1/p +≤ Cp +I + A +√n . +(77) +Proof. See [VW96, Theorem 2.14.2 and Theorem 2.14.5]. +References +[AHA98] Franz Aurenhammer, Friedrich Hoffmann, and Boris Aronov. Minkowski-type +theorems and least-squares clustering. 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An optimal trans- +port approach to causal inference. arXiv preprint arXiv:2108.05858, 2021. +[Ver18] Roman Vershynin. High-dimensional probability: An introduction with appli- +cations in data science, volume 47. Cambridge University Press, 2018. +[Vil09] C´edric Villani. Optimal transport: old and new, volume 338. Springer, 2009. +[VW96] Aad W Vaart and Jon A Wellner. Weak convergence and empirical processes +with applications to statistics. In Weak convergence and empirical processes, +pages 16–28. Springer, 1996. +[Wai19] Martin J Wainwright. High-dimensional statistics: A non-asymptotic view- +point, volume 48. Cambridge University Press, 2019. +33 + +[YDV+20] Karren Dai Yang, Karthik Damodaran, Saradha Venkatachalapathy, Ali C +Soylemezoglu, GV Shivashankar, and Caroline Uhler. Predicting cell lineages +using autoencoders and optimal transport. PLoS computational biology, +16(4):e1007828, 2020. +34 + diff --git a/ZtFIT4oBgHgl3EQfkyt3/content/tmp_files/load_file.txt b/ZtFIT4oBgHgl3EQfkyt3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c547588b0252d6bc324245b98b76a408c4e80ec7 --- /dev/null +++ b/ZtFIT4oBgHgl3EQfkyt3/content/tmp_files/load_file.txt @@ -0,0 +1,1028 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf,len=1027 +page_content='Minimax estimation of discontinuous optimal transport maps: The semi-discrete case Aram-Alexandre Pooladian1,∗, Vincent Divol3,∗, Jonathan Niles-Weed1,2 1Center for Data Science, New York University 2Courant Institute of Mathematical Sciences, New York University 3Universit´e Paris Dauphine - PSL aram-alexandre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='pooladian@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='edu,vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='divol@psl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='eu, jnw@cims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='edu January 27, 2023 Abstract We consider the problem of estimating the optimal transport map between two probability distributions, P and Q in Rd, on the basis of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We study a computationally efficient estimator initially proposed by [PNW21], based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate n−1/2, independent of dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 1 Introduction The theory of optimal transport (OT) defines a natural geometry on the space of probability measures [San15, Vil09] and has become ubiquitous in modern data-driven tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In this area, optimal transport maps are a central object of study: suppose P and Q are two probability distributions with finite second moments, with P having a density with respect to the Lebesegue measure on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, Brenier’s theorem (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1) states that there exists Pooladian and Divol contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='11302v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='ST] 26 Jan 2023 a convex function ϕ0 whose gradient defines a unique optimal transport map between P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This map is optimal in the sense that it minimizes the following objective function: ∇ϕ0 := argmin T∈T (P,Q) � 1 2∥x − T(x)∥2 dP(x) , (1) where T (P, Q) := {T : Rd → Rd | X ∼ P, T(X) ∼ Q} is the set of transport maps between P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The optimal value of the objective function in Equation (1) is called the (squared) 2-Wasserstein distance, written explicitly as S0(P, Q) = � 1 2∥x − ∇ϕ0(x)∥2 dP(x) , though a more general formulation is available (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Computing or approximating S0(P, Q) as well as ∇ϕ0 has found use in several academic communities, such as economics [CCG16, CGHH17, TGR21, GX21], computational biology [BSG+21, BKC22, LBG+22, SST+19, MSF+21, DSSS22, YDV+20], and computer vision [SPKS16, SDGP+15, FCVP17], among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Practitioners seldom have access to P or Q, but instead have access to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' samples X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , Xn ∼ P and Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , Yn ∼ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' On the basis of these samples, practitioners face both computational and statistical challenges when estimating ∇ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' From a theoretical perspective, the statistical task of estimating optimal transport maps has attracted much interest in the last few years [HR21, MVB+21, MBNWW21, DGS21, PNW21, DNWP22, GS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The first finite-sample analysis of this problem was performed by [HR21], who proposed an estimator for ∇ϕ0 under the assumption that ϕ0 is s + 1-times continuously differentiable, for s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' They showed that a wavelet-based estimator ˆϕW satisfies E∥∇ ˆϕW − ∇ϕ0∥2 L2(P) ≲ n− 2s 2s+d−2 log2(n) , and that this rate is minimax optimal up to logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Their analysis requires that P and Q have bounded densities with compact support Ω ⊆ Rd, and that ϕ0 be both strongly convex and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Implementing the estimator ˆϕW is computationally challenging even in moderate dimensions, and is practically infeasible for d > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Follow up work has proposed alternative estimators which improve upon ˆϕW either in computational efficiency or in the generality in which they apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Though these subsequent works go significantly beyond the setting considered by [HR21], none has eliminated the crucial assumption that ϕ0 is smooth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=', that the transport map ∇ϕ0 is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We highlight two estimators proposed in this line of work that are particularly practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' [MBNWW21] study the 1-Nearest Neighbor estimator ˆT1NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This estimator is obtained by solving the empirical optimal transport problem between the samples, which is then extended to a function defined on Rd using a projection scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' see Section 4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Given n samples from the source and target measures in Rd, ˆT1NN has a runtime of O(n3) via the Hungarian Algorithm [see PC19, Chapter 3], and, for d ≥ 5, achieves the rate E∥ ˆT1NN − ∇ϕ0∥2 L2(P) ≲ n− 2 d (2) whenever the optimal transport map ϕ0 is smooth and strongly convex, and under mild regularity conditions on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In another work, [PNW21] conducted a statistical analysis of 2 an estimator originally proposed by [SDF+18] based on entropic optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The efficiency of Sinkhorn’s algorithm for large-scale problems [Cut13, PC19] makes this estimator attractive from a computational perspective, and [PNW21] also give statistical guarantees, though these fall short of being minimax-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Despite this progress, none of the aforementioned results can be applied in situations where ∇ϕ0 is not Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' And in practice, even requiring the continuity of the transport map can be far too stringent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' It is indeed too much to hope for that an underlying data distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' over the space of images) has one single connected component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' this is supported by recent work that stipulates that the underlying data distribution is the union of disjoint manifolds of varying intrinsic dimension [BCR+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In such a setting, the transport map ∇ϕ0 will not be continuous, demonstrating the need of considering the problem of the statistical estimation of discontinuous transport maps to get closer to real-world situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As a first step, we choose to focus on the case where the target distribution Q = �J j=1 qjδyj is discrete while the source measure P has full support, often called the semi-discrete setting in the optimal transport literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In this setting, the optimal transport map ∇ϕ0 is constant over regions known as Laguerre cells (each cell corresponding to a different atom of the discrete measure), while displaying discontinuities on their boundaries (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Figure 1 provides such an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Semi-discrete optimal transport therefore provides a natural class of discontinuous transport maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Figure 1: An illustration of a semi-discrete optimal transport map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The support of P, the whole rectangle, is partitioned into regions, each of which is transported to one of the atoms of the discrete target measure Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The resulting map is discontinuous at the boundaries of each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We focus on this setting for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' First, it has garnered a lot of attention in recent years, in both computational and theoretical circles [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=', MSS21, ANWS22, CAN22], due in particular to its connection with the quantization problem [GL07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Second, the semi-discrete setting is intriguing from a statistical perspective: existing results show that statistical estimation problems involving semi-discrete optimal transport can escape the curse of dimensionality [FHN+19, dBL19, dBGSL22, HSM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For example, [HSM22, Theorem 3 O O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2] show that if Pn and Qn are empirical measures consisting of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' samples from P and Q, then the semi-discrete assumption implies E|S0(P, Q) − S0(Pn, Qn)| ≲ n−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' These results offer the tantalizing possibility that semi-discrete transport maps can be estimated at the rate n−1/2, in sharp contrast to the dimension-dependent rates obtained in bounds such as (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' However, the optimal rates of estimation for semi-discrete transport maps are not known, and no estimators with finite-sample convergence guarantees exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Main Contributions We show that the computationally efficient estimator ˆTε based on entropically regularized optimal transport, originally studied in [SDF+18, PNW21], provably estimates discontinuous semi-discrete optimal transport maps at the optimal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' More precisely, our contributions are the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For Q discrete and P with full support on a compact, convex set, we show that ˆTε achieves the following dimension-independent convergence rate to the optimal transport map (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1) E∥ ˆTε − ∇ϕ0∥2 L2(P) ≲ n−1/2 , (3) when the regularization parameter ε ≍ n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We further show (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1) that this rate is minimax optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As a by-product of our analysis, we give new parametric rates of convergence to the entropic Brenier map Tε, a result which improves exponentially on prior work in the dependence on ε (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Our proof technique requires several new results, including a novel stability bound for the entropic Brenier maps (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7), and a new stability result for the entropic dual Brenier potentials in the semi-discrete case (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We show that, unlike ˆTε, the 1-Nearest-Neighbor estimator is provably suboptimal in the semi-discrete setting (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2) by exhibiting a discrete measure Q such that the risk suffers from the curse of dimensionality: E∥ ˆT1NN − ∇ϕ0∥2 L2(P) ≳ n−1/d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In Section 4, we verify our theoretical findings on synthetic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We also show by simulation that the entropic estimator appears to perform well even outside the semi-discrete setting, suggesting it as a promising choice for estimating other types of discontinuous maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 4 2 Background on optimal transport 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 Optimal transport We define P(Ω) to be the space of probability measures whose support lies in a compact subset Ω ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' If a probability measure P has a density with respect to the Lebesgue measure on Rd with support Ω ⊆ Rd, then we write P ∈ Pac(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For two probability measures P, Q ∈ P(Ω), we define the (squared) 2-Wasserstein distance to be [Kan42] S0(P, Q) := min π∈Γ(P,Q) �� 1 2∥x − y∥2 dπ(x, y) , (4) where π ∈ Γ(P, Q) ⊆ P(Ω × Ω) such that for any event A, π(A × Ω) = P(A) , π(Ω × A) = Q(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We call Γ(P, Q) the set of couplings between P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In this work, we focus on the squared-Euclidean cost but Equation (4) is well-defined for convex, lower-semicontinuous costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' see [Vil09, San15] for more information on optimal transport under general costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Equation (4) is a convex optimization problem on the space of joint measures, and a minimizer, denoted π0, always exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' we call π0 an optimal plan from P to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Moreover, Equation (4) possesses the following dual formulation, S0(P, Q) = 1 2M2(P) + 1 2M2(Q) − inf (ϕ,ψ)∈Φ � ϕ dP + � ψ dQ (5) where M2(P) := � ∥x∥2 dP(x) (similarly for M2(Q)) and the functions (ϕ, ψ) ∈ Φ ⊆ L1(P) × L1(Q) satisfy ⟨x, y⟩ ≤ ϕ(x) + ψ(y) for all x, y ∈ Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As with the primal formulation, the infimum in Equation (5) is attained at functions (ϕ0, ψ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' These minimizers are called (optimal) Brenier potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In particular, at optimality, we have that these Brenier potentials are convex conjugates of one another, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' the Legendre transform of one of the potentials gives the other: ϕ∗ 0(y) := sup x {⟨x, y⟩ − ϕ0(x)} = ψ0(y) , (6) and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Apart from these two formulations of optimal transport under the squared-Euclidean cost, there exists a third, known as the Monge problem: T0 := argmin T∈T (P,Q) � 1 2∥x − T(x)∥2 dP(x) , (7) where T (P, Q) is the set of admissible transport maps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' for X ∼ P, T(X) ∼ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This optimization problem is non-convex in T, and a solution is not always guaranteed to exist for arbitrary P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The following theorem unifies these three formulations of optimal transport under the squared-Euclidean cost: 5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 (Brenier’s theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Bre91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let P ∈ Pac(Ω) and let Q ∈ P(Ω), then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' the solution to Equation (7) exists and is of the form T0 = ∇ϕ0, where ϕ0 solves Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' π0 is also uniquely defined as dπ0(x, y) = dP(x)δ{∇ϕ0(x)}(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' When we want to place emphasis on the underlying measures, we will write ϕ0 = ϕP→Q 0 , ψ0 = ψP→Q 0 and T0 = T P→Q 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 OT in the semi-discrete case In optimal transport, the semi-discrete setting refers to the case where P has as density with respect to the Lebesgue measure on Rd, and Q is a discrete measure supported on points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The following theorem characterizes the optimal transport map in this situation, which exhibits a particular structure compared to the general results in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let [J] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 (AHA98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' If P ∈ Pac(Ω) and Q is a discrete measure supported on the points y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ, then the optimal transport map ∇ϕ0 is given by ∇ϕ0(x) := argmax j∈[J] {⟨x, yj⟩ − ψ0(yj)} , (8) where ψ0 is the dual to ϕ0 in the sense of Equation (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Here, the optimal dual Brenier potential ψ0 can be identified with a vector in RJ, defined by the number of atoms, and the optimal Brenier potential is consequently given by ϕ0 := max j∈[J]{⟨x, yj⟩ − ψ0(yj)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Although ϕ0 is not differentiable, only subdifferentiable, we still use the gradient notation as ∇ϕ0 is well-defined P-almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The map ∇ϕ0 partitions the space into J convex polytopes Lj := ∇ϕ−1 0 ({yj}) called Laguerre cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' recall Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' From this definition, it is clear that for a given x ∈ Lj, x �→ ∇ϕ0(x) = yj is the optimal transport mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The difficulty in finding this map lies in determining the cells Lj, or equivalently the dual variables ψ0(yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 Entropic optimal transport Entropic regularization was introduced to both optimal transport and machine learning communities in the seminal paper by [Cut13], allowing approximate optimal transport distances to be computed at unprecedented speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Entropic optimal transport (EOT) is defined as the following regularized version of Equation (4): for ε > 0 Sε(P, Q) := min π∈Γ(P,Q) �� 1 2∥x − y∥2 dπ(x, y) + εKL(π∥P ⊗ Q) , (9) 6 where KL(µ∥ν) = � log dµ dν dµ when µ ∈ P(Ω) is absolutely continuous with respect to ν ∈ P(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This speedup is due to the elegant connection of (9) to Sinkhorn’s algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' we refer the interested reader to [PC19, Chapter 4] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The computational tractability of Sε compared to S0 when dealing with many samples lends itself to being a central object of study in its own right [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=', GCB+19, MNW19, CRL+20, RS22, GSLNW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Equation (9) admits the following dual formulation, which is now an unconstrained optimization problem [Gen19, MG20] Sε(P, Q) = 1 2M2(P) + 1 2M2(Q) − inf ϕ,ψ � � ϕ dP + � ψ dQ + ε �� (e(⟨x,y⟩−ϕ(x)−ψ(y))/ε − 1) dP(x) dQ(y) � , (10) where (ϕ, ψ) ∈ L1(P) × L1(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' When P and Q have finite second moments, Equation (9) admits a unique minimizer, πε and we have the existence of minimizers to Equation (10), which we denote as (ϕε, ψε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We call πε the entropic optimal plan and (ϕε, ψε) are called entropic Brenier potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The following optimality relation further relates these primal and dual solutions [Csi75]: dπε(x, y) := e(⟨x,y⟩−ϕε(x)−ψε(y))/ε dP(x) dQ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As a consequence, the following relationship holds at optimality: Sε(P, Q) = 1 2M2(P) + 1 2M2(Q) − � ϕε dP − � ψε dQ , and, moreover, we can define versions of ϕε and ψε such that the following relationships hold [see MNW19, NW22] over all x ∈ Rd and y ∈ Rd, respectively: ϕε(x) = ε log � e(⟨x,y⟩−ψε(y))/ε dQ(y) , (11) ψε(y) = ε log � e(⟨x,y⟩−ϕε(x))/ε dP(x) , (12) which are smoothed version of the Legendre transform, see Appendix A for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In what follows, we always assume that we have selected ϕε and ψε so that these identities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 Entropic Brenier Map If (X, Y ) ∼ πε, we may define the conditional probability πx ε of Y given that X = x, with density dπx ε dQ (y) ∝ exp ((⟨x, y⟩ − ψε(y))/ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (13) The barycentric projection of the optimal entropic coupling πε, or entropic Brenier map, is a central object of study in several works e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' [GKRS22, PNW21, dBGSLNW22, RS22], defined as Tε(x) = � y dπx ε(y) = ∇ϕε(x) , (14) 7 where πx ε is as in Equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Note that this quantity is well defined for all x ∈ Rd as long as the source and target measures have compact support;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' in particular, it applies to both discrete and continuous measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The second equality follows from Equation (11) and the dominated convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As in the unregularized case, we will write ϕε = ϕP→Q ε , ψε = ψP→Q ε and Tε = T P→Q ε when we want to emphasize on the dependency with respect to the underlying measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This particular barycentric projection was proposed as a tool for large-scale optimal transport by [SDF+18], but analyzed statistically for the first time by [PNW21] as an estimator for the optimal transport map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We mention some of their results to highlight the differences with our new results for the semi-discrete setting in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' First, they prove the following approximation result for Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3 (PNW21, Corollary 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let P, Q be compactly supported absolutely contin- uous measures on a compact set Ω ⊆ Rd with densities p and q, that are bounded away from 0 and ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Assume that ϕ0 is smooth and strongly convex, and that ϕ∗ 0 is at least C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, ∥Tε − ∇ϕ0∥2 L2(P) ≲ ε2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (15) Their main statistical result is the following theorem: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='4 (PNW21, Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Suppose the same assumptions as Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3, and let Pn and Qn denote the empirical measures of P and Q constructed from i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let ˆTε = T Pn→Qn ε denote the entropic Brenier map from Pn to Qn and let T0 = ∇ϕ0 be the optimal transport map from P to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, if ε ≍ n− 1 d′+3 E∥ ˆTε − T0∥2 L2(P) ≲ n− 3 2(d′+3) log(n) , (16) where d′ = 2⌈d/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Note that in particular the the rate of convergence of the entropic estimator critically depends on the ambient dimension d in the continuous-to-continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 Related work Characterizing the convergence of entropic objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' potentials, cost, plans) to their unregularized counterparts in the ε → 0 regime has been a topic of several works in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Convergence of the costs Sε to S0 with precise rates was investigated in [Pal19, CRL+20, CT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The works [CDPS17, L´eo12, BGN22, GNB22] study the convergence of the minimizers πε to π0 under varying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Convergence of the potentials in a very general setting was established in [NW22], though without a rate of convergence in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In the semi-discrete case, this gap was closed in [ANWS22] followed closely by [Del22], which gave non-asymptotic rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The Sinkhorn Divergence, a non-negative, symmetric version of Sε, was introduced in [GPC18], was statistically analysed in [GKRS22] and also in [GSLNW22, dBGSLNW22], and was connected to the entropic Brenier map in [PCNW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The recent preprint by [RS22] proved parametetric rates of estimation between the empirical entropic Brenier map and its population counterpart, though with an exponentially poor dependence on the regularization parameter (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Using covariance inequalities, the entropic Brenier potentials 8 were used give a new proof of Caffarelli’s contraction theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' see [CP22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' this approach was recently generalized in [Con22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Entropic optimal transport has also come into contact with the area of deep generative modelling through the following works [FGOP20, DBTHD21], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 3 Statistical performance of the entropic estimator in the semi-discrete setting Let Pn and Qn be the empirical measures associated with two n-samples from P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We make the following regularity assumptions on P, already introduced by [Del22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (A) The measure P has a compact convex support Ω ⊆ B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R), with a density p satisfying 0 < pmin ≤ p ≤ pmax < ∞ for positive constants pmin, pmax and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For example, P can be the uniform distribution over Ω, or a truncated Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Furthermore, we will need the following assumption on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (B) The discrete probability measure Q = �J j=1 qjδyj is such that qj ≥ qmin > 0 and yj ∈ B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R) for all j ∈ [J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The goal of this section is to prove the following theorem: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let P satisfy (A) and let Q satisfy (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let ˆTε = T Pn→Qn ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, for ε ≍ n−1/2 and n large enough, E∥ ˆTε − T0∥2 L2(P) ≲ n−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (17) Let Tε = T P→Q ε denote the entropic Brenier map associated to P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Our proof relies on the following bias-variance decomposition: E∥ ˆTε − T0∥2 L2(P) ≲ E∥ ˆTε − Tε∥2 L2(P)+∥Tε − T0∥2 L2(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Following the next two results (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5) and the preceding decomposi- tion, the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 is merely a balancing act in the regularization parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let P satisfy (A) and let Q satisfy (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, for ε small enough, ∥Tε − T0∥2 L2(P) ≲ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (18) The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 relies on the following qualitative picture: if a point x belongs to some Laguerre cell Lj, and is far away from the boundary of Lj, then the entropic optimal plan πε will send almost all of its mass towards the point yj = T0(x), sending an exponentially small amount of mass to the other points yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Such a picture is correct as long as x is at distance at least ε from the boundary of the Laguerre cell Lj, incurring a total error of order ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' A rigorous proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Note that this rate is slower than the rate appearing in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3 in the continuous- to-continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The following example shows that the dependency in ε is optimal in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2, indicating that the presence of discontinuities necessarily affects the approxima- tion properties of the entropic Brenier map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 9 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let P be a probability measure on R having a symmetric bounded density p continuous at 0, and let Q = 1 2(δ−1 + δ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Following [ANWS22, Section 3], one can check that the entropic Brenier map in this setting is the following scaled sigmoidal function Tε(x) = tanh(2x/ε) , whereas the optimal transport map T0(x) = sign(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, performing a computation ∥Tε − T0∥2 L2(P) = 2 � ∞ 0 (1 − tanh(2x/ε))2p(x) dx = ε � ∞ 0 (1 − tanh(u))2p(uε/2) du = εp(0)(log(4) − 1) + o(ε) , where in the last step we invoked the dominated convergence theorem, and computed the limiting integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Assumption (A) can be relaxed for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' More precisely, it can be replaced by Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='9 of [ANWS22], that hold for unbounded measures such as the normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Finally, we present the sample-complexity result: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let P satisfy (A) and let Q satisfy (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, for 0 < ε ≤ 1 such that log(1/ε) ≲ n/ log(n) E∥ ˆTε − Tε∥2 L2(P) ≲ ε−1n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (19) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In [RS22], the authors show that if P and Q are merely compactly supported with supp(P), supp(Q) ⊆ B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R), then E∥ ˆTε − Tε∥2 L2(P) ≲ ecR2/εε−1n−1 , (20) where c > 0 is some absolute positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Thus, under the additional structural assumptions of the semi-discrete formulation, we are able to significantly improve the rate of convergence between the empirical and population entropic Brenier maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5 relies on a novel stability result, reminiscent of [MBNWW21, Theorem 6], which is of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We provide the proof in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let µ, ν, µ′, ν′ be four probability measures supported in B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then the entropic maps T µ→ν ε and T µ′→ν′ ε satisfy ε 8R2∥T µ→ν ε − T µ′→ν′ ε ∥2 L2(µ) ≤ � (ϕµ′→ν′ ε − ϕµ→ν ε ) dµ + � (ψµ′→ν′ ε − ψµ→ν ε ) dν + εKL(ν∥ν′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The right side of the bound in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7 is equal to Sε(µ, ν) − Sε(µ′, ν′) + � f µ′→ν′ ε d(µ′ − µ) + � gµ′→ν′ ε d(ν′ − ν) + εKL(ν∥ν′) , where f µ′→ν′ ε = 1 2∥ · ∥2 − ϕµ′→ν′ ε and gµ′→ν′ ε = 1 2∥ · ∥2 − ψµ′→ν′ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7 is therefore the entropic analogue of the stability bounds of [MBNWW21, Theorem 6] and [GS22, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Unlike those results, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7 allows both the source and target measure to be modified, and does not require any smoothness assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 10 Proof sketch of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5 To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5, we first consider the one-sample setting, where we assume that we only have access to samples Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , Yn ∼ Q, but we have full access to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We then consider the one-sample entropic estimator T P→Qn ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7 with µ = µ′ := P, ν := Qn and ν′ := Q, yielding (see Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 for details) ε 8R2E∥T P→Qn ε − Tε∥2 L2(µ) ≤ E � � (ψε − ψP→Qn ε ) d(Qn − Q) + εKL(Qn∥Q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let χ2(P∥Q) denote the χ2-divergence between probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Young’s inequality (see Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1) and the inequality KL(Qn∥Q) ≤ χ2(Qn∥Q) yield the following bound: E∥T P→Qn ε − Tε∥2 L2(P) ≤ 8R2 ε �E[VarQ(ψP→Qn ε − ψε)] 2 + E[χ2(Qn∥Q)] 2 � + 8R2E[χ2(Qn∥Q)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' To complete our proof sketch, we use a new stability result on the entropic dual Brenier potentials, catered for the semi-discrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let µ be a measure that satisfies (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let ν, ν′ be two discrete probability measures supported on {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ}, with ν′ ≥ λν for some λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, for 0 < ε ≤ 1, Varν(ψµ→ν′ ε − ψµ→ν ε ) ≤ C λ2χ2(ν′∥ν), (21) where C depends on R, pmin and pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Moreover, a computation provided in Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 shows that E[χ2(Qn∥Q)] = J−1 n , which is enough to conclude the proof of the one-sample case, see Appendix E for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The two- sample setting is tackled using similar reasoning, where we ultimately prove in Appendix F that the risk E∥ ˆTε − T P→Qn ε ∥2 L2(P) is upper bounded by 8R2 ε E � (ϕP→Qn ε − ϕPn→Qn ε ) d(Pn − P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Such a quantity can again be related to the estimation of the dual potentials ψP→Qn ε and ψPn→Qn ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Using the same reasoning as before, we expect a parametric rate of convergence for this term as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Merging the two results completes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We refer to Appendix F for full details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 4 Comparing against the 1NN estimator 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 Rate optimality of the entropic Brenier map The upper bound of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5 shows that our estimator achieves the n−1/2 rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In fact, the following simple proposition tells us that this rate is optimal in the semi-discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 11 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let P be the uniform distribution on [−1/2, 1/2]d and for any J ≥ 2, let QJ denote the space of of probability measures with at most J atoms, supported on [−1/2, 1/2]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Define the minimax rate of estimation Rn(QJ) = inf ˆT sup Q∈QJ EQ⊗n[∥ ˆT − T P→Q 0 ∥2 L2(P)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, it holds that Rn(QJ) ≥ n−1/2/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let e be a vector of the canonical basis of Rd, scaled by 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Fix 0 < r < 1/2 and let Q0 = 1 2δ−e+ 1 2δe and Q1 = ( 1 2−r)δ−e+( 1 2+r)δe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' A computation gives ∥T P→Q0 ε −P→Q1 ε ∥2 L2(P) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Therefore, by Le Cam’s lemma [see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=', Wai19, Chapter 15], Rn(QJ,R) ≥ r 8(1 − dTV(Q⊗n 0 , Q⊗n 1 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (22) Let dH2(Q0, Q1) denote the (squared) Hellinger distance between measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We have dTV(Qn 0, Qn 1)2 ≤ dH2(Qn 0, Qn 1) ≤ ndH2(Q0, Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Furthermore, a computation gives dH2(Q0, Q1) = �� 1 2 − r − � 1 2 �2 + �� 1 2 + r − � 1 2 �2 = 2 − ( √ 1 + 2r + √ 1 − 2r) ≤ 4r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We obtain the conclusion by picking r = n−1/2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 The 1NN estimator is proveably suboptimal The 1-Nearest-Neighbor estimator, henceforth denoted ˆT1NN, was proposed by [MBNWW21] as a computational surrogate for estimating optimal transport maps in the low smoothness regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Written succinctly, their estimator is ˆT1NN(x) = �n i=1 1Vi(x)Yˆπ(i), where (Vi)n i=1 are Voronoi regions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Vi := {x ∈ Rd : ∥x − Xi∥ ≤ ∥x − Xk∥ , ∀ k ̸= i} , and ˆπ is the optimal transport plan between the empirical measures Pn and Qn, which amounts to a permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Computing the closest Xi to a new sample x has runtime O(n log(n)), though the complexity of this estimator is determined by computing the plan ˆπ, which takes O(n3) time via, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=', the Hungarian Algorithm [see PC19, Chapter 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' When ϕ0 is smooth and strongly convex, [MBNWW21] showed that, for d ≥ 5, E∥ ˆT1NN − ∇ϕ0∥2 L2(P) ≲ n−2/d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In contrast to the rate optimality of the entropic Brenier map, we now show that ˆT1NN is proveably suboptimal in the semi-discrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Not only does it fail to recover the minimax rate obtained by the entropic Brenier map, but its performance in fact degrades in comparison to the smooth case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' A proof appears in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' There exist a measure P satisfying (A) and a discrete measure Q satisfying (B) such that for d ≥ 3 E∥ ˆT1NN − T P→Q 0 ∥2 L2(P) ≳ n−1/d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3 Experiments We briefly verify our theoretical findings on synthetic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' To create the following plots, we draw two sets of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' points from P, (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , Xn) and (X′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , X′ n), and create target points Yi = T0(X′ i), where T0 is known to us in advance in order to generate the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Our estimators are computed on the data (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , Xn) and (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , Yn), and we evaluate the Mean-Squared error criterion MSE( ˆT) = ∥ ˆT − T0∥2 L2(P) of a given map estimator ˆT using Monte Carlo integration, using 50000 newly sampled points from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We plot the means across 10 repeated trials, accompanied by their standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 101 102 103 n 10 2 10 1 MSE T slope=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='485 T1NN slope=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='174 101 102 103 n 10 3 10 2 10 1 MSE T slope=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='675 T1NN slope=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='081 Figure 2: ˆTε versus ˆT1NN for: J = 2 and d = 10 (left), and J = 10 and d = 50 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 102 103 n 2 × 100 3 × 100 4 × 100 6 × 100 MSE T slope=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='294 T1NN slope=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='205 Figure 3: ˆTε versus ˆT1NN in d = 10 for estimating the splitting map (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 Semi-discrete example First consider P = Unif([0, 1]d) and create atoms {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ} by partitioning the points along the first coordinate for all j ∈ [J]: (yj)[1] = (j − 1/2) J , (yj)[2] = · · · = (yj)[d] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 13 We choose uniform qj = 1/J for j ∈ [J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In this case, it is easy to see that the optimal transport map T0(x) is uniquely defined by the first coordinate of x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Figure 2 illustrates the rate-optimal performance of the entropic Brenier map, and the proveably suboptimal performance of the 1-Nearest-Neighbor estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 Discontinuous example We turn our attention to a discontinuous transport map, where for x ∈ Rd, all the coordinates are fixed except for the first one T0(x) = 2sign(x[1]) ⊗ x[2] ⊗ · · · ⊗ x[d] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (23) We choose P = Unif([−1, 1]d) to exhibit a discontinuity in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Focusing on d = 10, we see in Figure 3 that the entropic map estimator avoids the curse of dimensionality and enjoys a faster convergence rate, with better constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 5 Conclusion Understanding optimal transport maps in the semi-discrete case is a natural stepping-stone to understanding the case for general discontinuous transport maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In this work, we propose a tractable, minimax optimal estimator of the Brenier map in the semi-discrete setting, where the rate of estimation is dimension independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' To prove our result, we require several new results and techniques, and, as a by-product of our analysis, give the first parametric rates of estimation the entropic Brenier map, without exponential dependence in the regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Our synthetic experiments indicate that the entropic Brenier map might be useful in estimating other variants of discontinuous transport maps, which constitutes an interesting direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Acknowledgements AAP would like to thank Tudor Manole for fruitful discussions, and gratefully thanks funding sources NSF Award 1922658, and Meta AI Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' JNW is thanks the Sloan Research Fellowship and NSF grant DMS-2210583 14 A Reminders on semi-discrete entropic optimal trans- port We recall in this section some known results on entropic optimal transport that will be needed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let µ, ν ∈ P(Ω), where Ω ⊂ B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R) is a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 (GCB+19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The entropic potential (ϕµ→ν ε , ψµ→ν ε ) have a bounded amplitude, in the sense that max x∈Ω ϕµ→ν ε − min x∈Ω ϕµ→ν ε ≤ cR (24) for some absolute constant c, and similarly for ψµ→ν ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Assume now that ν = �J j=1 νjδyj is a discrete measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In this situation, only the values of the dual potential ψµ→ν ε on the points y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We therefore consider ψµ→ν ε as a vector in RJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The potentials ϕµ→ν ε and ψµ→ν ε are dual of one another, in the sense of the ε-Legendre transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Given a finite measure ρ, the ε-Legendre transform of a function h with respect to ρ is given by Φρ ε(h)(y) = ε log � e(⟨x,y⟩−h(x))/ε dρ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (25) Relations (11) and (12) express that ϕµ→ν ε = Φν ε(ψµ→ν ε ) and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In the semi-discrete setting, it is also convenient to introduce the ε-Legendre transform with respect to the counting measure σ on {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For a vector ψ ∈ RJ, we have Φε(ψ)(x) := Φσ ε(ψ)(x) = ε log � e(⟨x,yj⟩−ψ(yj))/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (26) The Φε transform and the Φν ε transform are linked through the relation Φν ε(ψ) = Φε( ˜ψ) where ˜ψ(yj) = ψ(yj) − ε log νj, (27) while we call ˜ψ a shifted potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' With such a notation, the optimality condition on the potentials can be rephrased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let F µ→ν ε : ψ ∈ RJ → � Φε(ψ) + � ψ dν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (28) Then, the function F µ→ν ε is minimized at ˜ψµ→ν ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For ψ ∈ RJ and x ∈ Rd, we introduce the probability measure supported on {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ} given by ∀i ∈ [J], πx ε[ψ](yi) = e(⟨x,yi⟩−ψ(yi))/ε �J j=1 e(⟨x,yj⟩−ψ(yj))/ε = e(⟨x,yi⟩−Φε(ψ)(x)−ψ(yi))/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (29) A computation gives ∇F µ→ν ε (ψ) = � πx ε[ψ] dµ(x) − ν, so that at optimality, we have � πx ε[ ˜ψµ→ν ε ] dµ(x) = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (30) In this case, πx ε = πx ε [ ˜ψµ→ν ε ] is the conditional distribution of the second marginal of πε given that the first is equal to x, as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' More generally, for any potential ψ, the first order condition implies that ψ is equal to ˜ψ µ→νψ ε , the optimal dual potential between µ an νψ = � πx ε[ψ] dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 15 B Bound on the approximation error Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let i, j ∈ [J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We define the jth slack at x ∈ Li by 1 2∆ij(x) = −⟨x, yj⟩ + ϕ0(x) + ψ0(yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (31) As ϕ0 is the Legendre transform of ψ0, we have ∆ij(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' If the cells Li and Lj have a nonempty intersection, the set Hij(t) = {x ∈ Li : ∆ij(x) = t} represents the trace on Li of the hyperplane spanned by the boundary between Li and Lj, shifted by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' It is stated in [ANWS22] that for every nonnegative measurable function f : R → R, � Li f(∆ij(x))p(x) dx = 1 2∥yi − yj∥ � ∞ 0 f(t)hij(t) dt, (32) where hij(t) = � Hij(t) p(x) dHd−1(x) and Hd−1 is the (d − 1)-dimensional Hausdorff measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' In particular, wij = hij(0) is the (weighted) surface of the boundary between the ith and jth Laguerre cells (should it exist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Given x ∈ Li, let s(x) = minj̸=i 1 2∆ij(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' When the point x is sufficiently inside its Laguerre cell, the conditional probability πx ε becomes extremely concentrated around the point yi, as the next lemma shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Note that πx 0 = δyi when x ∈ Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let x ∈ Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For ε small enough, it holds that for every j ∈ [J], |πx ε(yj) − πx 0(yj)| ≤ ce−s(x)/ε, where c depends on J, the distances ∥yi − yj∥ and on the quantities wij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Such a result was already stated in [Del22, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2], although while requiring that the source measure P has a H¨older continuous density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Only assumption (A) is needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' According to [ANWS22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='6], for ε small enough, ε−1∥ ˜ψε − ψ0∥∞ ≤ C, (33) where ˜ψε is the shifted version of ψε (see (26)) and C depends on the distances ∥yi − yj∥ and on the wijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Following [Del22, Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2] and (29), we have for j ̸= i |πx ε(yj) − πx 0(yj)| = πx ε(yj) = e(⟨x,yj⟩− ˜ψε(yj))/ε �J j′=1 e(⟨x,yj′⟩− ˜ψε(yj′))/ε ≤ e2C e(⟨x,yj⟩−ψ0(yj))/ε �J j′=1 e(⟨x,yj′⟩−ψ0(yj′))/ε ≤ e2Ce−s(x)/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' A similar computation yields that |πx ε(yi) − πx 0(yi)| = |πx ε(yi) − 1| ≤ Je2Ce−s(x)/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We can bound for any x ∈ Li, ∥Tε(x) − T0(x)∥ = ∥ J � j=1 yj(πx ε(yj) − πx 0(yj))∥ ≤ c J � j=1 ∥yj∥e−s(x)/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (34) Therefore, letting C′ denote a constant, which may depend on J, whose value may change from line to line, we obtain ∥Tε − T0∥2 L2(P) = J � i=1 � Li ∥Tε(x) − T0(x)∥2 dP(x) ≤ C′ J � i=1 � Li J � j=1 e−2s(x)/ε dP(x) (35) ≤ C′ � i̸=j � Li e−∆ij(x)/ε dP(x) ≤ C′ � i̸=j 1 2∥yi − yj∥ � ∞ 0 e−t/εhij(t) dt , (36) 16 where in the second equality, we used the definition of s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Assumption (A) ensures that the functions hijs are bounded, which implies that the right-hand side in (36) is of order ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' C Stability of entropic transport plans Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Note that we may assume without loss of generality that ν ≪ ν′ and that KL(ν∥ν′) < ∞, for otherwise the bound is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For notational convenience, we omit the dependence on ε in the subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Write πµ,ν = γµ,ν(x, y)dµ(x)dν(y) for the entropic optimal plan between µ and ν, where γµ,ν = exp �1 ε(⟨x, y⟩ − ϕµ→ν(x) − ψµ→ν(y)) � , and analogously define γµ′,ν′ = exp �1 ε(⟨x, y⟩ − ϕµ′→ν′(x) − ψµ′→ν′(y)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Consider the measure γµ′,ν′(x, y) dµ(x) dν′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The first-order optimality condition for (ϕµ′→ν′, ψµ′→ν′) implies that � γµ′,ν′(y) dν′(y) = 1 ∀x ∈ Ω , (37) so that γµ′,ν′(x, y) dν′(y) is a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let us write dπx(y) = γµ,ν(x, y) dν(y) and dρx(y) = γµ′,ν′(x, y) dν′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We make the following observations: first, T µ→ν(x) = � y dπx(y) and T µ′→ν′(x) = � y dρx(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Second, the support of ρx lies inside B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' since any Lipschitz function f on B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R) satisfies supx f(x) − infx f(x) ≤ 2R, Hoeffding’s lemma [see BLM13, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2] implies that if f is Lipschitz and � f dρx = 0, then � etf dρx ≤ e2R2t2 ∀t ∈ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This implies [BG99, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1] that W1(πx, ρx)2 ≤ 8R2KL(πx∥ρx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (38) Third, Jensen’s inequality implies that for any coupling γ between πx and ρx, � ∥y − y′∥ dγ(y, y′) ≥ ���� � (y − y′) dγ(y, y′) ���� = ∥T µ→ν(x) − T µ′→ν′(x)∥ , (39) so that in particular, ∥T µ→ν(x) − T µ′→ν′(x)∥ ≤ W1(πx, ρx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Combining these facts, we obtain 1 8R2∥T µ→ν(x) − T µ′→ν′(x)∥2 ≤ KL(πx∥ρx) = � log � γµ,ν γµ′,ν′ (x, y) dν dν′(y) � γµ,ν(x, y) dν(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (40) 17 Integrating both sides of this equation with respect to µ yields 1 8R2∥T µ→ν(x) − T µ′→ν′(x)∥2 L2(µ) ≤ � log � γµ,ν γµ′,ν′ (x, y) dν dν′(y) � dπµ,ν(x, y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (41) Expanding the definition of γµ,ν and γµ′,ν′ and using that � log dν dν′(y) dπµ,ν(x, y) = � log dν dν′(y) dν(y) = KL(ν∥ν′) yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We now record two corollaries of this bound, which apply when either the source or the target measures of the entropic maps agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For any µ, ν, ν′ supported in B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R), 1 8R2∥T µ→ν ε − T µ→ν′ ε ∥2 L2(µ) ≤ ε−1 � (ψµ→ν′ ε − ψµ→ν ε ) d(ν − ν′) + KL(ν∥ν′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (42) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7 with µ = µ′, which yields (once again omitting the depen- dency in ε) 1 8R2∥T µ→ν ε −T µ→ν′ ε ∥2 L2(µ) ≤ ε−1 �� (ϕµ→ν′ − ϕµ→ν) dµ + � (ψµ→ν′ − ψµ→ν) dν � +KL(ν∥ν′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (43) By definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (ϕµ→ν′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' ψµ→ν′) minimizes the expression � ϕ dµ + � ψ dν′ + ε �� e(⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='y⟩−ϕ(x)−ψ(y))/ε dµ(x) dν′(y) − ε ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' recalling that �� e(⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='y⟩−ϕµ→ν′(x)−ψµ→ν′(y))/ε dµ(x) dν′(y) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' we have in particular � ϕµ→ν′ dµ + � ψµ→ν′ dν′ ≤ � ϕµ→ν dµ + � ψµ→ν dν′ + ε �� e(⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='y⟩−ϕµ→ν(x)−ψµ→ν(y))/ε dµ(x) dν′(y) − ε = � ϕµ→ν dµ + � ψµ→ν dν′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' where we have used that the first-order optimality condition for (ϕµ→ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' ψµ→ν) implies that �� e(⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='y⟩−ϕµ→ν(x)−ψµ→ν(y))/ε dµ(x) dν′(y) = 1 as well (see (11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This implies � (ϕµ→ν′ − ϕµ→ν) dµ ≤ − � (ψµ→ν′ − ψµ→ν) dν′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (44) Applying this inequality to (43) yields 1 8R2∥T µ→ν ε − T µ→ν′ ε ∥2 L2(µ) ≤ ε−1 � (ψµ→ν′ − ψµ→ν) d(ν − ν′) + KL(ν∥ν′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 18 Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For any µ, µ′, ν supported in B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R), 1 8R2∥T µ→ν ε − T µ′→ν ε ∥2 L2(µ) ≤ ε−1 � (ϕµ′→ν ε − ϕµ→ν ε ) d(µ − µ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (45) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7 with ν = ν′, yielding (dropping the dependency on ε) 1 8R2∥T µ→ν − T µ′→ν∥2 L2(µ) ≤ ε−1 �� (ϕµ′→ν − ϕµ→ν) dµ + � (ψµ′→ν − ψµ→ν) dν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (46) An argument analogous to the one used in the proof of Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 gives the inequality � ϕµ′→ν dµ′ + � ψµ′→ν dν ≤ � ϕµ→ν dµ′ + � ψµ→ν dν , (47) or, equivalently, � (ψµ′→ν − ψµ→ν) dν ≤ − � (ϕµ′→ν − ϕµ→ν) dµ′ , (48) and combining this inequality with (46) proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' D Strong convexity of the entropic semi-dual problem Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 (Strong convexity of F µ→ν ε ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let ν = �J j=1 νjδyj be a measure supported on {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ} ⊆ B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R) and let µ supported on a compact convex set Ω ⊆ B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R) with a density p satisfying pmin ≤ p ≤ pmax for some pmax ≥ pmin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For ψ ∈ RJ, define νψ = � πx ε(ψ) dµ(x) and assume that νψ ≥ λν for some 0 < λ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, we have for ε > 0 F µ→ν ε (ψ) − min ψ F µ→ν ε ≥ Cλ · Varν(ψ − ψµ→ν ε ), (49) where C = � e2R2 pmax pmin + ε �−1 pmin pmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As µ and ε are fixed, we will simply write ψν instead of ψµ→ν ε , and write similarly Fν = F µ→ν ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Recall the definition (26) of the shifted potential ˜ψν(yj) = ψν(yj) − ε log νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' According to [Del22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2], the functional Fν is minimized at the vector ˜ψν, with ∀v ∈ RJ, Varν(v) ≤ � e2R2 pmax pmin + ε � v⊤∇2Fν( ˜ψν)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (50) For t ∈ [0, 1], let ψt = ˜ψν + t(ψ − ˜ψν) and let νt = � πx ε(ψt) dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The potential ψt is the (shifted) entropic Brenier potential between µ and νt, so that it minimizes the functional Fνt (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Also, note that ∇2Fν does not depend on ν, so that v⊤∇2Fν(ψt)v = v⊤∇2Fνt(ψt)v ≥ � e2R2 pmax pmin + ε �−1 Varνt(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (51) Let v = ψ − ψµ→ν ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' A Taylor expansion of Fν gives Fν(ψ) − Fν( ˜ψν) = � 1 0 v⊤∇2Fν(ψt)v dt ≥ � e2R2 pmax pmin + ε �−1 � 1 0 Varνt(v) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (52) 19 Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Write νt = �J j=1 νt,jδyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, for all t ∈ [0, 1] and j ∈ [J], we have νt,j ≥ pmin pmaxν1−t 0,j νt 1,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This lemma is enough to conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Indeed, ν1 = νψ ≥ λν, so that it implies that Varνt(v) ≥ pmin pmaxλVarν(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' According to [Del22, Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1], Φε(ψt)(tx + (1 − t)y) ≤ tΦε( ˜ψµ→ν ε )(x) + (1 − t)Φε(ψ)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (53) Therefore, if we let ht(x) = e(⟨x,yj⟩−ψt(yj)−Φε(ψt)(x))/ε, then we have ht(tx + (1 − t)y) ≥ h0(x)th1(y)1−t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' By the Pr´ekopa-Leindler inequality, νt,j = � ht(x) dµ(x) ≥ pmin � X ht(x) dx ≥ pmin �� X h0(x) dx �t �� X h1(x) dx �1−t ≥ pmin pmax ν1−t 0,j νt 1,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As in the previous proof, we drop the ε and µ dependency in our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Write νk = �J j=1 νk,jδyj for k = 0, 1, and define as before the shifted potentials ˜ψνk(yj) = ψν1(yj) − ε log νk,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let θ > 0 be a parameter to fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' According to Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1, Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1, and using the inequality Fν1( ˜ψν1) ≤ Fν1( ˜ψν0), we have CλVarν0( ˜ψν1 − ˜ψν0) ≤ Fν0( ˜ψν1) − Fν0( ˜ψν0) ≤ Fν0( ˜ψν1) − Fν1( ˜ψν1) + Fν1( ˜ψν0) − Fν0( ˜ψν0) = � ( ˜ψν1 − ˜ψν0)( dν0 − dν1) ≤ θ 2Varν0( ˜ψν1 − ˜ψν0) + 1 2θχ2(ν1∥ν0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We pick θ = Cλ to conclude that Varν0( ˜ψν1 − ˜ψν0) ≤ 1 (Cλ)2χ2(ν1∥ν0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (54) Therefore, using the inequality | log(a/b)| ≤ |a − b|/ min{a, b} for a, b > 0, Varν0(ψ1 − ψ0) ≤ 2Varν0( ˜ψ1 − ˜ψ0) + 2 J � j=1 ν0,j � log �ν1,j ν0,j ��2 ≤ 2 (Cλ)2χ2(ν1∥ν0) + 2 J � j=1 ν0,j � ν1,j − ν0,j min{ν0,j, ν1,j} �2 ≤ 2 (Cλ)2χ2(ν1∥ν0) + 2 λ2 J � j=1 1 ν0,j (ν1,j − ν0,j)2 ≤ � 2 (Cλ)2 + 2 λ2 � χ2(ν1∥ν0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 20 E Control of the fluctuations in the one-sample case Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 (Sample complexity in the one-sample case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Assume that P satisfy (A) and that Q satisfy (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, it holds that E∥T P→Qn ε − Tε∥2 L2(P) ≲ ε−1n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' To ease notation, we write Tε,n = T P→Qn ε and ψε,n = ψP→Qn ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As explained in Section 3, the stability result Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='7 implies that E∥Tε,n − Tε∥2 L2(P) ≤ 8R2 ε �E[VarQ(ψε,n − ψε)] 2 + E[χ2(Qn∥Q)] 2 � + 8R2E[χ2(Qn∥Q)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (55) Write Q = �J j=1 qjδyj and Qn = �J j=1 ˆqjδyj, and introduce the event E = {∀j ∈ [J], ˆqj ≥ qj/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' If E is satisfied, we have Qn ≥ Q/2, so that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='9 yields VarQ(ψε,n − ψε) ≤ Cχ2(Qn∥Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (56) If E is not satisfied, we use the fact that the entropic potentials have a bounded amplitude (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1), to obtain that VarQ(ψε,n − ψε) ≤ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (57) Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let E be the event that Qn ≥ Q/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then P(Ec) ≤ Je−cqminn for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' By [Ver18, Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2], we have P(Ec) ≤ �J j=1 P(ˆqj < qj/2) ≤ Je−cqminn for some c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We obtain E∥ ˆTε,n − Tε∥2 L2(P) ≲ R2 ε E[χ2(Qn∥Q)] + R2 ε Je−cqminn ≲ ε−1n−1 (58) by Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' F Control of the fluctuations in the two-sample case The goal of this section is to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We will actually prove a more general result, and show that for any discrete measure ν = �J j=1 νjδyj supported on {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ} with νj ≥ νmin > 0 for all j ∈ [J], we have for log(1/ε) ≲ n/ log(n), E∥T Pn→ν ε − T P→ν ε ∥2 L2(P) ≲ ε−1n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (59) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5 follows from (59) by conditioning on Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let E be the event that Qn ≥ Q/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, by Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2, E∥ ˆTε − T P→Qn ε ∥2 L2(P) ≤ E � E[∥ ˆTε − T P→Qn ε ∥2 L2(P)|Qn]1{E} � + R2P(Ec) ≤ Cε−1n−1 + R2Je−cqminn ≲ ε−1n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We obtain Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='5 by combining this bound with Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 21 To prove (59), we first use Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 which yields E∥T Pn→ν ε − T P→ν ε ∥2 L2(P) ≤ 8R2ε−1E � (ϕPn→ν ε − ϕP→ν ε ) d(Pn − P) = 8R2ε−1E � (Φε( ˜ψPn→ν ε ) − Φε( ˜ψP→ν ε )) d(Pn − P), (60) where we recall that for a potential ψ, the shifted potential ˜ψ is given by ˜ψj = ψj − ε log νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The remainder of the proof consists in bounding this integral by using localization arguments and standard bounds on suprema of empirical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Our first goal is to show that the potential ψPn→ν ε is close to to the potential ψP→ν ε for the ∞-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' It will be convenient to work with the “L∞-variance” Var∞(ψ) = inf c∈R max j∈[J] |ψ(yj) − c|2 = �max ψ − min ψ 2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (61) As the measure ν is lower bounded, it holds that Varν(ψ) ≥ νminVar∞(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (62) Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 (Supremum of ε-Legendre transforms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let ψ0 be a fixed potential and let τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, for all j ∈ [J], E � sup Var∞(ψ−ψ0)≤τ 2 ���� � (πx ε(ψ)j − πx ε(ψ0)j) d(P − Pn)(x) ���� � ≤ C � J max{log(τ/ε), 1} n (63) E � sup Var∞(ψ−ψ0)≤τ 2 ���� � (Φε(ψ)(x) − Φε(ψ0))(x) d(P − Pn)(x) ���� � ≤ Cτ � J n (64) for some absolute constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let us prove the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The functional πx ε is invariant by translation: πx ε(ψ + c) = πx ε(ψ) for all c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This implies that sup Var∞(ψ−ψ0)≤τ 2 ���� � (Φε(ψ)(x) − Φε(ψ0))(x) d(P − Pn)(x) ���� = sup ∥ψ−ψ0∥∞≤τ ���� � (Φε(ψ)(x) − Φε(ψ0))(x) d(P − Pn)(x) ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For a metric space (A, d) and u > 0, we let N(u, A, d) be the covering number of A at scale u, that is the smallest number of balls of radius u needed to cover A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let B be the L∞-ball of radius τ in RJ, centered at ψ0, and let ∥ · ∥∞ denote the ∞-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For 0 < u ≤ τ, we have log N(u, B, ∥ · ∥∞) ≤ J log(τ/u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As the function ψ �→ πx ε (ψ)j is ε−1-Lipschitz continuous for every x ∈ Rd, we have for 0 < u ≤ τ/ε, log N(u, {x �→ πx ε(ψ)j : ψ ∈ B}, ∥ · ∥∞) ≤ J log(τ/(uε)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 22 Remarking furthermore that 0 ≤ πx ε(ψ)j ≤ 1 (so that the class of functions {x �→ πx ε(ψ)j : ψ ∈ B} admits the constant function 1 as an envelope function), we obtain the following control using Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3: E � sup ∥ψ−ψ0∥∞≤τ ���� � (πx ε(ψ)j − πx ε(ψ0)j)( dP − dPn)(x) ���� � ≤ c0 √n � c1 0 � J log 2N(u, {x �→ πx ε(ψ)j : ψ ∈ B}, ∥ · ∥∞) du ≤ � c2J max{log(τ/ε), 1} n , where c0, c1 and c2 are absolute constants, and the last line follows from arguing on whether c1 < τ/ε or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The second inequality follows from the same argument, using that the function ψ �→ Φε(ψ) is 1-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Indeed, the functional Φε satisfies Φε(ψ + c) = Φε(ψ) + c for all c ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then the set {ψ : Var∞(ψ − ψ0) ≤ τ 2} is equal to the set {ψ + c : ψ ∈ B, c ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As � c d(P − Pn) = 0, we can therefore once again restrict the supremum to vectors ψ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Furthermore, an envelope function of the class {Φε(ψ) − Φε(ψ0) : ψ ∈ B} is the constant function equal to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Therefore, by Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3, we obtain E � sup ∥ψ−ψ0∥∞≤τ ���� � (Φε(ψ) − Φε(ψ0))( dP − dPn) ���� � ≤ c0 √n � c1τ 0 � J log 2N(u, {Φε(ψ) : ψ ∈ B}, ∥ · ∥∞) du ≤ � c3Jτ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Assume that P satisfies (A) and let ν = �J j=1 νjδyj be a measure supported on {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , yJ} ⊂ B(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' R), with νj ≥ qmin for all j ∈ [J].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, for all 0 < ε ≤ 1 with log(1/ε) ≲ n/ log(n), it holds that EVar∞( ˜ψPn→ν ε − ˜ψP→ν ε ) ≲ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (65) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' To alleviate notation, we will write ψn = ψPn→ν ε and ψ0 = ψP→ν ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Similarly, we write Fn = F Pn→ν ε and F0 = F P→ν ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let νn = � πx ε (ψPn→ν ε ) dP(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Under the event E = {νn ≥ ν/2}, we have according to Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 and the fact that ˜ψn minimizes Fn, CνminVar∞( ˜ψn − ˜ψ0) ≤ CVarν( ˜ψn − ˜ψ0) ≤ F0( ˜ψn) − F0( ˜ψ0) ≤ F0( ˜ψn) − Fn( ˜ψn) + Fn( ˜ψ0) − F0( ˜ψ0) = � (Φε( ˜ψn) − Φε( ˜ψ0)) d(P − Pn) (66) Let us bound P(Ec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As ˜ψn is the minimum of Fn, we have ν = � πx ε( ˜ψn)j dPn(x) (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Therefore, we may write νn,j = � πx ε( ˜ψn)j dPn(x) + � πx ε( ˜ψn)j d(P − Pn)(x) = 23 νj + Zj, where Zj = � πx ε( ˜ψn)j d(P − Pn)(x) = � (πx ε( ˜ψn)j − πx ε( ˜ψ0)j) d(P − Pn)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Note that Var∞( ˜ψn − ˜ψ0) ≲ R2 (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1), so that by Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 and Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3, P(Ec) ≤ J � j=1 P(|Zj| > νj/2) ≤ J exp � −c √nqmin ( � J log(1/ε) + log n � ≲ n−1, (67) under the condition log(1/ε) ≲ n/ log(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For k ≥ 0, let ak = 2k/√n and fix some p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let Ba = sup Var∞(ψ− ˜ψ0)≤a2 ���� � (Φε(ψ) − Φε( ˜ψ0)) d(P − Pn) ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Assume that E is satisfied and that Var∞( ˜ψ0 − ˜ψn) ∈ [a2, b2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, according to (66), it holds that Bb ≥ ca2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Using Markov’s inequality, Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 and Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3, we bound EVar∞( ˜ψn − ˜ψ0) ≤ a2 0 + � k≥0 P(Var∞( ˜ψn − ˜ψ0) ∈ [a2 k, a2 k+1] and E)a2 k+1 + CP(Ec) ≲ n−1 + � k≥0 P � Bak+1 ≥ ca2 k � a2 k+1+ ≲ n−1 + � k≥0 E[Bp ak+1] a2p k a2 k+1+ ≲ n−1 + � k≥0 (2k/n)p (4k/n)p 4k+1 n + P(Ec) ≲ n−1 + � k≥0 22k−pk n ≲ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Under the same assumptions than Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2, it holds that E∥T Pn→ν ε − T P→ν ε ∥2 ∞ ≲ ε−1n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (68) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let Z = Var∞( ˜ψn − ˜ψ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let once again ak = 2k/√n for k ≥ 1, with a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Fix some p > 2, with q = p p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For a > 0, let Da = supVar∞(ψ− ˜ψ0)≤a2 ��� � (Φε(ψ) − Φε( ˜ψ0)) d(P − Pn) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' By H¨older inequality and Markov inequality, we obtain, E � (Φε( ˜ψn) − Φε( ˜ψ0)) d(P − Pn) ≤ � k≥0 E � 1{Z ∈ [a2 k, a2 k+1]} sup Var∞(ψ− ˜ψ0)≤a2 k+1 � (Φε(ψ) − Φε( ˜ψ0)) d(P − Pn) � ≤ E[Da1] + � k≥1 � P(Z ≥ a2 k) �1/q E � Dp ak+1 �1/p ≲ n−1 + � k≥0 �E[Z] a2 k �1/q 2k n ≲ � k≥0 2k(1−2/q) n ≲ n−1, where we use Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2, Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 and Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3 at the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Equation (60) then gives the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 24 G A lower bound for the performance of the 1NN es- timator In this section, we prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We let P be the Lebesgue measure on Ω = [0, 1]d, and let y0 = (0, 1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , 1/2) and y1 = (1, 1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We denote by Pn an empirical measure consisting of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' samples from Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As in Appendix F, we work in a general setting of a generic discrete target measure ν, which may either be fixed or may be a random measure independent of Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We let ν = � j=0,1 νjδyj for ν0, ν1 ≥ 1 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' this latter condition will hold with overwhelming probability if ν is an empirical measure Qn corresponding to n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' samples from Q = 1 2δy0 + 1 2δy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Following [MBNWW21], we define the one-nearest neighbor estimator ˆT1NN in this general context by ˆT1NN(x) = n � i=1 � j=0,1 1Vi(x)(nˆπ(Xi, yj)) , where ˆπ is the empirical optimal coupling between Pn and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We first examine the structure of the Brenier map T0 = ∇ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The considerations in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 imply that T0(x) = � y0 ⟨e1, x⟩ ≤ ν0 y1 ⟨e1, x⟩ > ν0 , where e1 is the first elementary basis vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The potential ϕ0 is not differentiable on the separating hyperplane ⟨e1, x⟩ = ν0, which has measure 0 under P, but we may arbitrarily assign points on this hyperplane to y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Similar arguments imply that the empirical transport plan ˆπ between Pn and ν has the following property: there exists a (random) threshold τ ∈ (0, 1) such that ˆπ(x, y0) = � 1 ⟨e1, x⟩ < τ 0 ⟨e1, x⟩ > τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The set ⟨e1, x⟩ = τ may not have measure 0 under Pn, and ˆπ(x, y0) may take values strictly between 0 and 1 on this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The following lemma shows that τ is close to ν0 with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For any t ≥ 0, P {τ ≥ ν0 + t} ≤ e−2nt2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' If τ ≥ ν0 + t, this implies that Pn({x : ⟨e1, x⟩ < ν0 + t}) ≤ ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' On the other hand, nPn({x : ⟨e1, x⟩ < ν0 + t} is a Bin(n, ν0 + t) random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The result then follows from Hoeffding’s inequality [BLM13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let us write H for the halfspace {x : ⟨e1, x⟩ ≤ ν0}, and ˆH for the halfspace {x : ⟨e1, x⟩ ≤ τ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let x be any point in Ω such that x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We are interested in the event that there exists an element Xi ∈ {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , Xn} such that a) x ∈ Vi and b) Xi ∈ ˆHc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Call this event E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' On this event, ˆT1NN(x) = y1 and T0(x) = y0, so ∥ ˆT1NN(x) − T0(x)∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 25 We therefore obtain E∥ ˆT1NN − T0∥2 L2(P) = E � ∥ ˆT1NN(x) − T0(x)∥2 dP(x) ≥ E � H ∥ ˆT1NN(x) − T0(x)∥21{E(x)} dP(x) ≳ E � H 1{E(x)} dP(x) = � H P {E(x)} dP(x) , where the final equality follows from the Fubin–Tonelli theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We now lower bound the probability of E(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let us write At for the event that τ < ν0 +t, for t > 0 to be specified, and write Ht for the halfspace {x : ⟨e1, x⟩ ≤ ν0 + t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Given any x ∈ H, write ∆ = d(x, Hc t ), and let B be a ball of radius 2∆ around x, intersected with Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Denote by F(x) the event that there are no samples in V = B ∩ Ht but there is at least one point in B ∩ Hc t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then F(x) ∩ At ⊆ E(x), since on F(x) the nearest neighbor to x must be a sample in Hc t , and on At we have Hc t ⊆ ˆHc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' P {F(x) ∩ At} ≥ (1 − vol(V ))n − (1 − vol(B))n − e−2nt2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We first compute P {F(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The probability that there are no samples in V is (1 − vol(V ))n, and this event may be written as the disjoint union of F(x) and the event that all of B is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The latter event has probability (1 − vol(B))n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Therefore (1 − vol(V ))n = P {F(x)} + (1 − vol(B))n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Since P {Ac t} ≤ e−2nt2, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Lemma G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Assume that ∆ > 0 and that d(x, ∂Ω) ≥ 2∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' There exist positive constants cd,0 < 1 and cd,1 such that vol(V ) ≤ cd,0 vol(B) (69) and vol(B) ≥ cd,1∆d (70) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' This is immediate from a scaling argument: since d(x, ∂Ω) ≥ 2∆, the set B is a Euclidean ball of radius 2∆, and the set V is a Euclidean ball of radius 2∆ minus a spherical dome cut off by a hyperlane at distance ∆ from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' When ∆ = 1, it is clear that the claimed inequalities hold, and the general case is obtained by dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We assume in what follows that d(x, ∂Ω) ≥ 2∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The inequalities (1 + x)n ≥ 1 + nx and ex ≤ 1 + x + x2, valid for all x ∈ [−1, 0] and n ≥ 1, imply that for any δ > 0 there exists a constant cd,δ > 0 such that if ∆ ≤ cd,δn−1/d, then we will have (1 − vol(V ))n ≥ 1 − ncd,0 vol(B) (71) (1 − vol(B))n ≤ e−n vol(B) ≤ 1 − (1 − δ)n vol(B) (72) 26 Choosing δ sufficiently small, we obtain the existence of a small cd,3 > 0 such that if ∆ ≤ cd,3n−1/d, then (1 − vol(V ))n − (1 − vol(B))n ≥ Cdn∆d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Define ∆n = cd,4n−1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Putting it all together, consider the set S = {x ∈ H ∩ Ω : ∆n/2 ≤ d(x, Hc t ) ≤ ∆n, d(x, ∂Ω) ≥ 2∆n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' The above considerations imply that P {E(x)} ≥ Cdn(∆n/2)d − e−2nt2 ≥ C′ d − e−2nt2 for all x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Choosing t to be a sufficiently large constant multiple of n−1/2, we obtain � H P {E(x)} dP(x) ≥ � S P {E(x)} dP(x) ≳d vol(S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Since t ≍ n−1/2, we will have that t ≪ ∆n for n sufficiently large (as d ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Therefore, for n large enough, the set S contains the set S′ = {x ∈ Ω : ν0−∆n+t ≤ ⟨e1, x⟩ ≤ ν0−∆n/2+t, 2∆n ≤ ⟨ej, x⟩ ≤ 1−2∆n ∀j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , d} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Since vol(S′) ≳d ∆n ≳ n−1/d, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' H Auxiliary lemmas Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='1 (Young’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let Q0, Q1 be probability measures with Q1 ≪ Q0 and let f be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, for θ > 0, � f( dQ0 − dQ1) ≤ θVarQ0(f) 2 + χ2(Q1∥Q0) 2θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (73) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Recall Young’s inequality: for a, b ∈ R, ab ≤ a2 2 + b2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' As the left-hand side is invariant by translation, we may assume without loss of generality that � f dQ0 = 0, so that VarQ0(f) = � f 2 dQ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We write � f( dQ0 − dQ1) = � ( √ θf) � 1 − dQ1 dQ0 � √ θ dQ0 ≤ θ 2 � f 2 dQ0 + 1 2θ � � 1 − dQ1 dQ0 �2 dQ0 = θVarQ0(f) 2 + χ2(Q1∥Q0) 2θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 (Expectation of empirical χ2-divergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let Q = �J j=1 qjδyj be a discrete measure supported on J atoms, and let Qn denote its empirical measure, consisting of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, E[χ2(Qn∥Q)] = J − 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (74) 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We can write Qn = �J j=1 ˆqjδyj, where ˆqj is a binomial random variable with parameters n and qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' We obtain χ2(Qn∥Q) = J � j=1 (ˆqj − qj)2 qj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Taking expectations, our bound reads E[χ2(Qn∥Q)] = J � j=1 Var(ˆqj) qj = J � j=1 qj(1 − qj) nqj = J − 1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='3 (Control of suprema of empirical processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' , Xn be an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' sample from some probability measure P on Rd, with Pn the associated empirical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Consider F a class of functions Rd → R with ∥f∥∞ ≤ A for all f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' For u > 0, let N(u) be the u-covering numbers of F, that is the minimal number of balls of radius u for the ∥ · ∥∞-metric required to cover F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Then, E � sup f∈F ���� � f d(Pn − P) ���� � ≤ C0 √n � C1A 0 � log 2N(u) du =: I √n (75) for two positive absolute constants C0 and C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Furthermore, for all t > 0, P � sup f∈F ���� � f d(Pn − P) ���� > t � ≤ exp � − C2 √nt I + A log n � , (76) for some positive absolute constant C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Eventually, for all p ≥ 2, E � sup f∈F ���� � f d(Pn − P) ���� p�1/p ≤ Cp I + A √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' (77) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' See [VW96, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content='2 and Theorem 2.' metadata={'source': 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[YDV+20] Karren Dai Yang, Karthik Damodaran, Saradha Venkatachalapathy, Ali C Soylemezoglu, GV Shivashankar, and Caroline Uhler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' Predicting cell lineages using autoencoders and optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' PLoS computational biology, 16(4):e1007828, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} +page_content=' 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFIT4oBgHgl3EQfkyt3/content/2301.11302v1.pdf'} diff --git a/ZtFST4oBgHgl3EQfAzi6/content/tmp_files/2301.13700v1.pdf.txt b/ZtFST4oBgHgl3EQfAzi6/content/tmp_files/2301.13700v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b4fdcdfa7f57b06f1157b98e0e5c76ccb2f48c6 --- /dev/null +++ b/ZtFST4oBgHgl3EQfAzi6/content/tmp_files/2301.13700v1.pdf.txt @@ -0,0 +1,1035 @@ +arXiv:2301.13700v1 [math.PR] 31 Jan 2023 +One step entropy variation in sequential +sampling of species for the +Poisson-Dirichlet Process +Servet Mart´ınez∗ +Javier Santib´a˜nez† +Departamento de Ingenier´ıa Matem´atica and Centro de +Modelamiento Matem´atico, UMI 2071 CNRS-UCHILE, +Facultad de Ciencias F´ısicas y Matem´aticas, Universidad +de Chile, Santiago, Chile. +February 1, 2023 +Abstract +We consider the sequential sampling of species, where observed +samples are classified into the species they belong to. We are partic- +ularly interested in studying some quantities describing the sampling +process when there is a new species discovery. We assume that the +observations and species are organized as a two-parameter Poisson- +Dirichlet Process, which is commonly used as a Bayesian prior in the +context of entropy estimation, and we use the computation of the +mean posterior entropy given a sample developed in [4]. Our main +result shows the existence of a monotone functional, constructed from +the difference between the maximal entropy and the mean entropy +throughout the sampling process. We show that this functional re- +mains constant only when a new species discovery occurs. +AMS Classification Number: 94A17 +Keywords: Entropy, Bayesian posterior distribution, Poisson-Dirichlet Pro- +cess, new species discovery. +∗E-mail address: smartine@dim.uchile.cl. +†E-mail address: jsantibanez@dim.uchile.cl. +1 + +1 +Introduction +Consider the sequential sampling of species, where one takes a random sample +from a population and classifies each observation according to the species (or +classes) to which they belong. Because the population is large, there are +some rare species that may not be observed. We intend to understand and +model the discovery of a new species in this context and to study related +informational quantities. Our main result shows that the two-step variation +of differences between the maximal entropy and the entropy allows us to +describe when a new species is discovered in the Poisson-Dirichlet Process +(PDP). It is worth mentioning that our work is purely statistical. +The two parameter PDP —introduced by Pitman and Yor in 1997 [15]— +supplies random partitions with an infinite number of components in [0, 1] +and serves to model the process of sampling species and the times at which +new species are discovered, see [11], [8] and [9]. This process has been used +in ecology, but also in genetic applications [7], natural language processing +[16] and finance [17]. In Section 2, we will introduce the PDP and some of +the basic properties that we shall use. +Entropy is a way to measure the diversity of communities in a sample and our +work focuses on studying some aspects of the posterior entropy of the process +of sampling species in the PDP. The computation of posterior entropy relies +on the fact that given the sample from a PDP, the posterior distribution is +a mixture of a finite Dirichlet distribution and a PDP. +Much of this paper concern with Bayesian entropy estimation, is due to the +results in [4], in which the prior and posterior mean entropies for the PDP +were computed and some of their properties stated. +This is discussed in +Section 3. In Proposition 3.1, we provide lower and upper bounds for the +entropy when the sample size is fixed. +The main purpose of this work is to obtain an increasing functional along +the process constructed with posterior mean entropy between two successive +steps of the PDP with parameters (α, θ). This functional is, +Lℓ = (θ + ℓ)( �Hmax +ℓ +− �Hℓ), +(1) +and satisfies the monotone property Lℓ+1 ≥ Lℓ. Here �Hℓ denotes the posterior +entropy when observing a sample at step ℓ and �Hmax +ℓ +is its maximum over +all samples of size ℓ. Our main result is Theorem 4.4 in Section 4, where we +show that Lℓ is increasing and the equality Lℓ+1 = Lℓ is attained only when +a new species is discovered. +2 + +We also show that the weighted difference of entropies satisfies +(θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ > 0. +The expression (18) obtained in Theorem 4.4, for the above difference of +weighted entropies, allows us to think of the entropy as a sum of the ‘dis- +covery values’ of the sampled species, plus an additive deterministic term +depending on ℓ, α and θ. On the other hand, the expression (17) allows us +to write straightforwardly the functional Lℓ as a sum of positive rewards for +‘reinforcing the knowledge’ of what it is known, and no additional additive +term is required. The discovery values and the reinforcement rewards are +expressed in terms of the digamma function. This is discussed in Remark +4.8. +We also study similar quantities in the frequentist framework and relations +in the same vein are shown in Proposition 4.2. +2 +Poisson-Dirichlet Process +This section is devoted to the definition of the PDP and to supply some of +its properties. We follow the articles [14], [5], [18], [13], [16] and [4]. Since +this is a well-known theory we only state those results directly related to our +work. +Let 0 ≤ α < 1 and θ > −α. Consider independent random variables βk ∼ +Beta(1 − α, θ + αk). Let π = (πk : k ≥ 1) be given by the two-parameter +Griffiths-Engen-McCloskey distribution, GEM(α, θ), +π1 := β1, +πk := βk +k−1 +� +j=1 +(1 − βj) +k ≥ 2, +which defines a probability vector a.s. Now consider a non-atomic probability +measure G defined on space X . Let (φk : k ≥ 1) be an i.i.d. sequence with +distribution as G, then are all different a.s. We assume φ = (φk : k ≥ 1) are +independent of π. The discrete random measure +Ξ(·) = +� +k≥1 +πkδφk(·) +(2) +is called the PDP with base measure G and parameters α and θ. The base +measure G is non-atomic, this is used to give different names to the species +in the process Ξ(·), but the unique fact that matters is that the species are +3 + +different, the exact names are not important, and this explains why we ignore +G and one simply notes PDP(α, θ). +The case α = 0 is called Dirichlet process and it can be constructed as an +infinite extension of a Dirichlet distribution. Examples on how PDP help to +model different phenomena can be seen in [14] and [12]. +Samples from a PDP are obtained from (2) in the following way. +For a +random measure Ξ(·) one takes an i.i.d. sequence of variables (Xn : n ≥ 1) +with values in X . Let Xℓ = (X1, . . . , Xℓ) be a sample of size ℓ collected in +a sequential way. By Kℓ we note the total number of different species of +the sample which are noted by X∗ +1, . . . , X∗ +Kℓ. For j = 1, . . . , Kℓ we note by +Nℓ +j the number of times that the species X∗ +j is observed in the sample, so +ℓ = �Kℓ +j=1 Nℓ +j. Further we do not take into account the order of the species in +the sample, if needed one can enumerate their frequencies in their decreasing +order. So, (Nℓ +j : j = 1, . . . , Kℓ) means the multiset of frequencies (that is a +set where the values can be repeated). +The conditional probability for a new observation Xℓ+1 is, see [5], +P(Xℓ+1 = • | Xℓ) = θ + αKℓ +θ + ℓ G(·) + +Kℓ +� +j=1 +Nℓ +j − α +θ + ℓ δX∗ +j . +(3) +So, the observation Xℓ+1 is part of the species X∗ +j already observed with +probability +Nℓ +j −α +θ+ℓ , and Xℓ+1 defines a new species with probability θ+αKℓ +θ+ℓ . In +this last case the new species Xℓ+1 = X∗ +Kℓ+1 is distributed as G independently +of the species already discovered, and ℓ+1 is said to be the discovery time of +a new species. That is, the transition probability (3) states the probability +of discovering a new species and gives a different name to it, the important +point is that it is different to the previous ones. +3 +Bayesian entropy +To define the Bayesian entropy one assumes a prior distribution and makes +the estimation of entropy based upon the posterior distribution given the +sample. We will introduce Bayesian entropy in the context of PDP following +closely, as mentioned in the introduction, the results in [4], and also [3] +and [6]. To do so, we need to recall the definition of entropy. Let π be a +distribution, the Shannon entropy is defined as +H(π) = − +∞ +� +i=1 +πi log(πi). +4 + +For further computations it is useful to introduce the digamma function and +some of its properties, which can be found in [1] and [2]. This function is the +logarithmic derivative of the Gamma function: +ψ(x) = d +dx log(Γ(x)) = Γ′(x) +Γ(x) , +where Γ(x) = +� ∞ +0 tx−1e−tdt. From Γ(x + 1) = xΓ(x), one gets ψ(x + 1) = +ψ(x) + 1/x for x > 0, that implies +xψ(x + 1) − (x − 1)ψ(x) = ψ(x) + 1, x > 0. +(4) +The digamma function is increasing for x > 0 and then xψ(x+1)−(x−1)ψ(x) +is also increasing for x > 0. Since ψ(2) > 0, then xψ(x + 1) > (x − 1)ψ(x) +when x ≥ 1. The digamma function admits the following bounds in terms of +the logarithmic function, see [2]: +log(x) − 1 +x ≤ ψ(x) ≤ log(x) − 1 +2x, +x > 0. +(5) +For x sufficiently big the digamma function can be approximated by +ψ(x) = log(x) − 1 +2x + o +�1 +x +� +. +(6) +3.1 +Entropy for the Poisson-Dirichlet Process +Let Xℓ = (X1, . . . , Xℓ) be a sample following a distribution π. The Bayesian +approach for estimating the entropy requires to assume a prior distribution +π and estimate the posterior distribution. The least square Bayes estimator +has the shape: E(H(π)|Xℓ). +When one takes a PDP as prior, the sample Xℓ should be obtained from the +random measure Ξ, given by (2). But, as we mentioned before, we can omit +any reference to G, so the sample is obtained from the weight distribution +π and we will refer to the process and its weight distribution indistinctly by +the same symbol, that is, the prior is π ∼ PDP(α, θ). In [4] the prior mean +of H(π) is proven to be, +E(H(π)) = ψ(θ + 1) − ψ(1 − α). +We are interested in finding the posterior mean of H(π), after seeing a sample. +To describe the posterior distribution consider the sample Xℓ with Kℓ dif- +ferent species and frequencies Nℓ +1, . . . , Nℓ +Kℓ. To simplify notation put Kℓ = k +5 + +and Nℓ +j = nj for j = 1, . . . , k. In [10] it was shown that the posterior distri- +bution πpost = (p1, . . . , pk, (1 − �k +j=1 pj)π′) is given by the mixture +(p1, . . . , pk, 1 − +k +� +j=1 +pj) +∼ +Dirichlet(n1 − α, . . . , nk − α, θ + αk) +π′ = (π′ +1, π′ +2, . . . ) +∼ +PDP(α, θ + αk). +Hence, the probability of belonging to some species X∗ +j already present in the +sample is pj for j = 1, . . . , k; and the probability to belong to a new species +is 1 − �k +j=1 pj, where the distribution of these probabilities depend on the +frequencies (nj) and k. In the event that a new species is discovered it will +be part of a specific species i with weight π′ +i. +The species X∗ +i related to the prior distribution π, is not the same as the +species X∗ +i in the posterior distribution πpost, because the index taken af- +ter observing the sample is arbitrary. But, this index discrepancy does not +cause any problem since the ordering of πi is not important in H(π) and the +transition probability for the discovery of a new species and for the species +that have been discovered in the past continues to have the weights given by +(3). Also, the posterior distribution of π is represented by a realization πpost +whose ordering is totally different from the ordering of π, this realization is +only one representation of the posterior distribution. +The Bayes estimator of the posterior mean of the entropy under the PDP +prior, at step ℓ, will be defined as +�Hℓ +P DP = E(H(π)|Xℓ). +We will write H instead of H(π) when there is no confusion, so �Hℓ +P DP = +E(H|Xℓ). In [4] it was shown that the posterior mean of H under the PDP +prior is, +�Hℓ +P DP = ψ(θ+ℓ+1)− θ + αk +θ + ℓ ψ(1−α)− +1 +θ + ℓ +k +� +i=1 +(ni −α)ψ(ni −α+1). (7) +Let �πℓ be the vector of empirical probabilities �πℓ +i = ni/ℓ, for i = 1, . . . , k, +and �πℓ +i = 0 for i > k, given by the sample Xℓ. The Maximum Likelihood +Estimator (MLE) of the entropy, at step ℓ, under multinomial likelihood, is +given by +�Hℓ +MLE = H(�πℓ) = − +∞ +� +i=1 +�πℓ +i log(�πℓ +i), +(8) +6 + +which is a biased estimator. In [4] it is shown that when Kℓ/ℓ converges in +probability to 0, then �Hℓ +P DP satisfies the following consistency property, +| �Hℓ +P DP − �Hℓ +MLE| → 0 as ℓ → ∞. +(9) +3.2 +Bounds for the posterior PDP entropy +Let us obtain lower and upper bounds for the entropy when the sample size +is fixed. This is made firstly when the number of species is fixed and after +over all possible number of species in the sample. +Proposition 3.1. For a sample Xℓ of a PDP(α, θ), with k different species +the entropy is upper and lower bounded by, +E(H|Xℓ) +≤ +ψ(θ+ℓ+1) − θ+αk +θ+ℓ ψ(1−α)− 1 +θ+ℓ +k +� +i=1 +(ni−α)ψ(ni−α+1); +E(H|Xℓ) +≥ +ψ(θ+ℓ+1) − θ+αk +θ+ℓ ψ(1−α)− 1 +θ+ℓ +k +� +i=1 +(ni−α)ψ(ni−α+1); +where the vectors of frequencies (ni : i = 1, . . . , k) and (ni : i = 1, . . . , k) of +the maximal entropy and the minimal entropy respectively, have the following +structures up to index permutation: +ni = ⌊ℓ/k⌋, i = 1, . . . , lk, +ni = ⌊ℓ/k⌋ + 1, i = lk + 1, . . . , lk + hk +where ⌊x⌋ is the biggest integer smallest or equal to x, hk = ℓ − k⌊ℓ/k⌋ and +lk = k − hk; and +nk = ℓ − (k − 1) and ni = 1, i = 1, . . . , k − 1. +Moreover, when one looks for the global bounds on all entropy maxima for +k ∈ {1, . . . , ℓ}, one finds that: the global maximum is attained when the ℓ +elements of the sample belong to different species and the global minimum is +attained when the ℓ elements of the sample belong to a unique species. This +is, +min +Yℓ E(H|Yℓ) ≤ E(H|Xℓ) ≤ max +Yℓ E(H|Yℓ) +with +max +Yℓ E(H|Yℓ) = ψ(θ+ℓ+1) − ψ(1 − α) − +ℓ +θ + ℓ, +(10) +min +Yℓ E(H|Yℓ) = ψ(θ+ℓ+1)−(θ+α)ψ(1−α) +θ+ℓ +−(ℓ−α)ψ(ℓ−α+1) +θ+ℓ +. (11) +7 + +Proof. We will take into account that −ψ(1 − α) > 0. Let us first prove the +extremal entropies for a fixed k. If k = 1 there nothing to examine because +n1 = ℓ and one simply computes the entropy. +Let k > 1. Take two species i ̸= j and set ni = n, nj = m. Assume n > 1. +We will fix when the entropy grows when one makes the change n → n − 1, +m → m + 1 and all other frequencies nl are equal, so the number of classes +continues to be k and the sum of their frequencies continues to be ℓ. This +change makes the entropy grow if and only if the following inequality holds +(we take into account that there is a minus in front of the third term at the +right hand side in (7)), +(n − 1 − α)ψ(n − α) + (m + 1 − α)ψ(m + 2 − α) +≤ +(n − α)ψ(n − α + 1) + (m − α)ψ(m − α + 1). +From (4) this is equivalent to +0 ≤ −ψ(m − α + 1) − 1 + ψ(n − α) + 1 = ψ(n − α) − ψ(m − α + 1). +But this is equivalent to m + 1 ≤ n. So, when this last inequality holds we +make the change n → n − 1 and m → m + 1. (Note that if n = m + 1 +the change leaves the set of frequencies invariant because the new pair is the +same, m, m + 1). Therefore the maximal entropy for k classes is attained by +the following structure of frequencies: +ni = ⌊ℓ/k⌋, i = 1, . . . , lk, +ni = ⌊ℓ/k⌋ + 1, i = lk + 1, . . . lk + hk +with hk = ℓ − k⌊ℓ/k⌋ and lk = k − hk. This is the frequencies are ’as equal +as possible’. +On the opposite when m + 1 ≥ n, the change n → n − 1, m → m + 1, makes +the entropy decrease. So, the minimal entropy structure of frequencies is +given by n1 = ℓ − (k − 1) and the rest of k − 1 species have frequency 1. +Therefore the first two inequalities of the Proposition are shown. +Now for obtaining the global maxima and minima we must see what happens +with the extreme solutions for different k’s. +This is based upon the following observation. Assume we have k < ℓ number +of species with frequencies (n1, · · · , nk) and nk > 1. Let us see what happens +when we change this structure of frequencies to one that contains k+1 species +and (n1, · · · , nk−1, nk − 1, 1), so with nk+1 = 1. We claim that this operation +makes the entropy strictly bigger. In fact by (7) the claim is equivalent to +−αψ(1−α)−(nk−1−α)ψ(nk−α)−(1−α)ψ(2−α) > −(nk−α)ψ(nk+1−α). +8 + +By using (4) this last inequality is equivalent to +−αψ(1 − α) − (1 − α)ψ(2 − α) + ψ(nk − α) + 1 > 0. +(12) +Since ψ(nk −α) ≥ ψ(2−α) it suffices to check the inequality (12) for nk = 2. +When in the expression at the left hand side in (12) we set nk = 2 we get, +α(ψ(2 − α) − ψ(1 − α)) + 1, +which is strictly positive, so (12) holds and the claim is satisfied. +Then, if one takes the maximal configuration for k < ℓ species, we know +that there exists a frequency, that we can assume is the k−th one, that +satisfies nk > 1. So, by making the above operation gives a configuration +of frequencies of a total number of species k + 1 and such that the entropy +increases strictly. +In particular the maximal entropy for k + 1 species is +strictly bigger than the maximal entropy for k species. Then, (10) is proven. +Finally when we make the above operation from the minimal configuration +of k species we retrieve the minimal configuration of the k + 1 species and so +the minimal entropy for k species is strictly lower than the minimal entropy +for k + 1 species. So, (11) follows. The result is shown. +Remark 3.2. From (11) and since −ψ(1 − α) > 0, we get +min +Yℓ ((θ+ℓ)E(H|Yℓ))≥(θ+ℓ)ψ(θ+ℓ+1)−(ℓ−α)ψ(ℓ−α+1), +where θ > −α. On the other hand for every real h > 0 we have (x+h) log(x+ +h + 1) − x log(x + 1) → ∞ as x → ∞. Then, by also using (6) we get that +min +Yℓ ((θ+ℓ)E(H|Yℓ)) → ∞ as ℓ → ∞. □ +The relation (9) shows a key property between the frequentist estimator based +on empirical probabilities and the Bayesian estimator based on the posterior +mean under the PDP prior, when ℓ → ∞. In next section we will study the +variation of weighted estimators when making a finite step ℓ to ℓ+1, showing +a property that is similar for both, the frequentist and the PDP cases. +4 +One step variation of entropy and discovery +of a new species +We will state and prove our main result: an equality proving that a weighted +variation between two successive steps of the posterior Bayesian entropy, is +9 + +nonnegative and only vanishes in the discovery times of a new species. This +is done in Section 4.2. +Related to this result, we previously study the variation of the entropy when +one only computes frequencies, and how it characterizes discovery time of +species. +4.1 +One step variation of entropy for frequencies +The framework is the following one: we collect a series of elements that are be- +ing classified in some class or species, at the moment when they are observed. +At step ℓ one has collected in a sequential way ℓ elements (X1, . . . , Xℓ) that +are grouped into a set of disjoint equivalence classes which are enumerated +in a sequential way as it first element is discovered. Let kℓ be the number of +classes at step ℓ and (nℓ +j : j = 1, . . . , kℓ) be the number of elements in these +classes, so ℓ = �kℓ +j=1 nℓ +j. +When a new element Xℓ+1 is observed, there are two possibilities: this ele- +ment is in a class of an element collected before or at ℓ, in this case kℓ+1 = kℓ +and if Xℓ+1 belongs to the class j then nℓ+1 +j += nℓ +j + 1. When Xℓ+1 is in +none of the classes of the previous elements then a new class is discovered, +so kℓ+1 = kℓ + 1, nℓ+1 +kℓ+1 = 1 at step ℓ + 1 and the frequencies of the classes +that do not contain Xℓ+1 remain unchanged from ℓ to ℓ + 1. The entropy at +step ℓ is +Hℓ = − +kℓ +� +j=1 +nℓ +j +ℓ log +� +nℓ +j +ℓ +� +. +This relation is entirely similar to (8). We set 0 log 0 = 0, so one can add an +empty class without changing the entropy. +Remark 4.1. In general the sequence (Hℓ : ℓ ≥ 1) is neither increasing nor +decreasing. For instance if the observations Xi, i = 1, . . . , 4 are such that +the pairs {X1, X3} and {X2, X4} belong to the same class, but the classes are +different, it holds log 2 = H2 = H4 > H3. □ +One has Hℓ ≤ log ℓ := Hmax +ℓ +, and the equality is attained only when kℓ = ℓ, +that is when each of the ℓ elements defines its own class. We also have Hℓ ≥ 0 +and it vanishes only when there is a unique class containing the ℓ elements. +In all the other cases both inequalities, the upper and lower bounds, are +strict. Also notice that H1 = 0. +Below we will consider the steps ℓ and ℓ + 1 of the sequence (Hℓ : ℓ ≥ 1). We +will note by jℓ+1 ∈ {1, . . . , kℓ+1} the index of class that contains observation +Xℓ+1. Then, nℓ+1 +jℓ+1 is the frequency of class X∗ +jℓ+1 = Xℓ+1 at step ℓ + 1. +10 + +Proposition 4.2. The functional given by +Lf +ℓ = ℓ(log ℓ − Hℓ), for ℓ ≥ 1 and Lf +0 = 0, +is a nondecreasing and nonnegative functional along the trajectory (Xℓ : ℓ ≥ +1) and it remains constant, Lf +ℓ+1 = Lf +ℓ , only when a new species is discovered +at ℓ + 1. More precisely, ∆f +ℓ+1 = Lf +ℓ+1 − Lf +ℓ satisfies +∀ℓ ≥ 1, +∆f +ℓ+1 = njℓ+1 log(njℓ+1) − (njℓ+1−1) log(njℓ+1−1) ≥ 0, +(13) +and ∆f +ℓ+1 = 0 only when a new class is discovered at ℓ + 1, that is +∆f +ℓ+1 = 0 ⇔ njℓ+1 = 1. +(14) +Moreover, +(ℓ + 1)Hℓ+1 − ℓHℓ +(15) +=(ℓ+1) log(ℓ+1)−ℓ log ℓ− +� +njℓ+1 log(njℓ+1)−(njℓ+1−1) log(njℓ+1−1) +� +≥0, +and vanishes only when Kℓ+1 = 1. +Proof. We will show (15) at the end of the proof. All the other properties +will follow when we show that ∆f +ℓ+1 satisfies the equality in (13). In fact, +the inequality ∆f +ℓ+1 ≥ 0 is a direct consequence of it because j log j − (j − +1) log(j − 1) ≥ 0. This implies that the functional Lf +ℓ is nondecreasing. Also +we have that j log j − (j − 1) log(j − 1) vanishes only if j = 1, and so (14) +is obtained and this ensures that the functional L remains constant only at +times when a new class is discovered. +Notice that ∆f +1 = Lf +1 − Lf +0 = 0 is consistent with the fact that at step 1 a +new class is discovered. +Let us show the equality in (13). To simplify notation, we note j∗ = jℓ+1 the +class containing Xℓ+1 at step ℓ + 1. Also we write � +j̸=j∗ to mean +� +1≤j≤kℓ+1,j̸=j∗. +In the rest of the proof we note nj = nℓ+1 +j +for j = 1, . . . , kℓ+1, so nj∗ is the +cardinality of the class X∗ +j∗. If at step ℓ + 1 one has j ̸= j∗ then the number +of elements of the class j is equal at steps ℓ and ℓ + 1. We have +(ℓ + 1)Hℓ+1 = − +kℓ+1 +� +j=1 +nj log nj + (ℓ + 1) log(ℓ + 1) +and then +(ℓ+1)(log(ℓ+1)−Hℓ+1)= +kℓ+1 +� +j=1 +nj log nj = +� +j̸=j∗ +nj log nj+nj∗ log nj∗. +11 + +Now, the frequency of class j∗ at step ℓ is nj∗ − 1, so in a similar way as we +did for the term ℓ + 1 we get +ℓ(log ℓ − Hℓ) = +� +j̸=j∗ +nj log nj + (nj∗ − 1) log(nj∗ − 1). +Then, ∆f +ℓ+1 = (ℓ + 1)(log(ℓ + 1) − Hℓ+1) − ℓ(log ℓ − Hℓ) satisfies the equality +in (13). +Finally the equality in (15) is directly obtained from the equality in (13). The +inequality ≥ 0 in this relation is a consequence of the increasing property +of the function (n + 1) log(n + 1) − n log n for n ≥ 1, which follows from +(1 + 1/n)n < (1 + 1/(n + 1))n+1 for all n ≥ 1 (and 0 log 0 = 0). +Consider the function κ(ℓ + 1) = (ℓ + 1) log(ℓ + 1) − ℓ log ℓ for ℓ ≥ 1. From +x − x2/2 ≤ log(1 + x) ≤ x for x ≥ 0, we get +1 +2ℓ − 1 +2ℓ2 ≤ κ(ℓ + 1) − (log ℓ + 1) ≤ 1 +ℓ , +and for large ℓ we have κ(ℓ + 1) ≈ log ℓ + 1 + o(1). +These bounds and +approximation can be applied for ∆f +ℓ+1 = κ(njℓ+1). +4.2 +One step variation of the Bayesian entropy +Let us consider the one step variation of Bayesian entropy for the PDP. +Consider an i.i.d. sequence (Xn : n ≥ 1) of elements in X chosen with a +random measure Ξ(·) of a PDP(α, θ) which fixes the family of finite samples +Xℓ = (X1, . . . , Xℓ), ℓ ≥ 1. +Remark 4.3. We note that the sequence of entropies (E(H|Xℓ) : ℓ ≥ 1) is +neither increasing nor decreasing. We can illustrate it with the same example +used in Remark 4.1. So, assume the observations Xi, i = 1, . . . , 4 are such +that the pairs {X1, X3} and {X2, X4} are in the same class, but the classes +are different. It can be checked that when 0 ≤ α < 1/2 and −α < θ < 1−3α, +it holds E(H|X2) > E(H|X3) and E(H|X4) > E(H|X3). □ +In the next result we will compute the one step variation of the posterior +entropy of a PDP(α, θ), when taking the sample Xℓ+1 = (Xℓ, Xℓ+1). We +recall relation (10) that gives the maximum entropy for samples of size ℓ, it +is +max +Yℓ E(H|Yℓ) = ψ(θ + ℓ + 1) − ψ(1 − α) − +ℓ +θ + ℓ. +12 + +From (4) we get +(θ + ℓ + 1)ψ(θ + ℓ + 2) − (θ + ℓ)ψ(θ + ℓ + 1) = ψ(θ + ℓ + 1) + 1, +and so, +(θ+ℓ+1) max +Yℓ+1 E(H|Yℓ+1)−(θ+ℓ) max +Yℓ E(H|Yℓ)=ψ(θ+ℓ+1)−ψ(1−α). (16) +Now we state our main result, satisfied by the functional given in (1). As +in the frequentist case we note by jℓ+1 the index of the species Xℓ+1, that is +such that Xℓ+1 = X∗ +jℓ+1. +Theorem 4.4. Let (Xn : n ≥ 1) be an i.i.d. sequence of a PDP(α, θ). The +functional (Lℓ : ℓ ≥ 0) given by L0 = 0 and +Lℓ = (θ + ℓ) +� +max +Yℓ E(H|Yℓ) − E(H|Xℓ) +� +for ℓ ≥ 1; +is a nondecreasing and nonnegative functional along the trajectory (Xℓ : ℓ ≥ +1) and it remains constant, Lℓ+1 = Lℓ, only when a new species is discovered +at ℓ + 1. More precisely, let +∆ℓ+1 = Lℓ+1 − Lℓ, +and note j∗ = jℓ+1 be the index of the species Xℓ+1 and nj∗ = nℓ+1 +j∗ +be the +frequency of this species at step ℓ + 1. Then, +∆ℓ+1 = ψ(nj∗ − α) − ψ(1 − α) ≥ 0 +(17) +and it vanishes only when nj∗ = 1, that is when a new species is discovered +at ℓ + 1. Moreover +(θ+ℓ+1)E(H|Xℓ+1)−(θ+ℓ)E(H|Xℓ) = ψ(θ+ℓ+1)−ψ(nj∗ −α) > 0. (18) +Proof. The relation (18) will be shown at the end of the proof. Note that +for the rest of the relations it suffices to show (17) because nj∗ ≥ 1 and ψ is +strictly increasing then the expression at the right hand side of (17) increases +strictly with nj∗ and it vanishes only when nj∗ = 1. So, let us show equality +(17). +The sequence of mean posterior entropies is noted by �Hℓ = E(H|Xℓ), ℓ ≥ 1. +From (7) we have +(θ+ℓ) �Hℓ = (θ+ℓ)ψ(θ+ℓ+1) − (θ+αkℓ)ψ(1−α) − +kℓ +� +i=1 +(nℓ +i −α)ψ(nℓ +i−α + 1). +13 + +Let us define, +ηℓ+1 = (θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ. +(19) +From the definitions of ∆ and η and equality (16) we get +∆ℓ+1 = ψ(θ + ℓ + 1) − ψ(1 − α) − ηℓ+1. +So, instead of proving results for Lℓ and ∆ℓ we will do it for ηℓ. +Let Kℓ+1 = kℓ+1. +We note by nj = nℓ+1 +j +the frequency of class X∗ +j for +j = 1, . . . , kℓ+1. We will show that the following relation holds for ℓ ≥ 1: +ηℓ+1 = ψ(θ+ℓ+1) − ψ(nj∗−α). +(20) +Since this implies (17), the result of the Theorem will be satisfied. +We first show the case kℓ+1 = kℓ + 1, so j∗ = kℓ+1 is the index of a new class +and nj∗ = nkℓ+1 = 1. The mean posterior entropy �Hℓ+1 is computed from +(7) but with the sample size ℓ + 1, the number of species kℓ+1 = kℓ + 1, the +frequencies nj are unchanged for j = 1, . . . , kℓ and the frequency for the new +species is nkℓ+1 = 1. Then, +(θ + ℓ + 1) �Hℓ+1 += +(θ + ℓ + 1)ψ(θ + ℓ + 2) − (θ + (kℓ + 1)α)ψ(1 − α) +− +kℓ+1 +� +i=1 +(ni − α)ψ(ni − α + 1). +Now we use (4) on x = θ + ℓ + 2 to get (θ + ℓ + 1)ψ(θ + ℓ + 2) = (θ + ℓ)ψ(θ + +ℓ + 1) + ψ(θ + ℓ + 1) + 1, decompose the first term at the right hand side, +separate the term kℓ + 1 in the sum and use nkℓ+1 = 1, to obtain, +(θ + ℓ + 1) �Hℓ+1 += +(θ + ℓ + 1)ψ(θ + ℓ + 1) + 1 − (θ + (kℓ + 1)α)ψ(1 − α) +− +kℓ +� +i=1 +(ni − α)ψ(ni − α + 1) − (1 − α)ψ(2 − α). +On the other hand, +(θ + ℓ) �Hℓ += +(θ + ℓ)ψ(θ + ℓ + 1) − (θ + αkℓ)ψ(1 − α) +− +kℓ +� +i=1 +(ni − α)ψ(ni − α + 1). +By using (1 − α)ψ(2 − α) = (1 − α)ψ(1 − α) + 1, we get +ηℓ+1 = (θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ = ψ(θ + ℓ + 1) − ψ(1 − α). +14 + +So, relation (20) is shown when kℓ+1 = kℓ + 1. +Let us show (20) when kℓ+1 = kℓ. For j ̸= j∗ we have nj = nℓ+1 +j += nℓ +j, and +for j∗ we have nℓ +j∗ = nj∗ − 1. We will simplify some notation on sums and +put � +i̸=j∗ = � +i=1,...,k,i̸=j∗. From, +(θ+ℓ+1) �Hℓ+1 += +(θ+ℓ+1)ψ(θ+ℓ+2) − (θ+αkℓ)ψ(1−α) +− +� +i̸=j∗ +(ni−α)ψ(ni−α+1) − (nj∗−α)ψ(nj∗−α + 1), +and +(θ + ℓ) �Hℓ = (θ+ℓ)ψ(θ+ℓ+1) − (θ+αkℓ)ψ(1−α) − +kℓ +� +i=1 +(ni−α)ψ(ni−α+1), +we obtain +ηℓ+1 += +(θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ += +(θ + ℓ + 1)ψ(θ + ℓ + 2) − (θ + ℓ)ψ(θ + ℓ + 1) +−(nj∗ − α)ψ(nj∗ − α + 1) + (nj∗ − 1 − α)ψ(nj∗ − α). +By using (4) in x = θ + ℓ + 1 and x = nj∗ − α we get, +(θ + ℓ + 1)ψ(θ + ℓ + 2) − (θ + ℓ)ψ(θ + ℓ + 1) = ψ(θ + ℓ + 1) + 1 and +−(nj∗ − α)ψ(nj∗ − α + 1) + (nj∗ − α − 1)ψ(nj∗ − α) = −ψ(nj∗ − α) − 1. +Therefore +ηℓ+1 = ψ(θ + ℓ + 1) − ψ(nj∗ − α), +and the relation (20) is shown for the case kℓ+1 = kℓ. +To finish the proof of the Theorem let us show (18). It follows from definition +(19), the relation (20), the inequality θ > −α and ψ is increasing. +Remark 4.5. Set �Hmax +ℓ += maxYℓ E(H|Yℓ). We have analyzed the variation, +∆ℓ+1 = (θ + ℓ + 1)( �Hmax +ℓ+1 − �Hℓ+1) − (θ + ℓ)( �Hmax +ℓ +− �Hℓ). +Note that any other weights would produces only trivial changes or would +lead to the analysis of the variation weighted with the entropy. In fact if one +considers +cℓ+1 = (θ + ℓ + 1)(aℓ+1 − �Hℓ+1) − (θ + ℓ)(aℓ − �Hℓ), +15 + +then cℓ+1 = ∆ℓ+1+(θ+ℓ+1)(aℓ+1− �Hmax +ℓ+1 )−(θ+ℓ)(aℓ − �Hmax +ℓ +), so it suffices +to add to ∆ℓ+1 a deterministic sequence depending on ℓ. If one considers +c′ +ℓ+1 = bℓ+1( �Hmax +ℓ+1 − �Hℓ+1) − bℓ( �Hmax +ℓ +− �Hℓ), +one gets +c′ +ℓ+1 += +bℓ +�bℓ+1 +bℓ +( �Hmax +ℓ+1 − �Hℓ+1) − ( �Hmax +ℓ +− �Hℓ) +� += +bℓ +�bℓ+1 +bℓ +− θ+ℓ+1 +θ + ℓ +� +( �Hmax +ℓ+1 − �Hℓ+1) + +bℓ +θ + ℓ∆ℓ+1. +When we modify both, the additive and the multiplicative terms, in ∆ℓ+1 we +get a combination of above situations. □ +Remark 4.6. In the frequentist case the weighted difference between maximal +entropies at steps ℓ + 1 and ℓ is, +df +ℓ+1 = (ℓ + 1)Hmax +ℓ+1 − ℓHmax +ℓ += (ℓ + 1) log(ℓ + 1) − ℓ log ℓ. +From (16), in the Bayesian PDP case the weighted difference of posterior +entropies is, +dℓ+1 = (θ+ℓ+1) �Hmax +ℓ+1 −(θ+ℓ) �Hmax +ℓ += ∆ℓ+1+ηℓ+1 = ψ(θ+ℓ+1)−ψ(1−α). +For big ℓ we have that df +ℓ+1 is of the order of log ℓ+1 while from (6) one gets +that dℓ+1 is of the order of log ℓ − ψ(1 − α) (we recall that −ψ(1 − α) > 0). +□. +Remark 4.7. Now, by applying the relations (5) and (6) satisfied by the +digamma function, from Theorem 4.4 we get the following bounds for the +weighted entropy variation ηℓ+1 = (θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ given by (18), +ηℓ+1 +≥ +log(θ+ℓ+1) − +1 +θ+ℓ+1 − log(nj∗−α) + +1 +2(nj∗ − α), +ηℓ+1 +≤ +log(θ+ℓ+1) − +1 +2(θ+ℓ+1) − log(nj∗−α) + +1 +nj∗−α. +When ℓ is sufficiently big one has, +ηℓ+1 ≈ log(θ + ℓ + 1) − +1 +2(θ + ℓ + 1) if kℓ+1 = kℓ + 1; +and if also nj∗ is also sufficiently big, then +ηℓ+1 ≈ log(θ+ℓ+1) − +1 +2(θ+ℓ+1) − log(nj∗−α) + +1 +2(nj∗−α) if kℓ+1 = kℓ. +16 + +Remark 4.8. One can check that (18) also holds for ℓ = 0, where for the +posterior mean entropy (7), when ℓ = 0, one takes k = 0, and so θ �H0 +P DP = +θψ(θ + 1) − θψ(1 − α). So, by applying the telescopic property to (18) we get +(θ + ℓ) �Hℓ = Cℓ(α, θ) − +ℓ +� +i=1 +ψ(n∗(i) − α), +where Cℓ(α, θ) = +��ℓ +i=1 ψ(θ + i) +� ++ θψ(θ + 1) − θψ(1 − α), and n∗(i) = +#{1 ≤ j ≤ i : Xj = Xi} is the frequency of the class of the species Xi at +step i. Therefore, the only part of the entropy depending on the sample is +− �ℓ +i=1 ψ(n∗(i) − α). The terms −ψ(n∗(i) − α) strictly decreases with n∗(i) +(note that −ψ(n∗(i)−α) is positive when n∗(i) = 1, negative if n∗(i) ≥ 3 and +the sign of −ψ(2 − α) depends on α ∈ [0, 1)). So, the terms −ψ(n∗(i) − α) +can be seen as the ‘discovery value’ of observing the species Xi at step i, and +so, up to the additive deterministic term, the entropy turns out to be the +‘discovery’ values at the successive steps of the sample. On the other hand, +from (17) we get that +Lℓ = +ℓ +� +i=1 +(ψ(n∗(i) − α) − ψ(1 − α)) +is a sum of positive rewards for reinforcing what is already known that is going +in the opposite direction of discovery. Thus the reward at step i, attains the +minimum 0 for the discovery of a new species. Differently to entropy, here +no additional deterministic term depending on ℓ, α and θ is required. +4.3 +A common framework for the frequentist and the +PDP cases +The equations (17) and (13) have the same shape, both are measuring the +weighted differences of the distance of successive entropies to the maximal +entropies and both formulae express that these differences only depend on +the updated frequency of the species of the new element. In fact this result +holds for the class of entropies that satisfy: +w(ℓ)Hℓ = u(a + ℓ) − b − +k +� +i=1 +(u(nℓ +i − c) + v). +(21) +Here w(ℓ) is a strictly positive function and increasing in ℓ and u is a real +function defined on N − c = {n − c : n ≥ 1} and it satisfies +u(n + 1 − c) − u(n − c) is increasing for n ≥ 1. +(22) +17 + +The quantities a, b, c, v are constants that satisfy the conditions +0 ≤ c < 1, −c ≤ a and 2u(1 − c) + v < u(2 − c). +(23) +Notice that Hℓ can be written as Hℓ with w(ℓ) = ℓ, u(x) = x log x and +a = b = c = v = 0; and �Hℓ can be also written in the form Hℓ with +w(ℓ) = θ + ℓ, u(x) = xψ(x + 1), a = θ, b = θψ(1 − α), c = α, v = αψ(1 − α). +In both cases 0 ≤ c < 1. The second part in (23) holds for the PDP because +θ > −α and the third part of (23) holds in the frequentist case because it is +equivalent to 2 log(1) ≤ log 2 and in the PDP case (23) becomes (1−α)ψ(2− +α) + αψ(1 − α) < ψ(2 − α) + 1 which is satisfied. In relation to (22), in the +PDP case it follows from xψ(x + 1) − (x − 1)ψ(x) increasing in x > 0 and +in the frequentist case (22) it is a consequence of (n + 2) log(n + 2) − (n + +1) log(n + 1) > (n + 1) log(n + 1) − n log n for n ≥ 0. +We will see that the conditions (22) and (23) are sufficient to show that the +properties proven for the variation of differences between maximal entropies +and entropies for the cases (Hℓ) and ( �Hℓ), also hold for the entropy (Hℓ) +written in (21). +In order to retrieve the results in Proposition 3.1 we need to analyze what +happens when, for two species i ̸= j with nℓ +i = n > 1 and nℓ +j = m, one makes +the change m → m + 1 and n → n − 1, and all other frequencies nl remain +equal. The entropy increases if and only if u(n−c−1)+u(m−c+1) ≤ u(n− +c)+u(m−c), or equivalently u(m−c+1)−u(m−c) ≤ u(n−c)−u(n−c−1). +From (22) this holds if and only if m + 1 ≤ n. +The second requirement has to do with the following change: for a class i ≤ k +with ni = n > 1 we set n → n − 1 and k → k + 1 so there is a new class with +nk+1 = 1. This change makes the entropy increase if u(n−c−1)+u(1−c)+v < +u(n − c) or equivalently if u(1 − c) + v < u(n − c) − u(n − c − 1) when +n > 1. From (22) we get that it suffices that the following inequality holds +2u(1 − c) + v < u(2 − c), which is the second condition in (23). +When these conditions take place the maximal entropy is attained when all +the classes are singletons, so +w(ℓ)Hmax +ℓ += u(a + ℓ) − b − +ℓ +� +i=1 +(u(1 − c) + v) +Hence, +w(ℓ + 1)Hmax +ℓ+1 − w(ℓ)Hmax +ℓ += u(a + ℓ + 1) − u(a + ℓ) − (u(1 − c) + v). +18 + +Let us consider +∆H +ℓ+1 = w(ℓ + 1) +� +Hmax +ℓ+1 − Hℓ+1 +� +− w(ℓ) (Hmax +ℓ +− Hℓ) . +If in the transition ℓ → ℓ + 1 the number of classes changes from k → k + 1 +one gets that +∆H +ℓ+1 = 0. +If in the transition ℓ → ℓ + 1 the number of classes is preserved, say k, and +the class j∗ adds in one unit we get +∆H +ℓ+1 = u(nℓ +j∗ − c + 1) − u(nℓ +j∗ − c) − (u(1 − c) + v). +We combine (22) with the third condition in (23), to deduce that when the +transition ℓ to ℓ+1 preserves the number of classes then ∆H +ℓ+1 > 0. Hence, the +results for the variation of the weighted differences of the maximal entropy +to the entropy hold for this class of entropies (21). +Finally, let us see what one requires to have +w(ℓ+1)Hℓ+1−w(ℓ)Hℓ = (u(a+ℓ+1)−u(a+ℓ))−(u(nℓ +j∗−c+1)−u(nℓ +j∗−c)) ≥ 0. +Since from (23) we have a ≥ −c and so the unique new condition is +u(n + a + 1) − u(n + a) ≥ u(m − c + 1) − u(m − c) for n ≥ m, +which is satisfied for both, the PDP and the frequentist case. +Acknowledgments. This work was supported by the Center for Mathemat- +ical Modeling ANID Basal PIA program FB210005. In addition, we would +like to thank the reviewer for their careful reading and valuable comments +and suggestions, which helped to clarify and improve the presentation of the +article. +References +[1] Abramowitz, Milton and Stegun, Irene A. Handbook of mathemati- +cal functions with formulas, graphs, and mathematical tables. (1972), +Dover. +[2] Alzer, Horst. 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College London. +21 + diff --git a/ZtFST4oBgHgl3EQfAzi6/content/tmp_files/load_file.txt b/ZtFST4oBgHgl3EQfAzi6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e9a8a4a80313602d19c0e8ab8e84045faca817a --- /dev/null +++ b/ZtFST4oBgHgl3EQfAzi6/content/tmp_files/load_file.txt @@ -0,0 +1,500 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf,len=499 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='13700v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='PR] 31 Jan 2023 One step entropy variation in sequential sampling of species for the Poisson-Dirichlet Process Servet Mart´ınez∗ Javier Santib´a˜nez† Departamento de Ingenier´ıa Matem´atica and Centro de Modelamiento Matem´atico, UMI 2071 CNRS-UCHILE, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Santiago, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' February 1, 2023 Abstract We consider the sequential sampling of species, where observed samples are classified into the species they belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We are partic- ularly interested in studying some quantities describing the sampling process when there is a new species discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We assume that the observations and species are organized as a two-parameter Poisson- Dirichlet Process, which is commonly used as a Bayesian prior in the context of entropy estimation, and we use the computation of the mean posterior entropy given a sample developed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Our main result shows the existence of a monotone functional, constructed from the difference between the maximal entropy and the mean entropy throughout the sampling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We show that this functional re- mains constant only when a new species discovery occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' AMS Classification Number: 94A17 Keywords: Entropy, Bayesian posterior distribution, Poisson-Dirichlet Pro- cess, new species discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' ∗E-mail address: smartine@dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='uchile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' †E-mail address: jsantibanez@dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='uchile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 1 1 Introduction Consider the sequential sampling of species, where one takes a random sample from a population and classifies each observation according to the species (or classes) to which they belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Because the population is large, there are some rare species that may not be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We intend to understand and model the discovery of a new species in this context and to study related informational quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Our main result shows that the two-step variation of differences between the maximal entropy and the entropy allows us to describe when a new species is discovered in the Poisson-Dirichlet Process (PDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' It is worth mentioning that our work is purely statistical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The two parameter PDP —introduced by Pitman and Yor in 1997 [15]— supplies random partitions with an infinite number of components in [0, 1] and serves to model the process of sampling species and the times at which new species are discovered, see [11], [8] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This process has been used in ecology, but also in genetic applications [7], natural language processing [16] and finance [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In Section 2, we will introduce the PDP and some of the basic properties that we shall use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Entropy is a way to measure the diversity of communities in a sample and our work focuses on studying some aspects of the posterior entropy of the process of sampling species in the PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The computation of posterior entropy relies on the fact that given the sample from a PDP, the posterior distribution is a mixture of a finite Dirichlet distribution and a PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Much of this paper concern with Bayesian entropy estimation, is due to the results in [4], in which the prior and posterior mean entropies for the PDP were computed and some of their properties stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='1, we provide lower and upper bounds for the entropy when the sample size is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The main purpose of this work is to obtain an increasing functional along the process constructed with posterior mean entropy between two successive steps of the PDP with parameters (α, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This functional is, Lℓ = (θ + ℓ)( �Hmax ℓ − �Hℓ), (1) and satisfies the monotone property Lℓ+1 ≥ Lℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Here �Hℓ denotes the posterior entropy when observing a sample at step ℓ and �Hmax ℓ is its maximum over all samples of size ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Our main result is Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='4 in Section 4, where we show that Lℓ is increasing and the equality Lℓ+1 = Lℓ is attained only when a new species is discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 2 We also show that the weighted difference of entropies satisfies (θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The expression (18) obtained in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='4, for the above difference of weighted entropies, allows us to think of the entropy as a sum of the ‘dis- covery values’ of the sampled species, plus an additive deterministic term depending on ℓ, α and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' On the other hand, the expression (17) allows us to write straightforwardly the functional Lℓ as a sum of positive rewards for ‘reinforcing the knowledge’ of what it is known, and no additional additive term is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The discovery values and the reinforcement rewards are expressed in terms of the digamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This is discussed in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We also study similar quantities in the frequentist framework and relations in the same vein are shown in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 2 Poisson-Dirichlet Process This section is devoted to the definition of the PDP and to supply some of its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We follow the articles [14], [5], [18], [13], [16] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Since this is a well-known theory we only state those results directly related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let 0 ≤ α < 1 and θ > −α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Consider independent random variables βk ∼ Beta(1 − α, θ + αk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let π = (πk : k ≥ 1) be given by the two-parameter Griffiths-Engen-McCloskey distribution, GEM(α, θ), π1 := β1, πk := βk k−1 � j=1 (1 − βj) k ≥ 2, which defines a probability vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Now consider a non-atomic probability measure G defined on space X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let (φk : k ≥ 1) be an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' sequence with distribution as G, then are all different a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We assume φ = (φk : k ≥ 1) are independent of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The discrete random measure Ξ(·) = � k≥1 πkδφk(·) (2) is called the PDP with base measure G and parameters α and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The base measure G is non-atomic, this is used to give different names to the species in the process Ξ(·), but the unique fact that matters is that the species are 3 different, the exact names are not important, and this explains why we ignore G and one simply notes PDP(α, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The case α = 0 is called Dirichlet process and it can be constructed as an infinite extension of a Dirichlet distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Examples on how PDP help to model different phenomena can be seen in [14] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Samples from a PDP are obtained from (2) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' For a random measure Ξ(·) one takes an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' sequence of variables (Xn : n ≥ 1) with values in X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let Xℓ = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , Xℓ) be a sample of size ℓ collected in a sequential way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' By Kℓ we note the total number of different species of the sample which are noted by X∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , X∗ Kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' For j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , Kℓ we note by Nℓ j the number of times that the species X∗ j is observed in the sample, so ℓ = �Kℓ j=1 Nℓ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Further we do not take into account the order of the species in the sample, if needed one can enumerate their frequencies in their decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, (Nℓ j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , Kℓ) means the multiset of frequencies (that is a set where the values can be repeated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The conditional probability for a new observation Xℓ+1 is, see [5], P(Xℓ+1 = • | Xℓ) = θ + αKℓ θ + ℓ G(·) + Kℓ � j=1 Nℓ j − α θ + ℓ δX∗ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (3) So, the observation Xℓ+1 is part of the species X∗ j already observed with probability Nℓ j −α θ+ℓ , and Xℓ+1 defines a new species with probability θ+αKℓ θ+ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In this last case the new species Xℓ+1 = X∗ Kℓ+1 is distributed as G independently of the species already discovered, and ℓ+1 is said to be the discovery time of a new species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' That is, the transition probability (3) states the probability of discovering a new species and gives a different name to it, the important point is that it is different to the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 3 Bayesian entropy To define the Bayesian entropy one assumes a prior distribution and makes the estimation of entropy based upon the posterior distribution given the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will introduce Bayesian entropy in the context of PDP following closely, as mentioned in the introduction, the results in [4], and also [3] and [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' To do so, we need to recall the definition of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let π be a distribution, the Shannon entropy is defined as H(π) = − ∞ � i=1 πi log(πi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 4 For further computations it is useful to introduce the digamma function and some of its properties, which can be found in [1] and [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This function is the logarithmic derivative of the Gamma function: ψ(x) = d dx log(Γ(x)) = Γ′(x) Γ(x) , where Γ(x) = � ∞ 0 tx−1e−tdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From Γ(x + 1) = xΓ(x), one gets ψ(x + 1) = ψ(x) + 1/x for x > 0, that implies xψ(x + 1) − (x − 1)ψ(x) = ψ(x) + 1, x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (4) The digamma function is increasing for x > 0 and then xψ(x+1)−(x−1)ψ(x) is also increasing for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Since ψ(2) > 0, then xψ(x + 1) > (x − 1)ψ(x) when x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The digamma function admits the following bounds in terms of the logarithmic function, see [2]: log(x) − 1 x ≤ ψ(x) ≤ log(x) − 1 2x, x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (5) For x sufficiently big the digamma function can be approximated by ψ(x) = log(x) − 1 2x + o �1 x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='1 Entropy for the Poisson-Dirichlet Process Let Xℓ = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , Xℓ) be a sample following a distribution π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The Bayesian approach for estimating the entropy requires to assume a prior distribution π and estimate the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The least square Bayes estimator has the shape: E(H(π)|Xℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' When one takes a PDP as prior, the sample Xℓ should be obtained from the random measure Ξ, given by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' But, as we mentioned before, we can omit any reference to G, so the sample is obtained from the weight distribution π and we will refer to the process and its weight distribution indistinctly by the same symbol, that is, the prior is π ∼ PDP(α, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In [4] the prior mean of H(π) is proven to be, E(H(π)) = ψ(θ + 1) − ψ(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We are interested in finding the posterior mean of H(π), after seeing a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' To describe the posterior distribution consider the sample Xℓ with Kℓ dif- ferent species and frequencies Nℓ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , Nℓ Kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' To simplify notation put Kℓ = k 5 and Nℓ j = nj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In [10] it was shown that the posterior distri- bution πpost = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , pk, (1 − �k j=1 pj)π′) is given by the mixture (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , pk, 1 − k � j=1 pj) ∼ Dirichlet(n1 − α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , nk − α, θ + αk) π′ = (π′ 1, π′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' ) ∼ PDP(α, θ + αk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Hence, the probability of belonging to some species X∗ j already present in the sample is pj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' and the probability to belong to a new species is 1 − �k j=1 pj, where the distribution of these probabilities depend on the frequencies (nj) and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In the event that a new species is discovered it will be part of a specific species i with weight π′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The species X∗ i related to the prior distribution π, is not the same as the species X∗ i in the posterior distribution πpost, because the index taken af- ter observing the sample is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' But, this index discrepancy does not cause any problem since the ordering of πi is not important in H(π) and the transition probability for the discovery of a new species and for the species that have been discovered in the past continues to have the weights given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Also, the posterior distribution of π is represented by a realization πpost whose ordering is totally different from the ordering of π, this realization is only one representation of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The Bayes estimator of the posterior mean of the entropy under the PDP prior, at step ℓ, will be defined as �Hℓ P DP = E(H(π)|Xℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will write H instead of H(π) when there is no confusion, so �Hℓ P DP = E(H|Xℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In [4] it was shown that the posterior mean of H under the PDP prior is, �Hℓ P DP = ψ(θ+ℓ+1)− θ + αk θ + ℓ ψ(1−α)− 1 θ + ℓ k � i=1 (ni −α)ψ(ni −α+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (7) Let �πℓ be the vector of empirical probabilities �πℓ i = ni/ℓ, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , k, and �πℓ i = 0 for i > k, given by the sample Xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The Maximum Likelihood Estimator (MLE) of the entropy, at step ℓ, under multinomial likelihood, is given by �Hℓ MLE = H(�πℓ) = − ∞ � i=1 �πℓ i log(�πℓ i), (8) 6 which is a biased estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In [4] it is shown that when Kℓ/ℓ converges in probability to 0, then �Hℓ P DP satisfies the following consistency property, | �Hℓ P DP − �Hℓ MLE| → 0 as ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (9) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='2 Bounds for the posterior PDP entropy Let us obtain lower and upper bounds for the entropy when the sample size is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This is made firstly when the number of species is fixed and after over all possible number of species in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' For a sample Xℓ of a PDP(α, θ), with k different species the entropy is upper and lower bounded by, E(H|Xℓ) ≤ ψ(θ+ℓ+1) − θ+αk θ+ℓ ψ(1−α)− 1 θ+ℓ k � i=1 (ni−α)ψ(ni−α+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' E(H|Xℓ) ≥ ψ(θ+ℓ+1) − θ+αk θ+ℓ ψ(1−α)− 1 θ+ℓ k � i=1 (ni−α)ψ(ni−α+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' where the vectors of frequencies (ni : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , k) and (ni : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , k) of the maximal entropy and the minimal entropy respectively, have the following structures up to index permutation: ni = ⌊ℓ/k⌋, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , lk, ni = ⌊ℓ/k⌋ + 1, i = lk + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , lk + hk where ⌊x⌋ is the biggest integer smallest or equal to x, hk = ℓ − k⌊ℓ/k⌋ and lk = k − hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' and nk = ℓ − (k − 1) and ni = 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Moreover, when one looks for the global bounds on all entropy maxima for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , ℓ}, one finds that: the global maximum is attained when the ℓ elements of the sample belong to different species and the global minimum is attained when the ℓ elements of the sample belong to a unique species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This is, min Yℓ E(H|Yℓ) ≤ E(H|Xℓ) ≤ max Yℓ E(H|Yℓ) with max Yℓ E(H|Yℓ) = ψ(θ+ℓ+1) − ψ(1 − α) − ℓ θ + ℓ, (10) min Yℓ E(H|Yℓ) = ψ(θ+ℓ+1)−(θ+α)ψ(1−α) θ+ℓ −(ℓ−α)ψ(ℓ−α+1) θ+ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (11) 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will take into account that −ψ(1 − α) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let us first prove the extremal entropies for a fixed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' If k = 1 there nothing to examine because n1 = ℓ and one simply computes the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Take two species i ̸= j and set ni = n, nj = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Assume n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will fix when the entropy grows when one makes the change n → n − 1, m → m + 1 and all other frequencies nl are equal, so the number of classes continues to be k and the sum of their frequencies continues to be ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This change makes the entropy grow if and only if the following inequality holds (we take into account that there is a minus in front of the third term at the right hand side in (7)), (n − 1 − α)ψ(n − α) + (m + 1 − α)ψ(m + 2 − α) ≤ (n − α)ψ(n − α + 1) + (m − α)ψ(m − α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From (4) this is equivalent to 0 ≤ −ψ(m − α + 1) − 1 + ψ(n − α) + 1 = ψ(n − α) − ψ(m − α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' But this is equivalent to m + 1 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, when this last inequality holds we make the change n → n − 1 and m → m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (Note that if n = m + 1 the change leaves the set of frequencies invariant because the new pair is the same, m, m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Therefore the maximal entropy for k classes is attained by the following structure of frequencies: ni = ⌊ℓ/k⌋, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , lk, ni = ⌊ℓ/k⌋ + 1, i = lk + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' lk + hk with hk = ℓ − k⌊ℓ/k⌋ and lk = k − hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This is the frequencies are ’as equal as possible’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' On the opposite when m + 1 ≥ n, the change n → n − 1, m → m + 1, makes the entropy decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, the minimal entropy structure of frequencies is given by n1 = ℓ − (k − 1) and the rest of k − 1 species have frequency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Therefore the first two inequalities of the Proposition are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Now for obtaining the global maxima and minima we must see what happens with the extreme solutions for different k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This is based upon the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Assume we have k < ℓ number of species with frequencies (n1, · · · , nk) and nk > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let us see what happens when we change this structure of frequencies to one that contains k+1 species and (n1, · · · , nk−1, nk − 1, 1), so with nk+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We claim that this operation makes the entropy strictly bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In fact by (7) the claim is equivalent to −αψ(1−α)−(nk−1−α)ψ(nk−α)−(1−α)ψ(2−α) > −(nk−α)ψ(nk+1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 8 By using (4) this last inequality is equivalent to −αψ(1 − α) − (1 − α)ψ(2 − α) + ψ(nk − α) + 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (12) Since ψ(nk −α) ≥ ψ(2−α) it suffices to check the inequality (12) for nk = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' When in the expression at the left hand side in (12) we set nk = 2 we get, α(ψ(2 − α) − ψ(1 − α)) + 1, which is strictly positive, so (12) holds and the claim is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Then, if one takes the maximal configuration for k < ℓ species, we know that there exists a frequency, that we can assume is the k−th one, that satisfies nk > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, by making the above operation gives a configuration of frequencies of a total number of species k + 1 and such that the entropy increases strictly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In particular the maximal entropy for k + 1 species is strictly bigger than the maximal entropy for k species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Then, (10) is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Finally when we make the above operation from the minimal configuration of k species we retrieve the minimal configuration of the k + 1 species and so the minimal entropy for k species is strictly lower than the minimal entropy for k + 1 species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, (11) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The result is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From (11) and since −ψ(1 − α) > 0, we get min Yℓ ((θ+ℓ)E(H|Yℓ))≥(θ+ℓ)ψ(θ+ℓ+1)−(ℓ−α)ψ(ℓ−α+1), where θ > −α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' On the other hand for every real h > 0 we have (x+h) log(x+ h + 1) − x log(x + 1) → ∞ as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Then, by also using (6) we get that min Yℓ ((θ+ℓ)E(H|Yℓ)) → ∞ as ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' □ The relation (9) shows a key property between the frequentist estimator based on empirical probabilities and the Bayesian estimator based on the posterior mean under the PDP prior, when ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In next section we will study the variation of weighted estimators when making a finite step ℓ to ℓ+1, showing a property that is similar for both, the frequentist and the PDP cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 4 One step variation of entropy and discovery of a new species We will state and prove our main result: an equality proving that a weighted variation between two successive steps of the posterior Bayesian entropy, is 9 nonnegative and only vanishes in the discovery times of a new species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This is done in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Related to this result, we previously study the variation of the entropy when one only computes frequencies, and how it characterizes discovery time of species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='1 One step variation of entropy for frequencies The framework is the following one: we collect a series of elements that are be- ing classified in some class or species, at the moment when they are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' At step ℓ one has collected in a sequential way ℓ elements (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , Xℓ) that are grouped into a set of disjoint equivalence classes which are enumerated in a sequential way as it first element is discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let kℓ be the number of classes at step ℓ and (nℓ j : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , kℓ) be the number of elements in these classes, so ℓ = �kℓ j=1 nℓ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' When a new element Xℓ+1 is observed, there are two possibilities: this ele- ment is in a class of an element collected before or at ℓ, in this case kℓ+1 = kℓ and if Xℓ+1 belongs to the class j then nℓ+1 j = nℓ j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' When Xℓ+1 is in none of the classes of the previous elements then a new class is discovered, so kℓ+1 = kℓ + 1, nℓ+1 kℓ+1 = 1 at step ℓ + 1 and the frequencies of the classes that do not contain Xℓ+1 remain unchanged from ℓ to ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The entropy at step ℓ is Hℓ = − kℓ � j=1 nℓ j ℓ log � nℓ j ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This relation is entirely similar to (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We set 0 log 0 = 0, so one can add an empty class without changing the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In general the sequence (Hℓ : ℓ ≥ 1) is neither increasing nor decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' For instance if the observations Xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , 4 are such that the pairs {X1, X3} and {X2, X4} belong to the same class, but the classes are different, it holds log 2 = H2 = H4 > H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' □ One has Hℓ ≤ log ℓ := Hmax ℓ , and the equality is attained only when kℓ = ℓ, that is when each of the ℓ elements defines its own class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We also have Hℓ ≥ 0 and it vanishes only when there is a unique class containing the ℓ elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In all the other cases both inequalities, the upper and lower bounds, are strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Also notice that H1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Below we will consider the steps ℓ and ℓ + 1 of the sequence (Hℓ : ℓ ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will note by jℓ+1 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , kℓ+1} the index of class that contains observation Xℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Then, nℓ+1 jℓ+1 is the frequency of class X∗ jℓ+1 = Xℓ+1 at step ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 10 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The functional given by Lf ℓ = ℓ(log ℓ − Hℓ), for ℓ ≥ 1 and Lf 0 = 0, is a nondecreasing and nonnegative functional along the trajectory (Xℓ : ℓ ≥ 1) and it remains constant, Lf ℓ+1 = Lf ℓ , only when a new species is discovered at ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' More precisely, ∆f ℓ+1 = Lf ℓ+1 − Lf ℓ satisfies ∀ℓ ≥ 1, ∆f ℓ+1 = njℓ+1 log(njℓ+1) − (njℓ+1−1) log(njℓ+1−1) ≥ 0, (13) and ∆f ℓ+1 = 0 only when a new class is discovered at ℓ + 1, that is ∆f ℓ+1 = 0 ⇔ njℓ+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (14) Moreover, (ℓ + 1)Hℓ+1 − ℓHℓ (15) =(ℓ+1) log(ℓ+1)−ℓ log ℓ− � njℓ+1 log(njℓ+1)−(njℓ+1−1) log(njℓ+1−1) � ≥0, and vanishes only when Kℓ+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will show (15) at the end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' All the other properties will follow when we show that ∆f ℓ+1 satisfies the equality in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In fact, the inequality ∆f ℓ+1 ≥ 0 is a direct consequence of it because j log j − (j − 1) log(j − 1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This implies that the functional Lf ℓ is nondecreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Also we have that j log j − (j − 1) log(j − 1) vanishes only if j = 1, and so (14) is obtained and this ensures that the functional L remains constant only at times when a new class is discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Notice that ∆f 1 = Lf 1 − Lf 0 = 0 is consistent with the fact that at step 1 a new class is discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let us show the equality in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' To simplify notation, we note j∗ = jℓ+1 the class containing Xℓ+1 at step ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Also we write � j̸=j∗ to mean � 1≤j≤kℓ+1,j̸=j∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In the rest of the proof we note nj = nℓ+1 j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , kℓ+1, so nj∗ is the cardinality of the class X∗ j∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' If at step ℓ + 1 one has j ̸= j∗ then the number of elements of the class j is equal at steps ℓ and ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We have (ℓ + 1)Hℓ+1 = − kℓ+1 � j=1 nj log nj + (ℓ + 1) log(ℓ + 1) and then (ℓ+1)(log(ℓ+1)−Hℓ+1)= kℓ+1 � j=1 nj log nj = � j̸=j∗ nj log nj+nj∗ log nj∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 11 Now, the frequency of class j∗ at step ℓ is nj∗ − 1, so in a similar way as we did for the term ℓ + 1 we get ℓ(log ℓ − Hℓ) = � j̸=j∗ nj log nj + (nj∗ − 1) log(nj∗ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Then, ∆f ℓ+1 = (ℓ + 1)(log(ℓ + 1) − Hℓ+1) − ℓ(log ℓ − Hℓ) satisfies the equality in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Finally the equality in (15) is directly obtained from the equality in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The inequality ≥ 0 in this relation is a consequence of the increasing property of the function (n + 1) log(n + 1) − n log n for n ≥ 1, which follows from (1 + 1/n)n < (1 + 1/(n + 1))n+1 for all n ≥ 1 (and 0 log 0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Consider the function κ(ℓ + 1) = (ℓ + 1) log(ℓ + 1) − ℓ log ℓ for ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From x − x2/2 ≤ log(1 + x) ≤ x for x ≥ 0, we get 1 2ℓ − 1 2ℓ2 ≤ κ(ℓ + 1) − (log ℓ + 1) ≤ 1 ℓ , and for large ℓ we have κ(ℓ + 1) ≈ log ℓ + 1 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' These bounds and approximation can be applied for ∆f ℓ+1 = κ(njℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='2 One step variation of the Bayesian entropy Let us consider the one step variation of Bayesian entropy for the PDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Consider an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' sequence (Xn : n ≥ 1) of elements in X chosen with a random measure Ξ(·) of a PDP(α, θ) which fixes the family of finite samples Xℓ = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , Xℓ), ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We note that the sequence of entropies (E(H|Xℓ) : ℓ ≥ 1) is neither increasing nor decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We can illustrate it with the same example used in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, assume the observations Xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , 4 are such that the pairs {X1, X3} and {X2, X4} are in the same class, but the classes are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' It can be checked that when 0 ≤ α < 1/2 and −α < θ < 1−3α, it holds E(H|X2) > E(H|X3) and E(H|X4) > E(H|X3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' □ In the next result we will compute the one step variation of the posterior entropy of a PDP(α, θ), when taking the sample Xℓ+1 = (Xℓ, Xℓ+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We recall relation (10) that gives the maximum entropy for samples of size ℓ, it is max Yℓ E(H|Yℓ) = ψ(θ + ℓ + 1) − ψ(1 − α) − ℓ θ + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 12 From (4) we get (θ + ℓ + 1)ψ(θ + ℓ + 2) − (θ + ℓ)ψ(θ + ℓ + 1) = ψ(θ + ℓ + 1) + 1, and so, (θ+ℓ+1) max Yℓ+1 E(H|Yℓ+1)−(θ+ℓ) max Yℓ E(H|Yℓ)=ψ(θ+ℓ+1)−ψ(1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (16) Now we state our main result, satisfied by the functional given in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' As in the frequentist case we note by jℓ+1 the index of the species Xℓ+1, that is such that Xℓ+1 = X∗ jℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let (Xn : n ≥ 1) be an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' sequence of a PDP(α, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The functional (Lℓ : ℓ ≥ 0) given by L0 = 0 and Lℓ = (θ + ℓ) � max Yℓ E(H|Yℓ) − E(H|Xℓ) � for ℓ ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' is a nondecreasing and nonnegative functional along the trajectory (Xℓ : ℓ ≥ 1) and it remains constant, Lℓ+1 = Lℓ, only when a new species is discovered at ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' More precisely, let ∆ℓ+1 = Lℓ+1 − Lℓ, and note j∗ = jℓ+1 be the index of the species Xℓ+1 and nj∗ = nℓ+1 j∗ be the frequency of this species at step ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Then, ∆ℓ+1 = ψ(nj∗ − α) − ψ(1 − α) ≥ 0 (17) and it vanishes only when nj∗ = 1, that is when a new species is discovered at ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Moreover (θ+ℓ+1)E(H|Xℓ+1)−(θ+ℓ)E(H|Xℓ) = ψ(θ+ℓ+1)−ψ(nj∗ −α) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The relation (18) will be shown at the end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Note that for the rest of the relations it suffices to show (17) because nj∗ ≥ 1 and ψ is strictly increasing then the expression at the right hand side of (17) increases strictly with nj∗ and it vanishes only when nj∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, let us show equality (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The sequence of mean posterior entropies is noted by �Hℓ = E(H|Xℓ), ℓ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From (7) we have (θ+ℓ) �Hℓ = (θ+ℓ)ψ(θ+ℓ+1) − (θ+αkℓ)ψ(1−α) − kℓ � i=1 (nℓ i −α)ψ(nℓ i−α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 13 Let us define, ηℓ+1 = (θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (19) From the definitions of ∆ and η and equality (16) we get ∆ℓ+1 = ψ(θ + ℓ + 1) − ψ(1 − α) − ηℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, instead of proving results for Lℓ and ∆ℓ we will do it for ηℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let Kℓ+1 = kℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We note by nj = nℓ+1 j the frequency of class X∗ j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , kℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will show that the following relation holds for ℓ ≥ 1: ηℓ+1 = ψ(θ+ℓ+1) − ψ(nj∗−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (20) Since this implies (17), the result of the Theorem will be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We first show the case kℓ+1 = kℓ + 1, so j∗ = kℓ+1 is the index of a new class and nj∗ = nkℓ+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The mean posterior entropy �Hℓ+1 is computed from (7) but with the sample size ℓ + 1, the number of species kℓ+1 = kℓ + 1, the frequencies nj are unchanged for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' , kℓ and the frequency for the new species is nkℓ+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Then, (θ + ℓ + 1) �Hℓ+1 = (θ + ℓ + 1)ψ(θ + ℓ + 2) − (θ + (kℓ + 1)α)ψ(1 − α) − kℓ+1 � i=1 (ni − α)ψ(ni − α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Now we use (4) on x = θ + ℓ + 2 to get (θ + ℓ + 1)ψ(θ + ℓ + 2) = (θ + ℓ)ψ(θ + ℓ + 1) + ψ(θ + ℓ + 1) + 1, decompose the first term at the right hand side, separate the term kℓ + 1 in the sum and use nkℓ+1 = 1, to obtain, (θ + ℓ + 1) �Hℓ+1 = (θ + ℓ + 1)ψ(θ + ℓ + 1) + 1 − (θ + (kℓ + 1)α)ψ(1 − α) − kℓ � i=1 (ni − α)ψ(ni − α + 1) − (1 − α)ψ(2 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' On the other hand, (θ + ℓ) �Hℓ = (θ + ℓ)ψ(θ + ℓ + 1) − (θ + αkℓ)ψ(1 − α) − kℓ � i=1 (ni − α)ψ(ni − α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' By using (1 − α)ψ(2 − α) = (1 − α)ψ(1 − α) + 1, we get ηℓ+1 = (θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ = ψ(θ + ℓ + 1) − ψ(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 14 So, relation (20) is shown when kℓ+1 = kℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Let us show (20) when kℓ+1 = kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' For j ̸= j∗ we have nj = nℓ+1 j = nℓ j, and for j∗ we have nℓ j∗ = nj∗ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will simplify some notation on sums and put � i̸=j∗ = � i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=',k,i̸=j∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From, (θ+ℓ+1) �Hℓ+1 = (θ+ℓ+1)ψ(θ+ℓ+2) − (θ+αkℓ)ψ(1−α) − � i̸=j∗ (ni−α)ψ(ni−α+1) − (nj∗−α)ψ(nj∗−α + 1), and (θ + ℓ) �Hℓ = (θ+ℓ)ψ(θ+ℓ+1) − (θ+αkℓ)ψ(1−α) − kℓ � i=1 (ni−α)ψ(ni−α+1), we obtain ηℓ+1 = (θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ = (θ + ℓ + 1)ψ(θ + ℓ + 2) − (θ + ℓ)ψ(θ + ℓ + 1) −(nj∗ − α)ψ(nj∗ − α + 1) + (nj∗ − 1 − α)ψ(nj∗ − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' By using (4) in x = θ + ℓ + 1 and x = nj∗ − α we get, (θ + ℓ + 1)ψ(θ + ℓ + 2) − (θ + ℓ)ψ(θ + ℓ + 1) = ψ(θ + ℓ + 1) + 1 and −(nj∗ − α)ψ(nj∗ − α + 1) + (nj∗ − α − 1)ψ(nj∗ − α) = −ψ(nj∗ − α) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Therefore ηℓ+1 = ψ(θ + ℓ + 1) − ψ(nj∗ − α), and the relation (20) is shown for the case kℓ+1 = kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' To finish the proof of the Theorem let us show (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' It follows from definition (19), the relation (20), the inequality θ > −α and ψ is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Set �Hmax ℓ = maxYℓ E(H|Yℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We have analyzed the variation, ∆ℓ+1 = (θ + ℓ + 1)( �Hmax ℓ+1 − �Hℓ+1) − (θ + ℓ)( �Hmax ℓ − �Hℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Note that any other weights would produces only trivial changes or would lead to the analysis of the variation weighted with the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In fact if one considers cℓ+1 = (θ + ℓ + 1)(aℓ+1 − �Hℓ+1) − (θ + ℓ)(aℓ − �Hℓ), 15 then cℓ+1 = ∆ℓ+1+(θ+ℓ+1)(aℓ+1− �Hmax ℓ+1 )−(θ+ℓ)(aℓ − �Hmax ℓ ), so it suffices to add to ∆ℓ+1 a deterministic sequence depending on ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' If one considers c′ ℓ+1 = bℓ+1( �Hmax ℓ+1 − �Hℓ+1) − bℓ( �Hmax ℓ − �Hℓ), one gets c′ ℓ+1 = bℓ �bℓ+1 bℓ ( �Hmax ℓ+1 − �Hℓ+1) − ( �Hmax ℓ − �Hℓ) � = bℓ �bℓ+1 bℓ − θ+ℓ+1 θ + ℓ � ( �Hmax ℓ+1 − �Hℓ+1) + bℓ θ + ℓ∆ℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' When we modify both, the additive and the multiplicative terms, in ∆ℓ+1 we get a combination of above situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In the frequentist case the weighted difference between maximal entropies at steps ℓ + 1 and ℓ is, df ℓ+1 = (ℓ + 1)Hmax ℓ+1 − ℓHmax ℓ = (ℓ + 1) log(ℓ + 1) − ℓ log ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From (16), in the Bayesian PDP case the weighted difference of posterior entropies is, dℓ+1 = (θ+ℓ+1) �Hmax ℓ+1 −(θ+ℓ) �Hmax ℓ = ∆ℓ+1+ηℓ+1 = ψ(θ+ℓ+1)−ψ(1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' For big ℓ we have that df ℓ+1 is of the order of log ℓ+1 while from (6) one gets that dℓ+1 is of the order of log ℓ − ψ(1 − α) (we recall that −ψ(1 − α) > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Now, by applying the relations (5) and (6) satisfied by the digamma function, from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='4 we get the following bounds for the weighted entropy variation ηℓ+1 = (θ + ℓ + 1) �Hℓ+1 − (θ + ℓ) �Hℓ given by (18), ηℓ+1 ≥ log(θ+ℓ+1) − 1 θ+ℓ+1 − log(nj∗−α) + 1 2(nj∗ − α), ηℓ+1 ≤ log(θ+ℓ+1) − 1 2(θ+ℓ+1) − log(nj∗−α) + 1 nj∗−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' When ℓ is sufficiently big one has, ηℓ+1 ≈ log(θ + ℓ + 1) − 1 2(θ + ℓ + 1) if kℓ+1 = kℓ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' and if also nj∗ is also sufficiently big, then ηℓ+1 ≈ log(θ+ℓ+1) − 1 2(θ+ℓ+1) − log(nj∗−α) + 1 2(nj∗−α) if kℓ+1 = kℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 16 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' One can check that (18) also holds for ℓ = 0, where for the posterior mean entropy (7), when ℓ = 0, one takes k = 0, and so θ �H0 P DP = θψ(θ + 1) − θψ(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, by applying the telescopic property to (18) we get (θ + ℓ) �Hℓ = Cℓ(α, θ) − ℓ � i=1 ψ(n∗(i) − α), where Cℓ(α, θ) = ��ℓ i=1 ψ(θ + i) � + θψ(θ + 1) − θψ(1 − α), and n∗(i) = #{1 ≤ j ≤ i : Xj = Xi} is the frequency of the class of the species Xi at step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Therefore, the only part of the entropy depending on the sample is − �ℓ i=1 ψ(n∗(i) − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The terms −ψ(n∗(i) − α) strictly decreases with n∗(i) (note that −ψ(n∗(i)−α) is positive when n∗(i) = 1, negative if n∗(i) ≥ 3 and the sign of −ψ(2 − α) depends on α ∈ [0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' So, the terms −ψ(n∗(i) − α) can be seen as the ‘discovery value’ of observing the species Xi at step i, and so, up to the additive deterministic term, the entropy turns out to be the ‘discovery’ values at the successive steps of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' On the other hand, from (17) we get that Lℓ = ℓ � i=1 (ψ(n∗(i) − α) − ψ(1 − α)) is a sum of positive rewards for reinforcing what is already known that is going in the opposite direction of discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Thus the reward at step i, attains the minimum 0 for the discovery of a new species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Differently to entropy, here no additional deterministic term depending on ℓ, α and θ is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='3 A common framework for the frequentist and the PDP cases The equations (17) and (13) have the same shape, both are measuring the weighted differences of the distance of successive entropies to the maximal entropies and both formulae express that these differences only depend on the updated frequency of the species of the new element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In fact this result holds for the class of entropies that satisfy: w(ℓ)Hℓ = u(a + ℓ) − b − k � i=1 (u(nℓ i − c) + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (21) Here w(ℓ) is a strictly positive function and increasing in ℓ and u is a real function defined on N − c = {n − c : n ≥ 1} and it satisfies u(n + 1 − c) − u(n − c) is increasing for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (22) 17 The quantities a, b, c, v are constants that satisfy the conditions 0 ≤ c < 1, −c ≤ a and 2u(1 − c) + v < u(2 − c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (23) Notice that Hℓ can be written as Hℓ with w(ℓ) = ℓ, u(x) = x log x and a = b = c = v = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' and �Hℓ can be also written in the form Hℓ with w(ℓ) = θ + ℓ, u(x) = xψ(x + 1), a = θ, b = θψ(1 − α), c = α, v = αψ(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In both cases 0 ≤ c < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The second part in (23) holds for the PDP because θ > −α and the third part of (23) holds in the frequentist case because it is equivalent to 2 log(1) ≤ log 2 and in the PDP case (23) becomes (1−α)ψ(2− α) + αψ(1 − α) < ψ(2 − α) + 1 which is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In relation to (22), in the PDP case it follows from xψ(x + 1) − (x − 1)ψ(x) increasing in x > 0 and in the frequentist case (22) it is a consequence of (n + 2) log(n + 2) − (n + 1) log(n + 1) > (n + 1) log(n + 1) − n log n for n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We will see that the conditions (22) and (23) are sufficient to show that the properties proven for the variation of differences between maximal entropies and entropies for the cases (Hℓ) and ( �Hℓ), also hold for the entropy (Hℓ) written in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In order to retrieve the results in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content='1 we need to analyze what happens when, for two species i ̸= j with nℓ i = n > 1 and nℓ j = m, one makes the change m → m + 1 and n → n − 1, and all other frequencies nl remain equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The entropy increases if and only if u(n−c−1)+u(m−c+1) ≤ u(n− c)+u(m−c), or equivalently u(m−c+1)−u(m−c) ≤ u(n−c)−u(n−c−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From (22) this holds if and only if m + 1 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' The second requirement has to do with the following change: for a class i ≤ k with ni = n > 1 we set n → n − 1 and k → k + 1 so there is a new class with nk+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This change makes the entropy increase if u(n−c−1)+u(1−c)+v < u(n − c) or equivalently if u(1 − c) + v < u(n − c) − u(n − c − 1) when n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' From (22) we get that it suffices that the following inequality holds 2u(1 − c) + v < u(2 − c), which is the second condition in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' When these conditions take place the maximal entropy is attained when all the classes are singletons, so w(ℓ)Hmax ℓ = u(a + ℓ) − b − ℓ � i=1 (u(1 − c) + v) Hence, w(ℓ + 1)Hmax ℓ+1 − w(ℓ)Hmax ℓ = u(a + ℓ + 1) − u(a + ℓ) − (u(1 − c) + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 18 Let us consider ∆H ℓ+1 = w(ℓ + 1) � Hmax ℓ+1 − Hℓ+1 � − w(ℓ) (Hmax ℓ − Hℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' If in the transition ℓ → ℓ + 1 the number of classes changes from k → k + 1 one gets that ∆H ℓ+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' If in the transition ℓ → ℓ + 1 the number of classes is preserved, say k, and the class j∗ adds in one unit we get ∆H ℓ+1 = u(nℓ j∗ − c + 1) − u(nℓ j∗ − c) − (u(1 − c) + v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' We combine (22) with the third condition in (23), to deduce that when the transition ℓ to ℓ+1 preserves the number of classes then ∆H ℓ+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Hence, the results for the variation of the weighted differences of the maximal entropy to the entropy hold for this class of entropies (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Finally, let us see what one requires to have w(ℓ+1)Hℓ+1−w(ℓ)Hℓ = (u(a+ℓ+1)−u(a+ℓ))−(u(nℓ j∗−c+1)−u(nℓ j∗−c)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Since from (23) we have a ≥ −c and so the unique new condition is u(n + a + 1) − u(n + a) ≥ u(m − c + 1) − u(m − c) for n ≥ m, which is satisfied for both, the PDP and the frequentist case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' This work was supported by the Center for Mathemat- ical Modeling ANID Basal PIA program FB210005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' In addition, we would like to thank the reviewer for their careful reading and valuable comments and suggestions, which helped to clarify and improve the presentation of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' References [1] Abramowitz, Milton and Stegun, Irene A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' Handbook of mathemati- cal functions with formulas, graphs, and mathematical tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' (1972), Dover.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' College London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtFST4oBgHgl3EQfAzi6/content/2301.13700v1.pdf'} diff --git a/_NAzT4oBgHgl3EQf_f58/vector_store/index.faiss b/_NAzT4oBgHgl3EQf_f58/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6cce9b745bef481e2fd1ed1da82683ce28b8fa1b --- /dev/null +++ b/_NAzT4oBgHgl3EQf_f58/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3ddb401ec3c6fc41f298b128c2538d0854e11f5f0dee6303315250f253625a14 +size 6488109 diff --git a/_NAzT4oBgHgl3EQf_f58/vector_store/index.pkl b/_NAzT4oBgHgl3EQf_f58/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a4524945b63bcd2ebf3ea712d70413d252a56e47 --- /dev/null +++ b/_NAzT4oBgHgl3EQf_f58/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b70a59b4ea6dfba785293523ebe198679723ba538c531729c60eacbdba95f841 +size 202505 diff --git a/_tFST4oBgHgl3EQfdjiO/content/tmp_files/2301.13807v1.pdf.txt b/_tFST4oBgHgl3EQfdjiO/content/tmp_files/2301.13807v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb0d3844ed81fb3a33d357605a951de99d5aa1a3 --- /dev/null +++ b/_tFST4oBgHgl3EQfdjiO/content/tmp_files/2301.13807v1.pdf.txt @@ -0,0 +1,2653 @@ +1 +Identifying the Hazard Boundary of +ML-enabled Autonomous Systems Using +Cooperative Co-Evolutionary Search +Sepehr Sharifi , Donghwan Shin , Member, IEEE, Lionel C. Briand , Fellow, IEEE, and Nathan Aschbacher +Abstract—In Machine Learning (ML)-enabled autonomous systems (MLASs), it is essential to identify the hazard boundary of ML +Components (MLCs) in the MLAS under analysis. Given that such boundary captures the conditions in terms of MLC behavior and +system context that can lead to hazards, it can then be used to, for example, build a safety monitor that can take any predefined +fallback mechanisms at runtime when reaching the hazard boundary. However, determining such hazard boundary for an ML +component is challenging. This is due to the space combining system contexts (i.e., scenarios) and MLC behaviors (i.e., inputs and +outputs) being far too large for exhaustive exploration and even to handle using conventional metaheuristics, such as genetic +algorithms. Additionally, the high computational cost of simulations required to determine any MLAS safety violations makes the +problem even more challenging. Furthermore, it is unrealistic to consider a region in the problem space deterministically safe or unsafe +due to the uncontrollable parameters in simulations and the non-linear behaviors of ML models (e.g., deep neural networks) in the +MLAS under analysis. To address the challenges, we propose MLCSHE (ML Component Safety Hazard Envelope), a novel method +based on a Cooperative Co-Evolutionary Algorithm (CCEA), which aims to tackle a high-dimensional problem by decomposing it into +two lower-dimensional search subproblems. Moreover, we take a probabilistic view of safe and unsafe regions and define a novel +fitness function to measure the distance from the probabilistic hazard boundary and thus drive the search effectively. We evaluate the +effectiveness and efficiency of MLCSHE on a complex Autonomous Vehicle (AV) case study. Our evaluation results show that MLCSHE +is significantly more effective and efficient compared to a standard genetic algorithm and random search. +Index Terms—ML-enabled Autonomous System, Hazard Boundary, System Safety Monitoring, Cooperative Co-Evolutionary Search. +! +1 +INTRODUCTION +A +UTONOMOUS systems are increasingly empowered by +being embedded with ML components (MLCs) for +various tasks, such as perception, localization, prediction, +planning and control. These components are inherently +different from conventional software components and pose +new challenges and safety risks that are not manageable by +traditional software engineering practices. The main reason +for this difference is that these components’ logic is not +captured by source code or specifications but their behavior +is rather determined by training. ML-enabled autonomous +systems (MLASs) have already led to fatalities in the case +of Autonomous Vehicles (AVs) [1]. This cannot be allowed +to continue, especially when human life or very expensive +equipment are involved. +Recent efforts have focused on making ML components +more reliable, robust and accurate through novel testing +methods [2, 3]. However, even a system with reliable com- +ponents can still lead to accidents [4]. For example, some +• +S. Sharifi and L. Briand are with the Department of Electrical and +Computer Engineering, University of Ottawa, Ottawa, Ontario, Canada, +K1N 5N6. L. Briand has also a faculty appointment with the SnT Centre +at the University of Luxembourg, Luxembourg. +E-mail: {s.sharifi, lbriand}@uottawa.ca +• +D. Shin is with Department of Computer Science, University of Sheffield, +Sheffield, United Kingdom, S1 4DP +E-mail: d.shin@sheffield.ac.uk +• +N. Aschbacher is with Auxon Corporation, Portland, Oregon, United +States and its subsidiary Auxon Technologies, Ottawa, Ontario, Canada +E-mail: nathan@auxon.io +accidents are caused as a result of unsafe component inter- +actions [4, 5]. Thus, the impact of ML components on safety +can only be studied in the context of the system they are +integrated into and in a specific operational context [4, 6]. +The inherent specificity of ML components favors the +use of safety monitors (also known as Run Time Assurance +or RTA mechanisms) [7]. Safety monitors, at run time, check +the inputs and outputs of a component that cannot be fully +trusted, e.g., an ML component, and will block its outputs +from being propagated to the rest of the system if they are +potentially hazardous. In such cases, systems usually fall +back on a trustworthy but less efficient component [8], or +take any other pre-designed fallback mechanisms, such as +stopping the AV on the shoulder of the road. To do this, +safety monitors have to observe the current state of the +system and compare it with its Operational Design Domain +(ODD) [9], to determine its deviation from ODD bounds +since it might lead to hazards. For instance, it is hazardous +to rely on the self-driving feature of an AV on a rainy night +if its ODD is characterized by normal dry operations during +daylight. Additionally, safety monitors have to know the +context of the system to determine whether the component +might contribute to a hazard. For example, misclassification +of an AV’s object detection component might not lead to any +hazards under a certain system context (henceforth called a +scenario), e.g., when an AV misidentifies an animal crossing +the road as a pedestrian and stops. Thus, identifying the +combinations of system contexts (i.e., scenarios) and ML +component’s behaviors (i.e., inputs and outputs) that will +arXiv:2301.13807v1 [cs.SE] 31 Jan 2023 + +2 +transition the system to a hazard state is an essential step in +developing safety monitors to be able to ensure the safety of +the ML-enabled system. +However, there are several challenges involved with +identifying the hazard boundary. First, the problem space +of scenarios and ML component behaviors is very large +and high-dimensional and is thus a challenge for more con- +ventional search metaheuristics such as Genetic Algorithms +(GA). Second, the violation of a given safety requirement +can only be determined if the system is executed within its +operational environment, which involves computationally +intensive simulations. The high computational cost, in addi- +tion to the large problem space, renders the problem even +more challenging. Last but not least, while safety can only be +evaluated by executing the system within an environment, +there are many environmental parameters that cannot be +controlled even via a high-fidelity simulator; for example, +the trajectory of pedestrians in CARLA [10], a well-known +AV simulator, is random. Furthermore, two similar MLAS +inputs may generate largely different outputs due to the +non-linear behavior of ML models, such as Deep Neural +Networks (DNNs). Therefore, we cannot assume that all +combinations of scenarios and ML component behaviors +within a region of the problem space have a uniform safety +outcome, i.e., the region is deterministically safe or unsafe. +Consequently, it is difficult to define hard boundaries be- +tween safe and unsafe regions. +To address the aforementioned challenges, we pro- +pose MLCSHE (ML Component Safety Hazard Envelope), +a novel Cooperative Co-Evolutionary Algorithm (CCEA)- +based approach that efficiently searches the problem space +by decomposing it into two sub-spaces (one for scenarios +and one for ML component behaviors) and parallelizing +the search of sub-spaces while taking the joint contribution +of both scenarios and ML component behaviors to the +autonomous system safety into account. Moreover, instead +of naively assuming that the hazard boundary is a clear +line that exists between the safe and unsafe regions, we +take a probabilistic view of the problem domain, i.e., at +any point within the scenario and ML component behavior +space, there is a probability of being safe. Based on this +probabilistic lens, we present a novel fitness function that +effectively guides the search towards the “probabilistic” +hazard boundary based on the probability of finding safe +scenario-behavior pairs within a given region. +Contributions. The contributions of this work is summa- +rized as follows: +• MLCSHE, a dedicated and tailored cooperative coevo- +lutionary search approach to approximate the hazard +boundary of an ML component, in a probabilistic way, +taking into account the combination of scenarios and +MLC behaviors. +• An application of MLCSHE to a complex Autonomous +Vehicle (AV) case study involving an industry-strength +simulator and an Autonomous Driving System with +deep learning components. Our implementation of +MLCSHE as well as other case study artefacts are +provided in our replication package (see Section 6.5). +• An evaluation of the effectiveness and efficiency of +MLCSHE through large scale experiments. +• A comparison of MLCSHE against baseline methods. +Paper Structure. The rest of the paper is structured as +follows. Section 2 provides background materials on CCEA. +Section 3 defines the problem of MLC hazard boundary +identification and details its challenges. Section 4 discusses +related work. Section 5 presents MLCSHE in detail. Section 6 +provides the empirical evaluation of MLCSHE and dis- +cusses the results. Section 7 concludes and suggests future +directions for research and improvement. +2 +BACKGROUND +In this section, an overview of Evolutionary Algorithms +(EAs) is provided. Then we focus on a specific family of EAs, +Cooperative Co-Evolutionary Algorithms (CCEAs), which +happens to be particularly useful in our context. Finally, the +key decision points involved in designing a CCEA, namely +collaborator selection and individual fitness assessment are +discussed. +EAs are a family of algorithms designed based on the +principles of evolutionary computation [11]. EAs are in- +spired by the concepts related to biological evolution and +have been applied to various optimization problems for +which standard mathematical optimization is not applica- +ble [11]. EAs use concepts such as individual, population, +fitness, selection and mutation to formalize an optimization +problem. Individuals usually represent solutions to the tar- +geted problem and are members of a population whose +fitness is evaluated (usually by a fitness function). Desirable +individuals, i.e., those with the highest fitness values, are +more likely to be selected to act as the parents of the +next generation. Using methods such as crossover (replacing +some parts of an individual with another one) and mutation +(adding randomness to some parts of an individual), indi- +viduals of the next generation population, i.e., next iteration +of the search, are created. +For +many +problems, +the +search +space +is +high- +dimensional such that a conventional EA would not be able +to solve it within a reasonable timeframe [12]. To address +this, Cooperative Coevolutionary Algorithms (CCEAs), +originally proposed by Potter and De Jong [13] in 1994, +decompose the original problem into lower-dimensional +subproblems, each of which can be solved in a separately +evolving population as in conventional EAs described pre- +viously. Since individuals from each subproblem population +must join together to form a complete solution to the original +problem, the fitness of an individual can only be evaluated +based on the joint fitness of the complete solution created +by joining the individual with representative individuals, +called collaborators, from other populations. By carefully +selecting collaborators and assessing individuals’ fitness, +CCEAs are known to be effective at solving even non- +separable problems [14, 15] where the fitness of an individual +of a subproblem population depends on the fitness of indi- +viduals of other populations. Furthermore, the decomposi- +tion of the original problem naturally allows parallelism to +increase search performance [12]. +Figure 1 depicts the process of an abstract CCEA. Each +population is initialized, either with randomly selected or +guessed values (usually provided by domain experts). Indi- +viduals of each population collaborate with individuals of +the other population(s) to form complete solutions. Then, + +3 +Evaluate +Complete +Solutions +No +Yes +stopping_condition? +New Generation +Evolve Individuals +Initialize +Populations +Output Fittest +Individual(s) +Evaluate +Individuals +Select +Collaborators +Fig. 1: An abstract coevolutionary algorithm (CCEA). +these complete solutions are evaluated via joint fitness as- +sessment functions. The joint assessments are then aggre- +gated to provide evaluations of individual fitness values. +If the stopping condition is reached (true), then the fittest +individuals are returned. Otherwise, the individuals go +through breeding (selection, crossover and mutation) to +create the next generation of the populations and go through +evaluations again. +Designing a CCEA includes three important decisions +in the following aspects: collaborator selection and individual +fitness assessment. +Collaborator Selection. One of the most important factors +affecting the performance of a CCEA is its collaborator se- +lection strategy. To evaluate the fitness of an individual, a +complete solution using the said individual and other indi- +viduals from other subproblems, also known as collaborators, +should be formed. The joint fitness of the complete solution +contributes to determining the fitness value of the individ- +ual. To assess the fitness of the individual, the algorithm has +to form one or multiple complete solutions with different +collaborators. However, ideally, to get closer to the global +optimum, all individuals of all other populations should be +used as collaborators [16], which is usually infeasible due +to resource constraints. Therefore, a strategy to efficiently +select collaborators is required. Various strategies have been +proposed in the literature, such as single best, tournament- +based, and random [12]. These strategies affect the algorithm +via controlling the selection pressure and the pool size of +the collaborators. +Some studies have proposed archive-based collaborator +selection to effectively reduce the number of collaborators to +join in individual fitness assessments while maintaining the +amount of information contained in the populations [12]. +The idea is to carefully select a population archive which is +a subset of a population to be used as collaborators. For +example, Panait et al. [15], have proposed iCCEA, which +aims to minimize the size of the population archives by +considering only the collaborators that are informative and +distinct. A collaborator in an archive is informative if adding +it to the archive changes the fitness ranking of the popula- +tion’s individuals. If there are multiple collaborators that can +change the ranking of the same individuals, the collaborator +that changes the ranking the most will be kept in the archive. +A collaborator in an archive is distinct if its (Euclidean) +distance from other collaborators in the archive is higher +than a pre-defined threshold. As a result, a population +archive keeps only a minimum number of collaborators +while attempting not to lose information in terms of collab- +orations between subproblem solutions. However, though +the population archive selected by iCCEA is minimal in +size, the algorithm proposed by the authors to update +the population archive in each generation has a high time +complexity (𝑂(𝑛3) where 𝑛 in the number of individuals in +the archive) and this severely impacts the performance of +the algorithm. Thus, simpler population archive selection +methods, e.g., elitist, random and best random, that are much +faster, are also widely used in practice. +Individual Fitness Assessment. The collaborator selection +strategy of a CCEA affects its individual fitness assessment +strategies as well. The only objective fitness assessment that +can be done on the individuals is based on their joint fitness +assessments with collaborators. Thus, all algorithms per- +form some form of aggregation on joint fitness assessments +related to an individual to determine its fitness value. Best, +worst or average joint fitness values are usually used for +individual fitness assessments. +3 +PROBLEM AND CHALLENGES +In this section, we provide a precise problem definition +regarding the identification of the boundaries of hazard +envelopes, focusing on the behavior of a Machine Learning +Component (MLC). While we use an AV as an example, +it can be easily generalised to any ML-based Autonomous +System (MLAS). +3.1 +Problem Definition +Consider an AV as an ML-enabled Autonomous System +(MLAS) including an ML component (MLC), namely an +image-based object detection component using DNNs. The +AV continuously observes its surrounding environments— +such as roads, traffic signs, buildings, and other moving +vehicles via sensors (e.g., camera) and generates driving +commands (e.g., steer left and decrease speed) to best satisfy +given functional and safety requirements (e.g., reach a given +destination point without colliding with other vehicles). +During testing of the AV, the environment is often simulated +by a high-fidelity driving simulator due to the high cost +and risk of real-world testing. Inside of the AV, whenever +new sensor data (e.g., an image taken from the camera) is +collected, it passes through the object detection component +to identify the positions of surrounding objects, if any, from +the (fused) sensor data (e.g., in the form of bounding boxes +in the given image), which will then be used to determine +proper driving commands. Under a certain driving scenario, +the AV might violate requirements such as “the AV shall +keep a minimum distance of 1.5 m from any vehicle in front.” +In such cases, the MLC could internally contribute to the vi- +olation. Therefore, identifying the boundaries of the hazard +envelopes of the AV in terms of the combination of driving +scenarios and MLC behaviors is important. Figure 2 pro- +vides a simplified illustration of an ML component’s hazard +envelope, defined in terms of scenarios and ML component +behaviors, where a safe region leading to no violations is + +4 +Fig. 2: An illustration of safe and unsafe regions and the +corresponding hazard boundary. +surrounded by an unsafe region leading to violations. The +goal is to identify, as precisely and completely as possible, +the boundaries between safe and unsafe regions, illustrated +by the dashed line in Figure 2. +More specifically, let 𝑠 be the AV including the MLC 𝑚 +for image-based object detection, operating in a simulated +driving environment. For a given scenario 𝑢, which consists +of all the static and dynamic entities of the environment +such as road shape, weather, and other vehicles, the simu- +lation result for 𝑠 and 𝑚, denoted by Π𝑢,𝑠,𝑚, is a sequence +⟨𝑒1, 𝑒2, . . . , 𝑒𝑇 ⟩ where 𝑇 is the duration of the execution and +𝑒𝑡 for 𝑡 = 1, . . . ,𝑇 is the snapshot (state) of the environment +at time step 𝑡. For each time 𝑡 ∈ {1, . . . ,𝑇}, 𝑠 takes an image +𝑖𝑛𝑠,𝑡 taken from the camera by observing 𝑒𝑡 and generates +a pre-processed (e.g., gray-scaled) image 𝑖𝑛𝑚,𝑡 for 𝑚. Then, +𝑚 produces the object information 𝑜𝑢𝑡𝑚,𝑡 (in the form of +bounding boxes) by processing 𝑖𝑛𝑚,𝑡 and 𝑠 produces driving +commands 𝑜𝑢𝑡𝑠,𝑡 by processing 𝑜𝑢𝑡𝑚,𝑡. The environment +snapshot 𝑒𝑡+1 for the next time step 𝑡 +1 is updated based on +𝑜𝑢𝑡𝑠,𝑡, and the whole process repeats until 𝑡 reaches 𝑇. The +behavior of 𝑚, denoted by 𝐵𝑚, is defined as the sequence +of input/output pairs such that, for an input/output pair +(𝑖𝑛𝑚, 𝑜𝑢𝑡𝑚) ∈ 𝐵𝑚, 𝑜𝑢𝑡𝑚 is the output produced by 𝑚 by +processing 𝑖𝑛𝑚. For a safety requirement 𝑟 (e.g., do not +collide with other vehicles), we can measure the degree of +the safety violation of 𝑠 for 𝑢 and 𝐵𝑚 in terms of 𝑟, denoted +by 𝑓 (𝑟, Π𝑢,𝑠,𝑚), by analyzing Π𝑢,𝑠,𝑚 = ⟨𝑒1, 𝑒2, . . . , 𝑒𝑇 ⟩ against +𝑟. If 𝑓 (𝑟, Π𝑢,𝑠,𝑚) > 𝜖 for a small threshold 𝜖 predefined for 𝑟, +we say that 𝑟 for 𝑠 is violated by (the combination of) 𝑢 and +𝐵𝑚. This means that we can decide the violation of 𝑟 (with +𝜖) given 𝑢 and 𝐵𝑚. +Given the above context, let (𝑢, 𝐵𝑚) be a point in a +space referred to as the input space, that is defined by (the +combination of) possible scenarios and MLC behaviors. For +each point (𝑢, 𝐵𝑚) in the input space, we can decide its +output (i.e., unsafe or safe) by checking whether it leads to +the violation of 𝑟 or not. The identification of the boundaries +of hazard envelopes attempts to find as many (𝑢, 𝐵𝑚) points +as possible that are close to the boundaries between safe +and unsafe regions in the space. Notice that we intentionally +left the precise definition of safe and unsafe regions unclear +since it is one of the challenges we address next. +3.2 +Challenges +The problem of hazard boundary identification for an MLC +in the MLAS under analysis, entails multiple major chal- +lenges. +As discussed in Section 3.1, both a scenario 𝑢 and an +MLC behavior 𝐵𝑚 collaboratively determine the violation +or satisfaction of a safety requirement 𝑟. As a result, there +are too many possible scenarios and MLC behaviors for the +input space to be exhaustively explored without resorting +to limiting assumptions that can bias the results [17]. One +might argue that unsafe regions of the input space could +be analytically identified using methods based on expert +knowledge, such as FTA [18] and HAZOP [19], to provide +clear insights into how hazards can occur. However, such +methods are not sufficient to address all possible ways haz- +ards can arise due to complex interactions between MLAS +components and the opacity of ML components. +Second, the satisfaction or violation of 𝑟 can only be +determined if the system is operated within its surrounding +environment. During testing, in addition to the first chal- +lenge above, this requires running a high-fidelity simulator +which is generally very resource-intensive. The high cost of +simulation highlights the need for an efficient and effective +method to search as much of the input space as possible +while focusing on the regions close to the boundary. +Lastly, recall it is unrealistic to consider a region 100% +safe or unsafe. This is explained by two main reasons. First, +simulators do not often enable full control of all relevant +parameters in the environment, thus randomly configuring +some of them. For example, the movement of pedestrians +is random in CARLA [20], a high-fidelity simulator. Second, +two inputs that are close in the input space may generate +different MLC outputs that are handled differently by the +rest of the system, e.g., due to the non-linear behavior of +other DNNs using the MLC outputs as their input, resulting +in different safety results (i.e., safe or unsafe). As a result, +we cannot assume a uniform and consistent safety outcome +for a region, making it difficult to define hard boundaries +between safe and unsafe regions. Rather, hazard envelop +boundaries (i.e., the dashed line in Figure 2) should be prob- +abilistic as they encompass regions with a given probability +threshold of violating a selected requirement. +To address the above challenges, we propose a novel +method using Cooperative Co-Evolutionary Algorithm +(CCEA) that efficiently address our objectives as an opti- +mization problem, within a large input space, by decom- +posing such problem into lower-dimensional subproblems. +Further, to recast our problem into a coevolutionary search +problem, we define a special fitness function that can assess +how far a candidate solution (i.e., a combination of 𝑢 and +𝐵𝑚) is from the boundary of a “probabilistic” unsafe region. +See Section 5 for details of our method. +4 +RELATED WORK +This section discusses existing studies related to the problem +of hazard envelope boundary identification. Depending on +the methods used, we found three categories: search-based +methods, sampling-based methods, and formal methods. + +Unsafe +Safe +S +Hazard +Boundary5 +Fig. 3: A possible application of DeepJanus to the systemic +hazard boundary detection problem. Connected dots are a +safe and unsafe pair. +4.1 +Search-based Methods +Search-based methods employ metaheuristics (search algo- +rithms) and convert the boundary identification problem +into a search problem guided by a fitness function that +evaluates how close a system input (e.g., test scenario) +is from the boundary. Fitness assessment for individual +system inputs often involves simulation executions to check +whether safety requirements are violated. +Although there are many search-based methods for test- +ing MLCs [3, 21], the problem of boundary identification +has received very little attention. Only recently, Riccio and +Tonella [22] proposed DeepJanus, the first search-based +method to identify the frontier of behavior (frontier) of MLCs, +i.e., a set of similar input pairs that trigger different behav- +iors (e.g., safe and unsafe) of the system. The discovered +frontier can allow developers to approximate a safe oper- +ating envelope for the MLC (by interpolating the pairs). +Also, the overlap of the estimated safe operating envelope +with the validity domain of the MLC, which is the domain +where the MLC is expected to behave according to its re- +quirement(s) [22], can facilitate the evaluation of the MLC’s +quality. Therefore, for example, DeepJanus can be useful in +distinguishing between the performance of two MLCs that +perform the same task. However, it cannot solve the issue +of identifying the hazard boundary, as the impact of MLCs +on safety can only be assessed when evaluating the entire +system in a given environmental context. Furthermore, as +illustrated in Figure 3, the interpolated frontier of behavior +and the hazard boundary of an MLC are not necessarily +the same. More precisely, a member of the frontier (i.e., a +pair of safe and unsafe inputs) does not necessarily lie in +proximity to the hazard boundary since the violations can +occur in probabilistic safe regions (as argued in Section 3.2). +Therefore, we need a novel method to identify the hazard +boundary of an MLC within a system, considering the +probabilistic nature of (un)safe inputs. +4.2 +Sampling-based Methods +Unlike search-based methods, which are guided by fitness +functions, sampling-based methods use repeated random +samplings (e.g., Monte Carlo methods) or statistical metrics +to identify certain system inputs that lead to safety vio- +lations. For example, Meltz and Guterman [23] proposed +SmARTest, which uses Monte Carlo methods to identify a +scenario domain (i.e., set of system inputs) that lead to safety +requirement violations determined by measuring Perfor- +mance Assessment Functions (PAFs) defined based on the re- +quirements’ Key Performance Indicators (KPIs). Sinha et al. +[24] proposed Neural Bridge Sampling (NBS), a method to +measure the probability of rare events, such as accidents, +using Monte Carlo methods. NBS decomposes the proba- +bility of a rare event into chained conditional probabilities, +which are tractable to compute using standard Monte Carlo +methods. This provides a better estimate than the naive +Monte Carlo or Adaptive Multi-Splitting (AMS) methods. +Sampling-based methods can efficiently identify safe +and unsafe inputs from the system’s input space. However, +they only consider system inputs (i.e., scenarios) and not +the effect of different MLC behaviors for the same scenario. +As discussed in Section 3.2, both the scenario and the +MLC behavior must be taken into account to determine the +conditions when MLC behavior leads to safety violations. +4.3 +Formal Methods +Formal methods rely on formal representations of the input +space, the system (including the MLC), and the output +space. Examples of such representations include hybrid +system or dynamical system formalisms [25]. Tools like +SMT solvers [26] and Mixed Integer Linear Programming +(MILP) [27] can be used to analyze whether the system +containing the MLC can reach an unsafe region given +its input space [2]. This is known as reachability analysis. +Reachability analysis is used to identify the MLC’s barrier +certificate, which is an invariant function that constrains the +state space of the system and ensures the satisfaction of +a safety property [28] while considering the closed-loop +behavior of the system. Barrier certificates can be seen as +an over-approximation of the hazard boundary of the MLC. +Ivanov et al. [29] proposed Verisig, which can be ap- +plied to Cyber-Physical Systems (CPSs) with DNN-based +feed-forward controllers with ReLU activation functions. +Verisig transforms the ReLU DNN into a hybrid system +representation and combines it with the rest of the system. +This recasts the problem as a hybrid system verification +problem. Given a set of system inputs, the outputs can be +approximated using Flow∗ [30], a nonlinear system reacha- +bility analyzer. Tuncali et al. [31] proposed another method +to identify barrier certificates of DNN-based, feed-forward +controllers, which is not limited to architectures with ReLU +activation functions. This method first identifies candidate +barrier certificates using simulations, then evaluates their +suitability using the dReal [32], an SMT solver for nonlinear +formulas in real numbers. Tran et al. [33] proposed NNV, a +method to perform closed-loop reachability analysis of con- +trol systems with Deep Reinforcement Learning (DRL) con- +trollers. These controllers have a feed-forward architecture +with ReLU/Saturation activation functions. NNV calculates +a low-error over-approximation of the output region, which +are reached by the system given its inputs. +Although the aforementioned methods provide guaran- +tees for the hazard boundary and cover all possible tra- + +Unsafe +Frontierof +Behavior +Safe6 +jectories of the system, they suffer from practicality and +scalability issues. For example, over-approximation of the +hazard boundary might incorrectly reduce (or even remove +in the worst case) the safe operating envelope of the system +by incorrectly considering some safe behaviors unsafe, thus +limiting the practicality of the methods [34]. Furthermore, +reachability analysis can only be applied to feed-forward +controllers with specific activation functions. Thus, it cannot +be used for practical MLCs that perform perception, obstacle +tracking, or prediction tasks with different DNN architec- +tures (e.g., recurrent neural networks). Also, reachability +analysis has not yet been applied to closed-loop, industrial +Cyber-Physical Systems (CPS) with feedback DNN con- +trollers [29, 31, 33]. In such a context, scalability is very likely +to become an acute problem. +4.4 +Remark on Differences in Objectives +A common goal underlying all the above-mentioned meth- +ods is to identify the hazard boundary of a given MLC +embedded within its containing system (MLAS). It could be +useful when the MLC under test is fixed, but as soon as the +MLC changes (e.g., via retraining), the previously identified +hazard boundary would be invalid, and the whole safety +verification exercise would have to be repeated. On the other +hand, in our research, we aim to identify the combinations +of conditions and MLC behaviors, without referring to a +specific MLC implementation, that could potentially lead +to hazards. Once characterized, such situations could then, +independently of a specific MLC implementation, be used +to monitor the operation of the system and MLC and warn +the user in case it is operating near to the hazard boundary. +In the following section, we propose a novel method that +addresses the challenges discussed in Section 3.2, and is +applicable to various types of MLCs, such as perception, +planning, and control, without making any assumptions +about their architecture. +5 +OUR APPROACH +In this section, we provide a solution to the problem de- +scribed in Section 3, i.e., the hazard boundary identification +of an MLC in the MLAS under analysis. Our key idea +is to recast the problem as a cooperative co-evolutionary +search problem where scenarios and MLC behaviors co- +evolve as two separate populations but contribute together +to find complete solutions (i.e., the combinations of scenar- +ios and MLC behaviors) close to the boundary. Then, we use +CCEAs, the algorithms that are well known to be effective +at solving search problems such as the one described in +Section 2. +In the following subsections, we first describe how +scenarios and MLC behaviors can be represented as two +separate populations in an search problem (Section 5.1). We +then define a novel fitness function of the search problem to +assess how close a complete solution is from the boundary +(Section 5.2). Finally, we present our novel method based +on CCEAs using the representation and the fitness function +(Section 5.3). +5.1 +Representations +We consider two populations, one for scenarios and another +one for MLC behaviors. The individuals of the MLC popu- +lation, subjected to evolutionary operators, are only repre- +sented as MLC-outputs (𝑜𝑢𝑡𝑚). This is due to the initial MLC- +input (𝑖𝑛𝑚) being (indirectly) determined by the scenario +whereas the next MLC-inputs are affected by previous MLC- +outputs. Therefore, 𝑖𝑛𝑚 is recorded in an archive of complete +solutions (i.e., 𝐴𝑐 in Algorithm 1; see Section 5.3 for details) +but not included in the representation of the behavior of +the MLC that can be manipulated by the search. Recording +the 𝑖𝑛𝑚 and 𝑜𝑢𝑡𝑚 sequences, along with their corresponding +scenarios (𝑢), is indeed crucial as it records unsafe behaviors +of an MLC (its input and output sequences) given a set +of environmental conditions (its scenarios). This information +enables the design of safety monitors that will prevent the +MLC from contributing to a systemic hazard via leveraging +the recorded information. +5.1.1 +MLC behaviors +One of the two populations considered for the search is +the set of MLC behaviors. The behavior of an MLC can be +expressed as a sequence of input and output tuples. The +inputs of an MLC are indirectly controlled by the environ- +mental input to the system (i.e., scenario parameters) and +the components of the system that process that input before +it is passed on to the MLC. Thus, the parameters that we +can directly manipulate during the search are the outputs +of the MLC. We represent an individual in the population +of the MLC behaviors as a sequence of MLC outputs where +the 𝑡-th element of the sequence denotes an MLC output at +time step 𝑡. +The output of an MLC depends on the task performed +by the MLC. For instance, in the case of a steering angle +estimator, the output is a single real value. Whereas, in +the case of an object classifier, the output is a vector of +probabilities (real values between 0 and 1), where each ele- +ment corresponds to a label. Finally, similar to our running +example, in the case of obstacle detection, the outputs in +an ML component (MLC) are detected obstacles, i.e., their +bounding box1, their label (such as pedestrian, vehicle, lamp +post, etc.), and their timestamp. Therefore, an mlco for a +simulation duration 𝑇 can be defined as a sequence of the +trajectories of detected obstacles during 𝑇 in the case of +obstacle detection. +Specifically, given the maximum number of detectable +objects 𝑛 and the simulation duration 𝑇, an mlco can be +defined as a sequence of 𝑛 trajectories ⟨trj1, . . . , trj𝑛⟩ where +trj𝑖 represents the trajectory of the 𝑖-th object (in terms of the +bounding boxes) for 𝑇. By allowing the search algorithm to +manipulate individual trajectories, an arbitrary mlco can be +generated for obstacle detection. +However, allowing the search algorithm to generate all +the bounding boxes for individual time steps will likely +yield an unrealistic trajectory randomly moving around +without a consistent direction, which we observed during +our initial trials. Therefore, it is better to allow the search +1. A bounding box specifies the area on the image processed by the +obstacle detector that contains the detected obstacle. It can be expressed +as (𝑥𝑚𝑖𝑛, 𝑥𝑚𝑎𝑥, 𝑦𝑚𝑖𝑛, 𝑦𝑚𝑎𝑥) corresponding to a specific 2D box. + +7 +algorithm to generate only the start and end bounding +boxes, and then generate the remaining bounding boxes for +intermediate time steps using linear interpolation between +the start and end boxes. Specifically, trj𝑖 can be defined as a +triple (class𝑖, start𝑖, end𝑖) where class𝑖 is the class of the 𝑖-th +object (e.g., car, bicycle, pedestrian), start𝑖 is the position and +the size of the bounding box of the 𝑖-th object at time step +𝑡 = 𝑡start, and 𝑒𝑛𝑑𝑖 is the position and the size of the bounding +box of the 𝑖-th object at time step 𝑡 = 𝑡end. For example, +start or end can be defined as a quintuple (𝑡, 𝑥𝑚𝑖𝑛, 𝑥𝑚𝑎𝑥, +𝑦𝑚𝑖𝑛, 𝑦𝑚𝑎𝑥), which are time and bounding box parameters +for the beginning or the end of the trajectory, respectively. +Then, for a given trajectory trj = (class, start, end), we can +easily generate the positions and sizes of bounding boxes +for intermediate time steps (i.e., 1 < 𝑡 < 𝑇) based on start +and end (using linear interpolation) whenever needed for a +simulation. +5.1.2 +Scenarios +A scenario can be represented as a heterogeneous vector of +real and integer values. For the case of an AV, a scenario +consists of the vehicle itself, the weather, the road and other +static (e.g., lamp posts and other obstacles) and dynamic +objects (e.g., pedestrians and other cars) [35]. Each of them +have many attributes of various types, namely float (e.g., +speed) and enumerated types (e.g., line pattern) which can +be encoded as integer values. +The size of a scenario individual is determined by the +simulator. Furthermore, a finer-grained level of simulation +control implies a larger scenario size as more parameters +have to be manipulated by the search algorithm. For in- +stance, one can manipulate all weather-related parameters +separately (10 parameters in the case of CARLA [20]) or +manipulate them using the weather preset parameter (1 +parameter) which sets the value of all granular weather +parameters according to high-level modalities, e.g., rainy +sunset, clear noon. +Figure 4 is the scenario domain model for our running +example. A Scenario consists of one or more Vehicles (in- +cluding the ego vehicle), zero or more Pedestrians and, +Mission and Weather. The attributes of the domain model +that act as the parameters for a scenario representation are +written in bold font in Figure 4. Therefore, a scenario can +be defined by the time of day, weather preset, map of the +town, start point of the ego vehicle, its target destination +and target velocity, the number of Pedestrians, and the +number and position of other Vehicles with respect to the +ego vehicle (e.g., in front, on the opposite lane). +Operational Design Domain (ODD). The Operational De- +sign Domain or ODD defines an operational envelope of the +AV, i.e., a set of bounds on the environmental parameters +of the system. For instance, highway driving is an ODD +for AVs which determines the type of the road, the average +speed of the surrounding vehicles, and the (lack of) pedes- +trians in the vicinity [9]. However, within an ODD, many +scenarios can still be defined, e.g., weather, the number of +cars, the length and shape of the road. Therefore, a search +can be done within an ODD, which sets the values or the +bounds of some parameters, such as the target speed of the +ego vehicle. The parameter bounds or values set by the ODD +will remain static for the duration of the search, e.g., a target +speed of 90kph in a highway driving ODD, or the angle of +the sunlight during a daytime driving ODD. +5.2 +Fitness Function +This section presents our proposed fitness function in detail. +Our aim is to design a fitness function that can effectively +guide the search towards the boundary of unsafe regions. +However, as mentioned in Section 3, we cannot assume that +a region is either 100% unsafe or safe. To address this, we +first define the notion of safe and unsafe inputs, followed by +probabilistic unsafe regions. +Definition 5.1 (Safe and Unsafe Inputs). An input is unsafe +if and only if it leads to the violation of a given requirement. +Otherwise, the input is safe. +Recall that an input is a combination of a scenario and +an MLC behavior in our context. +Definition 5.2 (Probabilistic Unsafe Region). Let 𝑋 be a set +of all possible inputs, representing the input space. Given a +threshold probability 𝑝th, a region 𝐺 ⊆ 𝑋 is 𝑝th-unsafe if and +only if the proportion of unsafe inputs in 𝐺 is higher than +or equal to 𝑝th. +For example, if we randomly draw an input from a 5%- +unsafe region, we have more than or equal to 5% chance +of leading to a safety violation. The value of 𝑝th should be +determined by a domain expert within a specific application +context. +Notice that the shape of a probabilistic unsafe region +is unknown, as is its boundary. Nevertheless, we can ap- +proximate how far an arbitrary input is from the boundary +by sampling its neighborhood. Specifically, for an input +𝑥 ∈ 𝑋, let 𝑝𝑥 be the proportion of unsafe inputs in the +neighborhood of 𝑥. If 𝑝𝑥 < 𝑝th, it implies that 𝑥 is not likely +to be located in a 𝑝th-unsafe region. Otherwise, if 𝑝𝑥 > 𝑝th, +it implies that 𝑥 is likely in a 𝑝th-unsafe region. Therefore, if +𝑝𝑥 is close to 𝑝th, it implies that 𝑥 is close to the boundary +of a 𝑝th-unsafe region. To leverage this idea, we define the +notion of neighborhood as follows: +Definition 5.3 (Neighborhood). For an input 𝑥 ∈ 𝑋 and a +non-negative real number 𝛿 ∈ R+, a neighborhood of 𝑥 with +the radius of 𝛿, denoted by 𝑁(𝑥, 𝛿), is defined as follows: +𝑁(𝑥, 𝛿) = {𝑥′ ∈ 𝑋| dist(𝑥, 𝑥′) ≤ 𝛿} +where dist(𝑥, 𝑥′) indicates the distance between 𝑥 and 𝑥′. +Notice that various distance functions dist can be +adopted depending on the nature of complete solutions. +For example, if a complete solution can be represented +as a heterogeneous vector composed of numerical, ordi- +nal, and categorical values, Heterogeneous Distance Metrics +(HDMs) [36] are good candidates to measure the distance +between two complete solutions. In Figure 2, a neighbor- +hood with a radius of 𝛿 is visualised as a circle between safe +and unsafe regions. +Based on Definition 5.3, let 𝑝𝑥,𝛿 be the proportion of +unsafe inputs in 𝑁(𝑥, 𝛿). Then, as discussed above, we can +use the difference between 𝑝𝑥,𝛿 and 𝑝th to approximate the +distance between 𝑥 and the boundary of a 𝑝th-unsafe region. +However, we cannot compute the exact value of 𝑝𝑥,𝛿 since + +8 +1 +1..* +1 +Scenario +Weather ++ preset: WeatherPreset ++ time_of_day: TimeOfDay +Vehicle ++ id: Integer ++ drive_control: VehicleControl ++ bounding_box: BoundingBox +Pedestrian ++ id: Integer ++ bounding_box: BoundingBox ++ control: WalkerControl +Mission ++ map: Map ++ start_point: Waypoint ++ target_destination: Waypoint ++ target_velocity: Float +0..* +<> +WeatherPreset +Clear +Cloudy +Wet +WetCloudy +SoftRain +MidRainy +HardRain +<> +TimeOfDay +Noon +Sunset +Night +Fig. 4: The scenario domain model for the running example. The model is based on the concepts provided in the Carla +World domain model [10, 20]. The scenario parameters are shown on the figure in bold font, i.e., the weather preset, the +attributes of Mission and the number of Actors such as vehicles and pedestrians. +𝑁(𝑥, 𝛿) has too many complete solutions to exhaustively +evaluate. Nevertheless, we can compute an estimate of 𝑝𝑥,𝛿, +denoted by ˆ𝑝𝑥, 𝛿, and its confidence interval since the con- +secutive trials of checking whether an input 𝑥′ ∈ 𝑁(𝑥, 𝛿) +is safe or not are assumed to be independent and can be +treated as Bernoulli Experiments. +Specifically, the probability distribution of 𝑝𝑥,𝛿 can be +modelled as a Binomial distribution, and we can compute +ˆ𝑝𝑥,𝛿 as follows: +ˆ𝑝𝑥, 𝛿 = +unsafe(𝑁(𝑥, 𝛿)) +evaluated(𝑁(𝑥, 𝛿)) +where evaluated(𝑁(𝑥, 𝛿)) is the number of inputs evaluated +(sampled) in 𝑁(𝑥, 𝛿) and unsafe(𝑁(𝑥, 𝛿)) is the number of +unsafe inputs among those evaluated. Furthermore, using +the Wilson Confidence Intervals [37], we can compute the +confidence interval of 𝑝𝑥, 𝛿, denoted by CI(𝑝𝑥, 𝛿), as follows: +CI(𝑝𝑥, 𝛿) = ˆ𝑝𝑥, 𝛿 ± 𝑧 × +√︄ +ˆ𝑝𝑥, 𝛿 × (1 − ˆ𝑝𝑥, 𝛿) +evaluated(𝑁(𝑥, 𝛿)) +where 𝑧 is determined by the standard normal distribution +for a given confidence level (e.g., for a 95% confidence level, +𝑧 = 1.96). +Based on CI(𝑝𝑥, 𝛿), we can assess the maximum differ- +ence2 between 𝑝𝑥, 𝛿 and 𝑝th as follows: +diff (𝑝𝑥, 𝛿, 𝑝th) = max �|UL(𝑝𝑥, 𝛿) − 𝑝th|, |LL(𝑝𝑥, 𝛿) − 𝑝th|� +where UL(𝑝𝑥, 𝛿) and LL(𝑝𝑥, 𝛿) are the upper and lower limits +of CI(𝑝𝑥, 𝛿), respectively. Using diff (𝑝𝑥, 𝛿, 𝑝th), we define our +fitness function as follows. +Definition 5.4 (Boundary-Seeking Fitness Function). For an +input 𝑥, a neighborhood radius 𝛿, and a threshold proba- +2. We consider the maximum difference to be conservative. +bility 𝑝th, the fitness value of 𝑥 given 𝛿 and 𝑝𝑡, denoted by +fitness(𝑥, 𝛿, 𝑝th), is defined as follows: +fitness(𝑥, 𝛿, 𝑝th) = +diff (𝑝𝑥,𝛿, 𝑝th) +max(𝑝th, (1 − 𝑝th)) +where the denominator is a normalisation factor, making the +range of the fitness value between 0 and 1. +In other words, we compute the fitness value of an input +𝑥 using the difference between 𝑝𝑥,𝛿 (i.e., the proportion of +unsafe inputs in the neighborhood of 𝑥 with the radius of 𝛿) +and 𝑝th (i.e., the probability threshold). +Note that the fitness function is meant to be mini- +mized and decreases as the difference between 𝑝th and +𝑝𝑥,𝛿 decreases. The fitness function also takes the number +of observations (i.e., evaluated inputs) within 𝑁(𝑥, 𝛿) into +account, as the size of CI(𝑝𝑥,𝛿) (i.e., the confidence interval +of 𝑝𝑥,𝛿) decreases when the number of observations in the +neighborhood increases, thereby also decreasing the value +of the fitness function. A sparsely populated neighborhood +therefore tends to yield high fitness values, which is what +we would expect as 𝑝𝑥,𝛿 in such neighborhoods comes with +much uncertainty. +To better illustrate how the boundary-seeking fitness +function distinguishes between inputs based on their prox- +imity to the boundary, let us consider an input space 𝑋 and +two inputs 𝑥1 ∈ 𝑋 and 𝑥2 ∈ 𝑋 where CI(𝑝𝑥1,𝛿) = 0.1 ± 0.05 +and CI(𝑝𝑥2,𝛿) = 0.5 ± 0.1 for a small 𝛿. This means that the +proportions of unsafe inputs around 𝑥1 and 𝑥2 are estimated +as 0.1 ± 0.05 and 0.5 ± 0.1, respectively. If we consider the +boundary of a 5%-unsafe region (i.e., 𝑝th = 0.05), we can say +that 𝑥1 is closer to the boundary than 𝑥2 since the proportion +of unsafe inputs around 𝑥1 is up to 15% while that around +𝑥2 is up to 60%. This is exactly captured by the fitness +function since diff (𝑝𝑥1,𝛿, 0.05) = 0.1 and diff (𝑝𝑥2,𝛿, 0.05) = +0.55, thus yielding fitness(𝑥1, 𝛿, 0.05) = +0.1 +0.95 = 0.105 and + +9 +fitness(𝑥2, 𝛿, 0.05) = +0.55 +0.95 = 0.579, showing that 𝑥1 is closer +to the boundary than 𝑥2. +5.3 +MLC Systemic Hazard Envelope (MLCSHE) Algo- +rithm +Based on the representations of scenarios and MLC behav- +iors described in Section 5.1 and the boundary-seeking fit- +ness function described in Section 5.2, this section proposes +a novel algorithm, MLC Systemic Hazard Envelope (MLCSHE) +[/mIlS/], based on CCEA as described at the beginning of +Section 5. +Algorithm 1 shows the pseudocode of MLCSHE. It takes +as input a population size 𝑛, a minimum number of joint +fitness assessments per individual 𝑘, a threshold probability +𝑝th to define a probabilistic unsafe region, a threshold dis- +tance 𝑑𝑎 to ensure the diversity of individuals and complete +solutions in archives, a maximum population archive size 𝑙, +a distance threshold 𝑑th to filter the complete solutions that +are distinct enough, and a boundary fitness threshold 𝑡𝑏 to +filter complete solutions close enough to the boundary; it +returns an archive 𝐴𝑏 of distinct complete solutions, with +the pairwise distance of more than 𝑑th, whose fitness values +are less than 𝑡𝑏 (i.e., close to the boundary of a 𝑝th-unsafe re- +gion), while 𝑘 and 𝑙 are parameters to control the algorithm’s +search behavior (detailed below). MLCSHE in essence is a +CCEA that uses population archives as described in Sec- +tion 2. However, it is different from other similar methods +as its goal is to return a set of complete solutions satisfying +certain properties (i.e., close to the boundary) rather than +returning a single-best complete solution. +Algorithm 1: MLC Hazard Envelope Search algorithm +(MLCSHE) +Input : Population Size 𝑛 +Minimum Number of Fitness Assessments per +Individual 𝑘 +Threshold Probability 𝑝th +Distance Threshold for Population Archives 𝑑𝑎 +Maximum Size of Population Archive 𝑙 +Distance Threshold for Post-processing 𝑑th +Boundary Fitness Threshold for +Post-processing 𝑡𝑏 +Output: Archive of Distinct Boundary Complete +Solutions 𝐴𝑏 +1 Population of MLC Output Sequences 𝑃𝑂 ← +initPopulation(𝑛) +2 Population of Scenarios 𝑃𝑆 ← initPopulation(𝑛) +3 Archive of MLC Output Sequences 𝐴𝑂 ← 𝑃𝑂 +4 Archive of Scenarios 𝐴𝑆 ← 𝑃𝑆 +5 Archive of Complete Solutions 𝐴𝑐 ← ∅ +6 while not(stopping condition) do +7 +𝑃𝑂, 𝑃𝑆, 𝐴𝑐 ← +assessFitness�𝑃𝑂, 𝑃𝑆, 𝐴𝑂, 𝐴𝑆, 𝑘, 𝐴𝑐, 𝑝th +� +8 +𝐴𝑂 ← updatePopulationArchive(𝑃𝑂, 𝑙, 𝑑𝑎) +9 +𝐴𝑆 ← updatePopulationArchive(𝑃𝑆, 𝑙, 𝑑𝑎) +10 +𝑃𝑂 ← Breed(𝑃𝑂) ∪ 𝐴𝑂 +11 +𝑃𝑆 ← Breed(𝑃𝑆) ∪ 𝐴𝑆 +12 Archive of Complete Solutions +𝐴𝑏 ← postProcess(𝐴𝑐, 𝑑th, 𝑡𝑏) +13 return 𝐴𝑏 +The algorithm first randomly initializes the population +of MLC Output sequences 𝑃𝑂 (line 1), the population of +scenarios 𝑃𝑆 (line 2), and their population archives, 𝐴𝑂 +(line 3) and 𝐴𝑆 (line 4), respectively. The algorithm also +initializes the archive of complete solutions 𝐴𝑐 as an empty +set (line 5). The algorithm then co-evolves 𝑃𝑂 and 𝑃𝑆 using +𝐴𝑂 and 𝐴𝑆, until the stopping condition is met (line 6), such +that it guides them towards the complete solutions that are +close to the boundary of a 𝑝th-unsafe region (lines 6–11). +During the co-evolution, the algorithm repeats the following +three steps: 1) assess the fitness values of individuals in both +𝑃𝑂 and 𝑃𝑆 and update 𝐴𝑐 to include complete solutions +with their joint fitness values evaluated by the simulator +(using function assessFitness at line 7, described in detail in +Algorithm 2); 2) update 𝐴𝑂 and 𝐴𝑆 based on the individual +fitness values, 𝑑, and 𝑙 (using function updatePopulation- +Archive at lines 8–9, described in detail in Algorithm 3); and +3) evolve 𝑃𝑂 and 𝑃𝑆 (using the function breed detailed at +the end of Section 5.3), and merging them with 𝐴𝑂 and 𝐴𝑆, +respectively, to make up the next generation of 𝑃𝑂 and 𝑃𝑆 +(lines 10–11). After the co-evolution, the algorithm creates a +set of complete solutions 𝐴𝑏 from 𝐴𝑐 such that the distance +between two arbitrary, complete solutions in 𝐴𝑏 is at least +𝑑th and the fitness value of every complete solution in 𝐴𝑏 +is less than 𝑡𝑏 (using function postProcess at line 12). The +algorithm ends by returning 𝐴𝑏 (line 13). +5.3.1 +Fitness Assessment +The function assessFitness is to first calculate the joint +fitness values of complete solutions, generated by joining +the individuals in 𝑃𝑂 and 𝑃𝑆 (with higher priorities to +the individuals in 𝐴𝑂 and 𝐴𝑆, respectively) such that each +individual is joined at least 𝑘 times, using the simulator. +To reduce the number of computationally intensive simu- +lations, complete solutions that are the same as the ones +in 𝐴𝐶 (i.e., generated in the previous generations) are not +simulated again. Then, the function assesses the fitness +value of each individual using the joint fitness values of +the complete solutions that contain the individual. +Specifically, Algorithm 2 shows the pseudocode of as- +sessFitness. It takes as input the population of MLC output +sequences 𝑃𝑂, the population of scenarios 𝑃𝑆, the popula- +tion archive of MLC output sequences 𝐴𝑂, the population +archive of scenarios 𝐴𝑆, the minimum number of fitness +assessments per individual 𝑘, the archive of previously +evaluated complete solutions 𝐴𝐶, the neighborhood radius +𝛿, and the threshold probability 𝑝th; it then returns 𝑃𝑂 and +𝑃𝑆 updated to include individual fitness values, and 𝐴𝐶 +updated to include newly generated complete solutions and +their joint fitness values. +The algorithm begins by initializing a set of populations +𝑃𝑆 as {𝑃𝑆, 𝑃𝑂} (line 1). It also initializes a set of complete +solutions 𝐶 by selecting and collaborating individuals from +𝑃𝑆 and 𝑃𝑂 using the collaborate function (line 2). This func- +tion first makes every individual of 𝑃𝑆 and 𝑃𝑂 collaborate +with every individual of 𝐴𝑂 and 𝐴𝑆, respectively, and if the +number of collaborations for each individual is less than 𝑘 +(i.e., when the size of 𝐴𝑂 and 𝐴𝑆 are less than 𝑘), randomly +selected individuals of 𝑃𝑂 \ 𝐴𝑂 and 𝑃𝑆 \ 𝐴𝑆 are used in +addition to 𝐴𝑂 and 𝐴𝑆, respectively, to ensure at least 𝑘 +collaborations for each individual. Then, for each complete +solution 𝑐 ∈ 𝐶 (line 3), if 𝑐 ∉ 𝐴𝐶, i.e., 𝑐 has not been pre- +viously evaluated (line 4), the algorithm evaluates 𝑐 using + +10 +Algorithm 2: assessFitness +Input : Population of MLC Output Sequences 𝑃𝑂 +Population of Scenarios 𝑃𝑆 +Archive of MLC Output Sequences 𝐴𝑂 +Archive of Scenarios 𝐴𝑆 +Minimum Number of Fitness Assessments per +Individual 𝑘 +Archive of Complete Solutions 𝐴𝑐 +Complete Solutions Pairwise Distance Matrix +𝐷𝑐 +Neighborhood Radius 𝛿 +Threshold Probability 𝑝th +Output: Updated Population of MLC Output +Sequences 𝑃𝑂 +Updated Population of Scenarios 𝑃𝑆 +Updated Archive of Complete Solutions 𝐴𝑐 +1 Set of Populations 𝑃𝑆 ← {𝑃𝑂, 𝑃𝑆} +2 Set of Complete Solutions +𝐶 ← collaborate(𝑃𝑂, 𝑃𝑆, 𝐴𝑂, 𝐴𝑆, 𝑘) +3 foreach Complete Solution 𝑐 ∈ 𝐶 do +4 +if 𝑐 ∉ 𝐴𝑐 then +5 +𝑐.isUnsafe ← simulate(c) +6 +𝐴𝑐 ← 𝐴𝑐 ∪ {𝑐} +7 foreach Complete Solution 𝑐 ∈ 𝐴𝑐 do +8 +𝑐.fitness ← computeBoundaryFitness(𝑐, 𝐴𝑐, 𝛿, 𝑝th) +9 foreach Population 𝑃 ∈ 𝑃𝑆 do +10 +foreach Individual 𝑖 ∈ 𝑃 do +11 +𝑖.fitness ← assessIndividualFitness(𝑖, 𝐴𝑐) +12 return 𝑃𝑂, 𝑃𝑆, 𝐴𝑐 +the high-fidelity simulator to identify if 𝑐 is unsafe (line 5) +and adds 𝑐 with its evaluated result into 𝐴𝑐 (line 6). Once +𝐴𝑐 is updated using 𝐶, for each complete solution 𝑐 ∈ 𝐴𝑐 +(line 7), the algorithm computes its joint fitness value (i.e., +the boundary-seeking fitness value) using 𝐴𝑐, 𝛿, and 𝑝th by +calculating the proportion of unsafe complete solutions in +the neighborhood of 𝑐 and its difference from the threshold +probability as described in Section 5.2 (line 8). Although +computing the neighborhood of 𝑐 requires many distance +computations, we can significantly reduce the computations +by reusing the distances among the complete solutions that +were originally in the input 𝐴𝑐. For each individual 𝑃 ∈ 𝑃𝑆 +and for each 𝑖 ∈ 𝑃 (lines 9–10), the algorithm sets the mini- +mum (i.e., the best since we aim to minimize fitness values) +joint fitness value of the complete solutions involving 𝑖 as +the individual fitness of 𝑖 (line 11). The algorithm ends by +returning the updated 𝑃𝑂, 𝑃𝑆, and 𝐴𝑐 (line 12). +5.3.2 +Update Population Archive +The function updatePopulationArchive updates the popu- +lation archives 𝐴𝑂 and 𝐴𝑆, for the next generation. They +play a key role in guiding the search algorithm since every +other individual has to form a complete solution with them, +whose joint fitness will be assessed afterwards. +There are many ways to update the population archive +such as the ones proposed in iCCEA and pCCEA [12]. +However, as mentioned in Section 2, they can be inefficient +due to additional fitness evaluations for updating popu- +lation archives, making them impractical for our problem +involving computationally expensive simulations for fitness +evaluation. Instead, we can consider more efficient archive +update strategies as follows: selecting individuals with the +best fitness values (Best), selecting the best individual plus +random individuals (BestRandom), or randomly selecting in- +dividuals (Random) [12]. Our preliminary evaluation results +on a widely used benchmark problem known as the MTQ +(Maximum of Two Quadratics) [38] showed that both Best and +BestRandom work similarly well for updating population +archives in MLCSHE. To ensure the diversity of individuals +in each population archive and maximize exploration, we +choose BestRandom with a similarity threshold (i.e., the +distance threshold 𝑑th) that filters out individuals deemed +too similar to be included in a population archive. The +pseudocode for updating a population archive is provided +in Algorithm 3. +The algorithm takes as input a target population 𝑃, a +maximum size of a population archive 𝑙, and a threshold +distance (i.e., the minimum distance between arbitrary two +individuals in a population archive) 𝑑th; it returns a popu- +lation archive 𝐴𝑃 of 𝑃 such that |𝐴𝑃| ≤ 𝑙 and 𝑑(𝑖, 𝑗) ≥ 𝑑th, +based on the distance function 𝑑 as described in Section 5.2, +for all 𝑖, 𝑗 ∈ 𝐴𝑃 if 𝑖 ≠ 𝑗. +Algorithm 3: updatePopulationArchive +Input : Population 𝑃 +Maximum Size of Population Archive 𝑙 +Threshold Distance 𝑑th +Output: Population Archive 𝐴𝑃 +1 Archive 𝐴𝑃 ← {popBestFitnessIndividual(𝑃)} +2 while |𝐴𝑃| < 𝑙 or |𝑃| > 0 do +3 +Individual 𝑖 ← randomPop(𝑃) +4 +if isDistinct(𝑖, 𝐴𝑃, 𝑑th) then +5 +𝐴𝑃 ← 𝐴𝑃 ∪ {𝑖} +6 return 𝐴𝑃 +The algorithm starts by initializing a population archive +𝐴𝑃 for 𝑃 using the individual with the best fitness value +among all the individuals in 𝑃 (using function popBestFit- +nessIndividual at line 1). While |𝐴𝑃| < 𝑙 or |𝑃| > 0, the +algorithm iteratively pops a random individual 𝑖 from 𝑃 +(line 3) and add 𝑖 into 𝐴𝑃 (line 5) if 𝑖 is distinct from all +individuals in 𝐴𝑃 with based on the distance threshold of +𝑑th (line 4). The algorithm ends by returning 𝐴𝑃 (line 6). +5.3.3 +Evolution +As illustrated in Algorithm 1, after updating 𝐴𝑂 and 𝐴𝑆, 𝑃𝑂 +and 𝑃𝑆 undergo evolution to generate their next generation +using a breed operation, which entails three main steps: +1) Selection. MLCSHE selects the candidate individuals +for breeding via the standard tournament selection tech- +nique, i.e., the most widely used selection technique for +evolutionary algorithms [39]. It is simple yet effective +since it only requires the rank ordering of individuals +in terms of their fitness values. +2) Crossover. Selected individuals of each population act as +parents to create offspring individuals using a crossover +operation [39]. For our problem, the widely used uni- +form crossover technique [11, 39] is used, since there is no +preference for specific points in individuals as crossover +points. + +11 +3) Mutation. Finally, offspring individuals are mutated +via the introduction of stochastic noise [39]. Since the +individuals are heterogeneous vectors with both float +and integer values, the standard Gaussian and integer +randomization mutation techniques are applied to indi- +vidual elements, respectively [11]. Through these mu- +tations, all valid individuals can be considered during +the search, making it possible to find the global opti- +mum. There is a chance a mutated individual might be +invalid. For example, the x-coordinates (𝑥𝑚𝑖𝑛 and 𝑥𝑚𝑎𝑥) +of a bounding box used to define a detected obstacle +can be out of the camera frame width bounds (from +0 to 800 pixels). Such invalid cases are handled by a +simple repair function in our implementation, which is +available in the replication package (see Section 6.5). +We want to note that there are hyperparameter values for +selection, crossover, and mutation (e.g., the tournament size, +crossover and mutation rates) that could affect breeding +performance. More details on tuning hyperparameter values +in our evaluation is provided in Section 6.2.1. +6 +EVALUATION +In this section, we report on the empirical evaluation of +MLCSHE when applied to an open-source MLAS. Specifi- +cally, we provide answers for the following research ques- +tions: +RQ1 (Effectiveness) How effectively can MLCSHE find the +MLAS hazard envelop boundary compared to baseline +boundary search approaches? +RQ2 (Efficiency) How efficiently can MLCSHE find the +MLAS hazard envelop boundary compared to baseline +boundary search approaches? +To answer RQ1, we investigate how many complete solu- +tions (i.e., combinations of scenarios and MLC behaviors) +that are close to the boundaries are found by different +boundary search approaches, including MLCSHE, given +a same time budget. To answer RQ2, we investigate how +quickly complete solutions that are close to the boundaries +are found by different boundary search approaches. The re- +sults of RQ1 and RQ2 may depend on the distance threshold +between complete solutions and the boundaries3 (𝑑th and 𝑡𝑏 +in algorithm 1). Thus, we also consider the effects of the +distance thresholds while answering RQ1 and RQ2. +6.1 +Evaluation Subjects +We use Pylot [40], one of the highest ranking component- +based AV on the CARLA Autonomous Driving Leader- +board [41], at the time of the evaluation. The leaderboard +evaluates AV according to 11 metrics that are designed to +assess safe driving performance such as collision, red light +infractions and route completion. Pylot’s high performance +on the leaderboard makes it a good candidate to consider +as an evaluation case study. Furthermore, Pylot is one of +the only high-ranking AV that is open-source, has been +deployed on a real-life vehicle [40]. +3. Recall that the fitness of a complete solution is defined based on +its distance from the hazard boundary. Throughout the rest of the +section boundary distance threshold and fitness threshold are used +interchangeably. +We also use CARLA [10], a high-fidelity open-source +AV simulator. CARLA allows us to control various static +and dynamic elements in driving environments. Based on +the controllable elements, following a previous study using +CARLA [42], we consider the following seven scenario ele- +ments: road curve and length, start and end points on maps, +the density of pedestrians, time of day, and weather condi- +tion. The detailed explanation for the scenario elements and +their values (ranges) are available in the supporting material +(see Section 6.5). +For the ML component under test in Pylot, we tar- +get a DNN-based obstacle detection module. The obstacle +detection module takes digital images of the front-facing +camera and detects obstacles in the images in terms of their +location and size (captured as a bounding box), their type, +and the uncertainty associated with the predicted label. The +output of the obstacle detection module is then used by +an obstacle prediction module that predicts the trajectories +of the detected obstacles for future timestamps, followed +by planning and control modules that generate driving +commands considering the obstacles’ predicted trajectories. +Thus, we let the boundary search methods manipulate the +parameters that define the output sequence of the target +MLC (i.e., the object detection module) during the execution +of a simulation. Specifically, for each obstacle, there are 11 +parameters: the label of the detected obstacles (pedestrian or +vehicle), the start and end time the obstacles are detected, +and the 2D coordinates (i.e., 𝑥𝑚𝑖𝑛, 𝑥𝑚𝑎𝑥, 𝑦𝑚𝑖𝑛 and 𝑦𝑚𝑎𝑥) that +define the start and end bounding boxes of the trajecto- +ries on the input image. Additional details regarding the +scenario and MLC output are available in the supporting +material (see Section 6.5). +The above-mentioned categorical (e.g., road curve, +weather condition, and obstacle’s label) and numeric (e.g., +obstacle’s position) parameters defining scenarios and MLC +outputs are used to define a distance function dist which +measures the distance between two complete solutions as +discussed in Section 5.2. Given that these parameters are +heterogeneous, we use an HDM. Specifically, dist is defined +as the average of the normalized Hamming distance [43] +of categorical values and the normalized City Block dis- +tance [43] of numeric values, where the latter are normalized +by their maximum range of values that each parameter can +take. For instance, the 𝑦-coordinates of the MLC outputs +range from 0 to 600 due to the height of the camera frame, +and thus they are divided by 600 to be normalized. As +we have many pairwise distance calculations during the +search, we opted for these distance metrics since they are +computationally efficient compared to alternatives. Given +its definition above, dist ranges between 0 and 1. If dist = 1 +between two complete solutions, it means all the categorical +values of the two are different, and the differences in all the +numeric values of the two are the maximum. +Among various AV safety requirements used in the +literature [35, 44, 45], considering the capability of CARLA +and the major functionality of our target MLC (i.e., the +object detection module), we focus on the following safety +requirement: “AV should have a distance no less than 𝑑𝑚𝑖𝑛 +from the vehicle in front.” To detect safety violations, if any, +during the simulation of a complete solution (i.e., the com- +bination of a scenario and an MLC behavior), we measure + +12 +the distance between the ego vehicle and the vehicle in +front for each simulation time step. Based on feedback from +domain experts and values provided in [42, 46], the value +of 1.5 m was chosen for 𝑑𝑚𝑖𝑛 for the evaluation, which is +reasonable as minimum distance behind a stopped car in +the front since AVs that have much quicker reaction times +than humans [47, 48]. If the distance is less then 𝑑𝑚𝑖𝑛 at any +time, the violation is detected and the complete solution is +marked as unsafe. +Due to the execution time of individual simulations in +CARLA, which is around five minutes on average, the +total computing time for the evaluation is more than 1800 +hours (75 days). To address this issue, we conduct our +evaluation on two machines, M1 and M2. Machine M1 is +a desktop computer with 2.6 GHz Intel i7-10750H CPU, +NVidia GeForce RTX 2070 with Max-Q Design GPU (with +8 GB memory), and 32 GB RAM, running Ubuntu 20.04. +Machine M2 is a g4dn.2xlarge node configured as NVIDIA +GPU-Optimized AMI (version 22.06.0) in Amazon Elastic +Cloud (EC2) with eight virtual cores, NVIDIA T4 GPU +(with 16GB memory), and 32 GB RAM, running Ubuntu +20.04. Specifically, we use M1 for Random Search (RS) and +standard Genetic Algorithm (GA), while MLCSHE is run on +M2. Note that since we keep the number of simulations, +as opposed to the execution time, constant over all the +experiments, the experiments on M1 and M2 are comparable +(see Section 6.2.1 for details). +6.2 +RQ1: Effectiveness +6.2.1 +Methodology +To answer RQ1, we execute MLCSHE and other comparable +methods to generate sets of complete solutions that are +close to the boundary and measure their boundary search +effectiveness in terms of Distinct Boundary Solutions (DBS) +capturing the number of distinct complete solutions close to +the boundary. Specifically, given a distance threshold 𝑑th (for +the distinctiveness of complete solutions) and a boundary +fitness threshold 𝑡𝑏 (for the closeness to the boundary), +let 𝐶𝑉 be the set of complete solutions generated by a +boundary-seeking method 𝑉, satisfying the following con- +ditions4: (1) the pairwise distance between two arbitrary +complete solutions in 𝐶𝑉 is more than 𝑑th and (2) the fitness +value of every complete solution in 𝐶𝑉 is less than 𝑡𝑏. Then, +DBS of 𝑉 is defined as DBS(𝑉) = |𝐶𝑉 | (i.e., the size of 𝐶𝑉 ). To +better understand how DBS varies depending on different +𝑑th and 𝑡𝑏 thresholds, we vary 𝑑th from 0.1 to 0.9 in steps of +0.1 and 𝑡𝑏 from 0.05 to 0.20 in steps of 0.05. Recall that both +distance and fitness values are normalized (𝑑th, 𝑡𝑏 ∈ [0, 1]). +For the other methods to compare with MLCSHE, as +discussed in section 4, we could not find any other work +that has been proposed to address the problem targeted +by this paper. Note that DeepJanus is incomparable to +MLCSHE, as discussed in Section 4, because: 1) its goal is +to study an MLC’s safety under various conditions, which +is different than the goal of this research effort, i.e., finding +the conditions under which an MLC’s behavior can impact +the safety of the system; 2) the boundary identified by +DeepJanus consists of safe-unsafe pairs that can exist in +4. 𝐶𝑉 is computed via the post-processing function postProcess shown +in Algorithm 1. +probabilistic safe or unsafe regions. Thus, we compare MLC- +SHE against two baseline methods, namely Random Search +(RS) and standard Genetic Algorithm (GA) [11]. RS randomly +generates complete solutions, and GA evolves complete +solutions without considering two separate populations of +scenarios and MLC behaviors. In all the methods (including +MLCSHE), the fitness function is the same as defined in +Section 5.2. The results of RS will show how difficult the +search problem is. Furthermore, the comparison between +MLCSHE and GA will show how effective our CCEA-based +method is compared to a standard search method. +For all methods, we set the total number of simulations +as the search budget to 1,300 (i.e., around 2.5 days to run +with two parallel simulations per run), which was a large +enough number to see the convergence of the effectiveness +metrics on our preliminary evaluation. Since most of the +execution cost is dedicated to running simulations, the com- +putation budget of the experiments is mainly determined by +the number of simulations. Thus, we use the total number +of simulations as the search budget. Note that, for MLCSHE +and GA, the actual number of simulations could be slightly +more than the predefined total number since population- +based method check if the search budget is exhausted only +after the completion of one generation. In addition to the +search budget, to ensure the comparability, we set the same +boundary threshold probability (i.e., 𝑝𝑡) and the same maxi- +mum number of obstacle trajectories per mlco to 0.9 and +2, respectively, for all the methods. This makes a complete +solution have 7 (𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜) + 2 × 11 (𝑚𝑙𝑐𝑜) = 29 dimensions. +MLCSHE and GA have additional hyperparameters. For +GA, we used recommended values in [49]; the population +size, the mutation rate, and the crossover rate are set to +60, 0.01, and 0.85, respectively. However, since there are no +suggested values for CCEAs, for which there is much less +experience, we decided to tune them on two benchmark +problems, namely MTQ and Onemax, that are widely used +in evaluating CCEAs [12]. As a result, we used the following +hyperparameters for MLCSHE: population size = 10, maxi- +mum population archive size = 3, mutation rate = 1.0, and +crossover rate = 0.5. The reason for the high mutation rate is +to compensate for the individuals in the population archives +that are directly passed to the next generation without +mutation and crossover in CCEAs. Similarly, regarding the +distance threshold for population archives (𝑑𝑎) in MLCSHE, +we set it to 0.4 based on the two benchmark results. +To account for the randomness of the search-based meth- +ods, we repeat the experiments for each method 10 times. +To evaluate the statistical significance of the difference in +effectiveness metrics of different search methods, we use +the Mann-Whitney U test [50]. To measure the effect size +of the differences, we measure Vargha and Delaney’s ˆ𝐴𝐴𝐵, +where 0 ≤ ˆ𝐴𝐴𝐵 ≤ 1 [51]. Typically, the value of ˆ𝐴𝐴𝐵 indicates +a small, medium, and large difference (effect size) between +populations 𝐴 and 𝐵 when it is higher than 0.56, 0.64, and +0.71, respectively. +6.2.2 +Results +Table 1 reports the DBS achieved by MLCSHE, RS and GA +over 10 runs at various distance threshold (𝑑th) and fitness +threshold (𝑡𝑏) values. + +13 +TABLE 1: DBS values for different search methods at differ- +ent values of 𝑡𝑏 and 𝑑th. +Average DBS±0.5 × 𝐶𝐼0.95 +𝑡𝑏 = 0.05 +𝑡𝑏 = 0.10 +𝑡𝑏 = 0.15 +𝑡𝑏 = 0.20 +𝑑th = +0.1 +RS +0.0 ± 0.0 +0.5 ± 0.4 +5.1 ± 2.0 +19.0 ± 8.8 +GA +37.2 ± 8.9 +66.7 ± 11.4 +82.8 ± 10.8 +94.1 ± 15.4 +MLCSHE +39.6 ± 17.2 +166.8 ± 33.2 +247.5 ± 29.2 +315.4 ± 35.1 +𝑑th = +0.2 +RS +0.0 ± 0.0 +0.5 ± 0.4 +4.6 ± 1.5 +17.0 ± 7.3 +GA +9.2 ± 2.5 +18.8 ± 3.3 +26.0 ± 2.6 +29.8 ± 2.8 +MLCSHE +19.4 ± 6.8 +57.5 ± 6.7 +82.0 ± 5.2 +101.6 ± 6.5 +𝑑th = +0.3 +RS +0.0 ± 0.0 +0.4 ± 0.3 +2.6 ± 0.6 +6.7 ± 2.1 +GA +3.7 ± 0.9 +8.0 ± 1.0 +11.2 ± 1.2 +12.3 ± 1.1 +MLCSHE +7.9 ± 1.8 +17.9 ± 1.1 +23.1 ± 1.7 +25.6 ± 2.2 +𝑑th = +0.4 +RS +0.0 ± 0.0 +0.4 ± 0.3 +1.9 ± 0.4 +3.3 ± 0.7 +GA +1.7 ± 0.4 +3.1 ± 0.6 +4.4 ± 0.4 +4.9 ± 0.5 +MLCSHE +3.5 ± 0.8 +7.4 ± 0.5 +8.7 ± 1.0 +9.3 ± 1.1 +𝑑th = +0.5 +RS +0.0 ± 0.0 +0.4 ± 0.3 +1.2 ± 0.2 +1.5 ± 0.4 +GA +1.1 ± 0.2 +2.3 ± 0.5 +2.6 ± 0.5 +2.8 ± 0.4 +MLCSHE +1.9 ± 0.4 +3.5 ± 0.3 +4.0 ± 0.4 +4.0 ± 0.4 +𝑑th = +0.6 +RS +0.0 ± 0.0 +0.4 ± 0.3 +1.1 ± 0.2 +1.2 ± 0.4 +GA +1.1 ± 0.2 +1.2 ± 0.2 +1.3 ± 0.3 +1.5 ± 0.4 +MLCSHE +1.3 ± 0.3 +1.9 ± 0.3 +2.0 ± 0.3 +2.2 ± 0.4 +𝑑th = +0.7 +RS +0.0 ± 0.0 +0.4 ± 0.3 +1.0 ± 0.0 +1.1 ± 0.2 +GA +1.0 ± 0.0 +1.0 ± 0.0 +1.2 ± 0.2 +1.3 ± 0.3 +MLCSHE +1.1 ± 0.2 +1.3 ± 0.3 +1.3 ± 0.3 +1.5 ± 0.3 +𝑑th = +0.8 +RS +0.0 ± 0.0 +0.4 ± 0.3 +1.0 ± 0.0 +1.0 ± 0.0 +GA +1.0 ± 0.0 +1.0 ± 0.0 +1.0 ± 0.0 +1.0 ± 0.0 +MLCSHE +1.1 ± 0.2 +1.1 ± 0.2 +1.1 ± 0.2 +1.2 ± 0.2 +𝑑th = +0.9 +RS +0.0 ± 0.0 +0.4 ± 0.3 +1.0 ± 0.0 +1.0 ± 0.0 +GA +1.0 ± 0.0 +1.0 ± 0.0 +1.0 ± 0.0 +1.0 ± 0.0 +MLCSHE +1.0 ± 0.0 +1.0 ± 0.0 +1.0 ± 0.0 +1.0 ± 0.0 +For all three boundary-seeking methods, DBS values +increase as fitness threshold (𝑡𝑏) values increase. This is +expected since increasing the value of 𝑡𝑏 results in more +boundary solutions to consider. Further, DBS values plum- +met as the distance threshold (𝑑th) values rise, as expected. +Fig. 5 depicts how the DBS values of the different meth- +ods vary with increasing 𝑑th for different 𝑡𝑏 values. In each +plot, the x-axis is 𝑑th and the y-axis is the average DBS over +10 repeats. The DBS values for MLCSHE, GA, and RS are +marked with circles, triangles, and squares, respectively. The +95% confidence intervals for the average DBS values are also +shown as error bars. +First, RS achieves extremely low DBS values when com- +pared to the other two methods in all cases. This implies that +the problem of identifying MLAS boundaries is sufficiently +challenging for RS not to be able to satisfactorily address it. +Regarding MLCSHE and GA, we can see that the DBS +values drop rapidly with increasing 𝑑th since identifying +complete solutions that are distinct enough with respect +to higher distance thresholds becomes quickly more chal- +lenging. Further, given how we normalized, it is unrealistic +to expect many complete solutions near the boundary with +normalized pairwise distances above 0.5 (i.e., 𝑑th > 0.5); this +is especially true considering that the hazard boundary is +expected to span over a rather limited region in the input +space of the system under test as the safe input region is of- +ten much smaller than the overall input space. However, for +realistic cases (i.e., when 𝑑th ≤ 0.5), MLCSHE significantly +outperforms GA in terms of DBS, except when 𝑑th = 0.1 +and 𝑡𝑏 = 0.05—that is when the very low thresholds make +it infeasible to find many complete solutions that are both +distinct enough from each other and close enough to the hazard +boundary—for which the average DBS of MLCSHE is only +slightly higher than that of GA. In short, this means that +MLCSHE is significantly more effective than GA at finding +complete solutions, at least when the distance threshold is +low enough (𝑑th ≤ 0.5) to be able to find them. Furthermore, +for the same 𝑑th value, the gap between MLCSHE and GA +increases as 𝑡𝑏 increases. +A plausible explanation for these results is that MLC- +SHE uses a cooperative co-evolutionary algorithm which +decomposes a high-dimensional problem into two lower- +dimensional sub-problems, making the search more effec- +tive than GA. Furthermore, MLCSHE takes advantage of +population archives that not only carry information re- +garding the highest-performing individuals but also enforce +diversity among archive members. +Our visual observations are supported by the results of +the statistical comparisons provided in Table 2. Columns +𝐴 and 𝐵 indicate the search methods being compared. +Columns 𝑝 and +ˆ𝐴𝐴𝐵 indicate statistical significance and +effect size, respectively, when comparing A and B in terms +of DBS at different 𝑡𝑏 and 𝑑th values. MLCSHE outperforms +both RS and GA in terms of DBS, when 𝑑th ≤ 0.5, except +when 𝑡𝑏 = 0.05 and 𝑑th = 0.1 for the reasons explained +above. Given a significance level of 𝛼 = 0.01, the differ- +ences between MLCSHE and other methods are significant +(𝑝-value < 0.01) when 𝑑th ≤ 0.5, at all 𝑡𝑏 values. Moreover, +ˆ𝐴𝐴𝐵 is always greater than 0.71 when 𝐴 = MLCSHE and +𝑑th ≤ 0.5, indicating that MLCSHE always has a large effect +size when compared to other search methods. +For realistic distance thresholds (𝑑th ≤ 0.5), MLC- +SHE is significantly more effective than GA and RS +with high effect size, meaning that MLCSHE finds +significantly more diverse regions near the hazard +boundary. +6.3 +RQ2: Efficiency +6.3.1 +Methodology +To answer RQ2, we follow the same methodology as for +RQ1, including the hyperparameters and 10 repeats for +each method, except for the search (simulation) budget. +Specifically, we measure DBS across different methods while +varying the simulation budget from 10% (130 simulations) +to 100% (1300 simulations) in steps of 10%. We then report +and analyze how the effectiveness values of different meth- +ods vary over time. +6.3.2 +Results +Based on the data we collected in our experiment, we +analyzed how all different threshold values for 𝑑th and 𝑡𝑏 +affect the relationship between the percentage of simulation +budget consumed and the average DBS values for 10 runs +across MLCSHE, GA, and RS. Though the trends remain +similar for different 𝑡𝑏 values, 𝑑th affects them significantly. +Furthermore, it makes sense to focus on 𝑑th ≤ 0.5 as we +already found in RQ1 that it is unrealistic to expect many +distinct boundary solutions when 𝑑th > 0.5. Therefore, in +Fig. 6, we selected three 𝑑th values (0.1, 0.3, and 0.5) that, +together, are representative of the overall trends, whereas + +14 +0.2 +0.4 +0.6 +0.8 +Distance Threshold (dth) +0 +10 +20 +30 +40 +50 +DBS +MLCSHE +GA +RS +(a) 𝑡𝑏 = 0.05 +0.2 +0.4 +0.6 +0.8 +Distance Threshold (dth) +0 +50 +100 +150 +200 +DBS +MLCSHE +GA +RS +(b) 𝑡𝑏 = 0.10 +0.2 +0.4 +0.6 +0.8 +Distance Threshold (dth) +0 +50 +100 +150 +200 +250 +DBS +MLCSHE +GA +RS +(c) 𝑡𝑏 = 0.15 +0.2 +0.4 +0.6 +0.8 +Distance Threshold (dth) +0 +50 +100 +150 +200 +250 +300 +350 +DBS +MLCSHE +GA +RS +(d) 𝑡𝑏 = 0.20 +Fig. 5: The relationship between 𝑑th (distance threshold) and DBS (distinct boundary solutions) along with their confidence +intervals (shown as error bars) for MLCSHE, GA, and RS for different 𝑡𝑏 (fitness threshold) values. +0 +25 +50 +75 +100 +% Simulation Budget +0 +50 +100 +150 +200 +250 +300 +Average DBS +a) dth = 0.1 +0 +25 +50 +75 +100 +% Simulation Budget +0 +5 +10 +15 +20 +25 +Average DBS +b) dth = 0.3 +0 +25 +50 +75 +100 +% Simulation Budget +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Average DBS +c) dth = 0.5 +MLCSHE +GA +RS +Fig. 6: Plots of DBS vs. % simulation budget for MLCSHE, GA, and RS 𝑑th = 0.1, 0.3, 0.5 and 𝑡𝑏 = 0.20. + +15 +TABLE 2: Statistical comparison of DBS values for different search methods at different values of 𝑡𝑏 and 𝑑th. +Comparison +𝐷𝐵𝑆 +𝐴 +𝐵 +𝑡𝑏 = 0.05 +𝑡𝑏 = 0.10 +𝑡𝑏 = 0.15 +𝑡𝑏 = 0.20 +𝑝 +ˆ𝐴𝐴𝐵 +𝑝 +ˆ𝐴𝐴𝐵 +𝑝 +ˆ𝐴𝐴𝐵 +𝑝 +ˆ𝐴𝐴𝐵 +𝑑th = +0.1 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +1.46 × 10−4 +1.00 +1.78 × 10−4 +1.00 +1.83 × 10−4 +1.00 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +1.00 +0.51 +4.40 × 10−4 +0.97 +1.83 × 10−4 +1.00 +1.83 × 10−4 +1.00 +𝑅𝑆 +𝐺𝐴 +− +− +1.46 × 10−4 +0.00 +1.78 × 10−4 +0.00 +1.83 × 10−4 +0.00 +𝑑th = +0.2 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +1.44 × 10−4 +1.00 +1.73 × 10−4 +1.00 +1.77 × 10−4 +1.00 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +4.88 × 10−2 +0.77 +1.79 × 10−4 +1.00 +1.81 × 10−4 +1.00 +1.73 × 10−4 +1.00 +𝑅𝑆 +𝐺𝐴 +− +− +1.44 × 10−4 +0.00 +1.73 × 10−4 +0.00 +6.32 × 10−3 +0.14 +𝑑th = +0.3 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +1.38 × 10−4 +1.00 +1.63 × 10−4 +1.00 +1.70 × 10−4 +1.00 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +3.35 × 10−3 +0.89 +1.74 × 10−4 +1.00 +1.72 × 10−4 +1.00 +1.75 × 10−4 +1.00 +𝑅𝑆 +𝐺𝐴 +− +− +1.36 × 10−4 +0.00 +1.62 × 10−4 +0.00 +5.45 × 10−3 +0.13 +𝑑th = +0.4 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +1.22 × 10−4 +1.00 +1.55 × 10−4 +1.00 +1.66 × 10−4 +0.99 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +2.79 × 10−3 +0.89 +1.46 × 10−4 +1.00 +1.50 × 10−4 +1.00 +1.92 × 10−4 +0.99 +𝑅𝑆 +𝐺𝐴 +− +− +1.29 × 10−4 +0.00 +1.78 × 10−4 +0.01 +1.01 × 10−2 +0.17 +𝑑th = +0.5 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +1.10 × 10−4 +1.00 +8.63 × 10−5 +1.00 +1.57 × 10−4 +0.99 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +8.09 × 10−3 +0.81 +3.29 × 10−3 +0.88 +1.77 × 10−3 +0.90 +2.05 × 10−3 +0.89 +𝑅𝑆 +𝐺𝐴 +− +− +1.99 × 10−4 +0.02 +1.10 × 10−3 +0.09 +1.94 × 10−3 +0.10 +𝑑th = +0.6 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +3.00 × 10−4 +0.96 +6.47 × 10−4 +0.91 +2.64 × 10−3 +0.87 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +3.01 × 10−1 +0.60 +9.29 × 10−3 +0.81 +7.63 × 10−3 +0.81 +3.41 × 10−2 +0.77 +𝑅𝑆 +𝐺𝐴 +− +− +3.63 × 10−3 +0.16 +3.01 × 10−1 +0.40 +1.95 × 10−1 +0.36 +𝑑th = +0.7 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +2.79 × 10−3 +0.86 +7.67 × 10−2 +0.65 +6.36 × 10−2 +0.70 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +3.68 × 10−1 +0.55 +7.67 × 10−2 +0.65 +6.51 × 10−1 +0.55 +3.98 × 10−1 +0.60 +𝑅𝑆 +𝐺𝐴 +− +− +5.02 × 10−3 +0.20 +1.67 × 10−1 +0.40 +3.01 × 10−1 +0.40 +𝑑th = +0.8 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +4.43 × 10−3 +0.82 +3.68 × 10−1 +0.55 +1.67 × 10−1 +0.60 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +3.68 × 10−1 +0.55 +3.68 × 10−1 +0.55 +3.68 × 10−1 +0.55 +1.67 × 10−1 +0.60 +𝑅𝑆 +𝐺𝐴 +− +− +5.02 × 10−3 +0.20 +1.00 +0.50 +1.00 +0.50 +𝑑th = +0.9 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝑅𝑆 +− +− +5.02 × 10−3 +0.80 +1.00 +0.50 +1.00 +0.50 +𝑀𝐿𝐶𝑆𝐻𝐸 +𝐺𝐴 +1.00 +0.50 +1.00 +0.50 +1.00 +0.50 +1.00 +0.50 +𝑅𝑆 +𝐺𝐴 +− +− +5.02 × 10−3 +0.20 +1.00 +0.50 +1.00 +0.50 +we fix 𝑡𝑏 at 0.20. The remaining plots are available in the +supporting material (see Section 6.5). +Overall, MLCSHE leads to significantly higher DBS once +the consumed budget is above 10%. We suspect that the +results during the first 10% of the simulation budget is +related to the initial overhead of MLCSHE: since MLCSHE +simulates all possible complete solutions that can be gener- +ated by joining the scenario and MLC output populations +in the first generation, it could complete only one search +generation while GA could complete two or more gener- +ations. However, MLCSHE continues to find new distinct +complete solutions near the boundary as the budget in- +creases, whereas GA quickly starts to stagnate and reach +a plateau. As a result, after only spending 20% of the total +budget, MLCSHE always significantly outperforms GA. +Note that all the search methods start to converge earlier +with higher 𝑑th values; for example, in Fig. 6(c), all the +methods already achieved their best DBS values before +consuming 40% of the simulation budget. This is because, +as mentioned in Section 6.2.2, few or no complete solutions +with normalized pairwise distances above 0.5 can be found +near the hazard boundary. +We also want to note that, even though we had to set the +maximum simulation budget to 1,300 simulations due to the +large size of experiments and the unavoidable limitations in +computational resources, the DBS values of MLCSHE keep +increasing until the budget is exhausted for realistic distance +thresholds (𝑑th ≤ 0.5). This suggests that MLCSHE is able +to find considerably more boundary solutions with more +simulation budget, when available. +MLCSHE is significantly more efficient than GA +and RS: MLCSHE finds significantly more diverse +regions that overlap with the hazard boundary at a +faster rate than GA and RS. +6.4 +Threats to Validity +This section discusses potential threats to the validity of our +results. +As mentioned in Section 6.2.1, the actual number of +executed simulations is slightly higher than the allocated +simulation budget (1,300) for the population-based methods +(i.e., MLCSHE and GA). Although the same budget should +be used for different methods for a fair comparison, the +deviations are so small (less than 5% of the allocated budget) +that they cannot significantly impact the results in terms of +effectiveness and efficiency. +The hyperparameter values for GA can affect the results. +To mitigate this threat, as mentioned in Section 6.2.1, we +relied on the values recommended by Mirjalili [49], which +are commonly used in the literature. +One notable factor to the generalizability of our results is +related to the fact that we have relied only on a specific ADS +(Pylot) and simulator (Carla). However, Carla is a widely +used open-source, high-fidelity simulator, and Pylot was +the only component-based AV among those high-ranking + +16 +in the Carla leaderboard [41] at the time of our evaluation. +Moreover, running the experiments on Pylot and Carla +took more than 75 days of execution, even with paralleliza- +tion, making it infeasible to consider additional evaluation +subjects. Nonetheless, further studies involving other ML- +enabled Autonomous Systems in autonomous driving as +well as other domains, such as aerospace, agriculture, and +manufacturing, are required. +The specific encoding of scenarios and MLC outputs +would be another generalizability factor since it determines +the search space, which could significantly affect the effec- +tiveness and efficiency of each search method. However, +for the large search space problems that are common in +practice, we expect MLCSHE to fare increasingly better +than GA and RS since MLCSHE is designed to decom- +pose high-dimensional problems into lower-dimensional +subproblems. +6.5 +Data Availability +The search algorithms (i.e., MLCSHE, GA, RS), the parallel +simulation execution module, and the postprocess script are +all implemented in Python. The replication package, includ- +ing the aforementioned implementations, the instructions to +set up and configure Pylot and CARLA, the detailed de- +scriptions of the initial conditions used in the experiments, +and the detailed results, is available at [52]. +7 +CONCLUSION AND FUTURE WORK +In this paper, we presented MLCSHE, a cooperative co- +evolutionary search algorithm to effectively and efficiently +approximate the systemic hazard boundary of a machine +learning component embedded in an ML-enabled au- +tonomous system, given a system-level safety requirement. +We address the challenge of the high-dimensional search +space and expensive high-fidelity simulations by using +cooperative coevolutionary search, which decomposes the +problem into two smaller subproblems. We rely on a prob- +abilistic fitness function that guides the search towards +the boundary of probabilistic unsafe regions. We apply +the method to an AV case study, where we run large- +scale experiments with parallel simulations to evaluate the +effectiveness and efficiency of MLCSHE. The evaluation +results show that MLCSHE is significantly more effective +and efficient than random search and a standard genetic +algorithm in identifying diverse boundary regions. +As part of the future work, we plan to apply MLCSHE +to other AVs as well as other ML-enabled autonomous sys- +tems in various domains such as agriculture or aerospace. +Furthermore, we plan to use the hazard boundary approxi- +mated using MLCSHE in developing and evaluating safety +monitors, and guiding the testing of ML components being +integrated in ML-enabled autonomous systems. +ACKNOWLEDGMENTS +The authors are very grateful to Auxon Corporation for their +financial support and to Zachary Pierce for his insightful +feedback. 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Available: https: +//doi.org/10.6084/m9.figshare.21965021 + diff --git a/_tFST4oBgHgl3EQfdjiO/content/tmp_files/load_file.txt b/_tFST4oBgHgl3EQfdjiO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3cd02e7db4735d256519ebfc5218046bc63ffc4c --- /dev/null +++ b/_tFST4oBgHgl3EQfdjiO/content/tmp_files/load_file.txt @@ -0,0 +1,1897 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf,len=1896 +page_content='1 Identifying the Hazard Boundary of ML-enabled Autonomous Systems Using Cooperative Co-Evolutionary Search Sepehr Sharifi , Donghwan Shin , Member, IEEE, Lionel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Briand , Fellow, IEEE, and Nathan Aschbacher Abstract—In Machine Learning (ML)-enabled autonomous systems (MLASs), it is essential to identify the hazard boundary of ML Components (MLCs) in the MLAS under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Given that such boundary captures the conditions in terms of MLC behavior and system context that can lead to hazards, it can then be used to, for example, build a safety monitor that can take any predefined fallback mechanisms at runtime when reaching the hazard boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, determining such hazard boundary for an ML component is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This is due to the space combining system contexts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', scenarios) and MLC behaviors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', inputs and outputs) being far too large for exhaustive exploration and even to handle using conventional metaheuristics, such as genetic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Additionally, the high computational cost of simulations required to determine any MLAS safety violations makes the problem even more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, it is unrealistic to consider a region in the problem space deterministically safe or unsafe due to the uncontrollable parameters in simulations and the non-linear behaviors of ML models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', deep neural networks) in the MLAS under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To address the challenges, we propose MLCSHE (ML Component Safety Hazard Envelope), a novel method based on a Cooperative Co-Evolutionary Algorithm (CCEA), which aims to tackle a high-dimensional problem by decomposing it into two lower-dimensional search subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Moreover, we take a probabilistic view of safe and unsafe regions and define a novel fitness function to measure the distance from the probabilistic hazard boundary and thus drive the search effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We evaluate the effectiveness and efficiency of MLCSHE on a complex Autonomous Vehicle (AV) case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Our evaluation results show that MLCSHE is significantly more effective and efficient compared to a standard genetic algorithm and random search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Index Terms—ML-enabled Autonomous System, Hazard Boundary, System Safety Monitoring, Cooperative Co-Evolutionary Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 1 INTRODUCTION A UTONOMOUS systems are increasingly empowered by being embedded with ML components (MLCs) for various tasks, such as perception, localization, prediction, planning and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' These components are inherently different from conventional software components and pose new challenges and safety risks that are not manageable by traditional software engineering practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The main reason for this difference is that these components’ logic is not captured by source code or specifications but their behavior is rather determined by training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' ML-enabled autonomous systems (MLASs) have already led to fatalities in the case of Autonomous Vehicles (AVs) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This cannot be allowed to continue, especially when human life or very expensive equipment are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Recent efforts have focused on making ML components more reliable, robust and accurate through novel testing methods [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, even a system with reliable com- ponents can still lead to accidents [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, some S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Sharifi and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Briand are with the Department of Electrical and Computer Engineering, University of Ottawa, Ottawa, Ontario, Canada, K1N 5N6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Briand has also a faculty appointment with the SnT Centre at the University of Luxembourg, Luxembourg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' E-mail: {s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='sharifi, lbriand}@uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='ca D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Shin is with Department of Computer Science, University of Sheffield, Sheffield, United Kingdom, S1 4DP E-mail: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='shin@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='uk N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Aschbacher is with Auxon Corporation, Portland, Oregon, United States and its subsidiary Auxon Technologies, Ottawa, Ontario, Canada E-mail: nathan@auxon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='io accidents are caused as a result of unsafe component inter- actions [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, the impact of ML components on safety can only be studied in the context of the system they are integrated into and in a specific operational context [4, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The inherent specificity of ML components favors the use of safety monitors (also known as Run Time Assurance or RTA mechanisms) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Safety monitors, at run time, check the inputs and outputs of a component that cannot be fully trusted, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', an ML component, and will block its outputs from being propagated to the rest of the system if they are potentially hazardous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In such cases, systems usually fall back on a trustworthy but less efficient component [8], or take any other pre-designed fallback mechanisms, such as stopping the AV on the shoulder of the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To do this, safety monitors have to observe the current state of the system and compare it with its Operational Design Domain (ODD) [9], to determine its deviation from ODD bounds since it might lead to hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For instance, it is hazardous to rely on the self-driving feature of an AV on a rainy night if its ODD is characterized by normal dry operations during daylight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Additionally, safety monitors have to know the context of the system to determine whether the component might contribute to a hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, misclassification of an AV’s object detection component might not lead to any hazards under a certain system context (henceforth called a scenario), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', when an AV misidentifies an animal crossing the road as a pedestrian and stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, identifying the combinations of system contexts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', scenarios) and ML component’s behaviors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', inputs and outputs) that will arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='13807v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='SE] 31 Jan 2023 2 transition the system to a hazard state is an essential step in developing safety monitors to be able to ensure the safety of the ML-enabled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, there are several challenges involved with identifying the hazard boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' First, the problem space of scenarios and ML component behaviors is very large and high-dimensional and is thus a challenge for more con- ventional search metaheuristics such as Genetic Algorithms (GA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Second, the violation of a given safety requirement can only be determined if the system is executed within its operational environment, which involves computationally intensive simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The high computational cost, in addi- tion to the large problem space, renders the problem even more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Last but not least, while safety can only be evaluated by executing the system within an environment, there are many environmental parameters that cannot be controlled even via a high-fidelity simulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' for example, the trajectory of pedestrians in CARLA [10], a well-known AV simulator, is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, two similar MLAS inputs may generate largely different outputs due to the non-linear behavior of ML models, such as Deep Neural Networks (DNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, we cannot assume that all combinations of scenarios and ML component behaviors within a region of the problem space have a uniform safety outcome, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the region is deterministically safe or unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Consequently, it is difficult to define hard boundaries be- tween safe and unsafe regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To address the aforementioned challenges, we pro- pose MLCSHE (ML Component Safety Hazard Envelope), a novel Cooperative Co-Evolutionary Algorithm (CCEA)- based approach that efficiently searches the problem space by decomposing it into two sub-spaces (one for scenarios and one for ML component behaviors) and parallelizing the search of sub-spaces while taking the joint contribution of both scenarios and ML component behaviors to the autonomous system safety into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Moreover, instead of naively assuming that the hazard boundary is a clear line that exists between the safe and unsafe regions, we take a probabilistic view of the problem domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', at any point within the scenario and ML component behavior space, there is a probability of being safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Based on this probabilistic lens, we present a novel fitness function that effectively guides the search towards the “probabilistic” hazard boundary based on the probability of finding safe scenario-behavior pairs within a given region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The contributions of this work is summa- rized as follows: MLCSHE, a dedicated and tailored cooperative coevo- lutionary search approach to approximate the hazard boundary of an ML component, in a probabilistic way, taking into account the combination of scenarios and MLC behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' An application of MLCSHE to a complex Autonomous Vehicle (AV) case study involving an industry-strength simulator and an Autonomous Driving System with deep learning components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Our implementation of MLCSHE as well as other case study artefacts are provided in our replication package (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' An evaluation of the effectiveness and efficiency of MLCSHE through large scale experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' A comparison of MLCSHE against baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Paper Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Section 2 provides background materials on CCEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Section 3 defines the problem of MLC hazard boundary identification and details its challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Section 4 discusses related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Section 5 presents MLCSHE in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Section 6 provides the empirical evaluation of MLCSHE and dis- cusses the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Section 7 concludes and suggests future directions for research and improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 2 BACKGROUND In this section, an overview of Evolutionary Algorithms (EAs) is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then we focus on a specific family of EAs, Cooperative Co-Evolutionary Algorithms (CCEAs), which happens to be particularly useful in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Finally, the key decision points involved in designing a CCEA, namely collaborator selection and individual fitness assessment are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' EAs are a family of algorithms designed based on the principles of evolutionary computation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' EAs are in- spired by the concepts related to biological evolution and have been applied to various optimization problems for which standard mathematical optimization is not applica- ble [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' EAs use concepts such as individual, population, fitness, selection and mutation to formalize an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Individuals usually represent solutions to the tar- geted problem and are members of a population whose fitness is evaluated (usually by a fitness function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Desirable individuals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', those with the highest fitness values, are more likely to be selected to act as the parents of the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Using methods such as crossover (replacing some parts of an individual with another one) and mutation (adding randomness to some parts of an individual), indi- viduals of the next generation population, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', next iteration of the search, are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For many problems, the search space is high- dimensional such that a conventional EA would not be able to solve it within a reasonable timeframe [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To address this, Cooperative Coevolutionary Algorithms (CCEAs), originally proposed by Potter and De Jong [13] in 1994, decompose the original problem into lower-dimensional subproblems, each of which can be solved in a separately evolving population as in conventional EAs described pre- viously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Since individuals from each subproblem population must join together to form a complete solution to the original problem, the fitness of an individual can only be evaluated based on the joint fitness of the complete solution created by joining the individual with representative individuals, called collaborators, from other populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' By carefully selecting collaborators and assessing individuals’ fitness, CCEAs are known to be effective at solving even non- separable problems [14, 15] where the fitness of an individual of a subproblem population depends on the fitness of indi- viduals of other populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, the decomposi- tion of the original problem naturally allows parallelism to increase search performance [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Figure 1 depicts the process of an abstract CCEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Each population is initialized, either with randomly selected or guessed values (usually provided by domain experts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Indi- viduals of each population collaborate with individuals of the other population(s) to form complete solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then, 3 Evaluate Complete Solutions No Yes stopping_condition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' New Generation Evolve Individuals Initialize Populations Output Fittest Individual(s) Evaluate Individuals Select Collaborators Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 1: An abstract coevolutionary algorithm (CCEA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' these complete solutions are evaluated via joint fitness as- sessment functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The joint assessments are then aggre- gated to provide evaluations of individual fitness values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' If the stopping condition is reached (true), then the fittest individuals are returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Otherwise, the individuals go through breeding (selection, crossover and mutation) to create the next generation of the populations and go through evaluations again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Designing a CCEA includes three important decisions in the following aspects: collaborator selection and individual fitness assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Collaborator Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' One of the most important factors affecting the performance of a CCEA is its collaborator se- lection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To evaluate the fitness of an individual, a complete solution using the said individual and other indi- viduals from other subproblems, also known as collaborators, should be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The joint fitness of the complete solution contributes to determining the fitness value of the individ- ual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To assess the fitness of the individual, the algorithm has to form one or multiple complete solutions with different collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, ideally, to get closer to the global optimum, all individuals of all other populations should be used as collaborators [16], which is usually infeasible due to resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, a strategy to efficiently select collaborators is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Various strategies have been proposed in the literature, such as single best, tournament- based, and random [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' These strategies affect the algorithm via controlling the selection pressure and the pool size of the collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Some studies have proposed archive-based collaborator selection to effectively reduce the number of collaborators to join in individual fitness assessments while maintaining the amount of information contained in the populations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The idea is to carefully select a population archive which is a subset of a population to be used as collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, Panait et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' [15], have proposed iCCEA, which aims to minimize the size of the population archives by considering only the collaborators that are informative and distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' A collaborator in an archive is informative if adding it to the archive changes the fitness ranking of the popula- tion’s individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' If there are multiple collaborators that can change the ranking of the same individuals, the collaborator that changes the ranking the most will be kept in the archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' A collaborator in an archive is distinct if its (Euclidean) distance from other collaborators in the archive is higher than a pre-defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As a result, a population archive keeps only a minimum number of collaborators while attempting not to lose information in terms of collab- orations between subproblem solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, though the population archive selected by iCCEA is minimal in size, the algorithm proposed by the authors to update the population archive in each generation has a high time complexity (𝑂(𝑛3) where 𝑛 in the number of individuals in the archive) and this severely impacts the performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, simpler population archive selection methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', elitist, random and best random, that are much faster, are also widely used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Individual Fitness Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The collaborator selection strategy of a CCEA affects its individual fitness assessment strategies as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The only objective fitness assessment that can be done on the individuals is based on their joint fitness assessments with collaborators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, all algorithms per- form some form of aggregation on joint fitness assessments related to an individual to determine its fitness value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Best, worst or average joint fitness values are usually used for individual fitness assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 3 PROBLEM AND CHALLENGES In this section, we provide a precise problem definition regarding the identification of the boundaries of hazard envelopes, focusing on the behavior of a Machine Learning Component (MLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' While we use an AV as an example, it can be easily generalised to any ML-based Autonomous System (MLAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Problem Definition Consider an AV as an ML-enabled Autonomous System (MLAS) including an ML component (MLC), namely an image-based object detection component using DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The AV continuously observes its surrounding environments— such as roads, traffic signs, buildings, and other moving vehicles via sensors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', camera) and generates driving commands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', steer left and decrease speed) to best satisfy given functional and safety requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', reach a given destination point without colliding with other vehicles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' During testing of the AV, the environment is often simulated by a high-fidelity driving simulator due to the high cost and risk of real-world testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Inside of the AV, whenever new sensor data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', an image taken from the camera) is collected, it passes through the object detection component to identify the positions of surrounding objects, if any, from the (fused) sensor data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', in the form of bounding boxes in the given image), which will then be used to determine proper driving commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Under a certain driving scenario, the AV might violate requirements such as “the AV shall keep a minimum distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 m from any vehicle in front.” In such cases, the MLC could internally contribute to the vi- olation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, identifying the boundaries of the hazard envelopes of the AV in terms of the combination of driving scenarios and MLC behaviors is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Figure 2 pro- vides a simplified illustration of an ML component’s hazard envelope, defined in terms of scenarios and ML component behaviors, where a safe region leading to no violations is 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 2: An illustration of safe and unsafe regions and the corresponding hazard boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' surrounded by an unsafe region leading to violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The goal is to identify, as precisely and completely as possible, the boundaries between safe and unsafe regions, illustrated by the dashed line in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' More specifically, let 𝑠 be the AV including the MLC 𝑚 for image-based object detection, operating in a simulated driving environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For a given scenario 𝑢, which consists of all the static and dynamic entities of the environment such as road shape, weather, and other vehicles, the simu- lation result for 𝑠 and 𝑚, denoted by Π𝑢,𝑠,𝑚, is a sequence ⟨𝑒1, 𝑒2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' , 𝑒𝑇 ⟩ where 𝑇 is the duration of the execution and 𝑒𝑡 for 𝑡 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' ,𝑇 is the snapshot (state) of the environment at time step 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For each time 𝑡 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' ,𝑇}, 𝑠 takes an image 𝑖𝑛𝑠,𝑡 taken from the camera by observing 𝑒𝑡 and generates a pre-processed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', gray-scaled) image 𝑖𝑛𝑚,𝑡 for 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then, 𝑚 produces the object information 𝑜𝑢𝑡𝑚,𝑡 (in the form of bounding boxes) by processing 𝑖𝑛𝑚,𝑡 and 𝑠 produces driving commands 𝑜𝑢𝑡𝑠,𝑡 by processing 𝑜𝑢𝑡𝑚,𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The environment snapshot 𝑒𝑡+1 for the next time step 𝑡 +1 is updated based on 𝑜𝑢𝑡𝑠,𝑡, and the whole process repeats until 𝑡 reaches 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The behavior of 𝑚, denoted by 𝐵𝑚, is defined as the sequence of input/output pairs such that, for an input/output pair (𝑖𝑛𝑚, 𝑜𝑢𝑡𝑚) ∈ 𝐵𝑚, 𝑜𝑢𝑡𝑚 is the output produced by 𝑚 by processing 𝑖𝑛𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For a safety requirement 𝑟 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', do not collide with other vehicles), we can measure the degree of the safety violation of 𝑠 for 𝑢 and 𝐵𝑚 in terms of 𝑟, denoted by 𝑓 (𝑟, Π𝑢,𝑠,𝑚), by analyzing Π𝑢,𝑠,𝑚 = ⟨𝑒1, 𝑒2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' , 𝑒𝑇 ⟩ against 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' If 𝑓 (𝑟, Π𝑢,𝑠,𝑚) > 𝜖 for a small threshold 𝜖 predefined for 𝑟, we say that 𝑟 for 𝑠 is violated by (the combination of) 𝑢 and 𝐵𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This means that we can decide the violation of 𝑟 (with 𝜖) given 𝑢 and 𝐵𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Given the above context, let (𝑢, 𝐵𝑚) be a point in a space referred to as the input space, that is defined by (the combination of) possible scenarios and MLC behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For each point (𝑢, 𝐵𝑚) in the input space, we can decide its output (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', unsafe or safe) by checking whether it leads to the violation of 𝑟 or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The identification of the boundaries of hazard envelopes attempts to find as many (𝑢, 𝐵𝑚) points as possible that are close to the boundaries between safe and unsafe regions in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Notice that we intentionally left the precise definition of safe and unsafe regions unclear since it is one of the challenges we address next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 Challenges The problem of hazard boundary identification for an MLC in the MLAS under analysis, entails multiple major chal- lenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1, both a scenario 𝑢 and an MLC behavior 𝐵𝑚 collaboratively determine the violation or satisfaction of a safety requirement 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As a result, there are too many possible scenarios and MLC behaviors for the input space to be exhaustively explored without resorting to limiting assumptions that can bias the results [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' One might argue that unsafe regions of the input space could be analytically identified using methods based on expert knowledge, such as FTA [18] and HAZOP [19], to provide clear insights into how hazards can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, such methods are not sufficient to address all possible ways haz- ards can arise due to complex interactions between MLAS components and the opacity of ML components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Second, the satisfaction or violation of 𝑟 can only be determined if the system is operated within its surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' During testing, in addition to the first chal- lenge above, this requires running a high-fidelity simulator which is generally very resource-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The high cost of simulation highlights the need for an efficient and effective method to search as much of the input space as possible while focusing on the regions close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Lastly, recall it is unrealistic to consider a region 100% safe or unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This is explained by two main reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' First, simulators do not often enable full control of all relevant parameters in the environment, thus randomly configuring some of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, the movement of pedestrians is random in CARLA [20], a high-fidelity simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Second, two inputs that are close in the input space may generate different MLC outputs that are handled differently by the rest of the system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', due to the non-linear behavior of other DNNs using the MLC outputs as their input, resulting in different safety results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', safe or unsafe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As a result, we cannot assume a uniform and consistent safety outcome for a region, making it difficult to define hard boundaries between safe and unsafe regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Rather, hazard envelop boundaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the dashed line in Figure 2) should be prob- abilistic as they encompass regions with a given probability threshold of violating a selected requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To address the above challenges, we propose a novel method using Cooperative Co-Evolutionary Algorithm (CCEA) that efficiently address our objectives as an opti- mization problem, within a large input space, by decom- posing such problem into lower-dimensional subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Further, to recast our problem into a coevolutionary search problem, we define a special fitness function that can assess how far a candidate solution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', a combination of 𝑢 and 𝐵𝑚) is from the boundary of a “probabilistic” unsafe region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' See Section 5 for details of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 4 RELATED WORK This section discusses existing studies related to the problem of hazard envelope boundary identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Depending on the methods used, we found three categories: search-based methods, sampling-based methods, and formal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Unsafe Safe S Hazard Boundary5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 3: A possible application of DeepJanus to the systemic hazard boundary detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Connected dots are a safe and unsafe pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Search-based Methods Search-based methods employ metaheuristics (search algo- rithms) and convert the boundary identification problem into a search problem guided by a fitness function that evaluates how close a system input (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', test scenario) is from the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Fitness assessment for individual system inputs often involves simulation executions to check whether safety requirements are violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Although there are many search-based methods for test- ing MLCs [3, 21], the problem of boundary identification has received very little attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Only recently, Riccio and Tonella [22] proposed DeepJanus, the first search-based method to identify the frontier of behavior (frontier) of MLCs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', a set of similar input pairs that trigger different behav- iors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', safe and unsafe) of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The discovered frontier can allow developers to approximate a safe oper- ating envelope for the MLC (by interpolating the pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Also, the overlap of the estimated safe operating envelope with the validity domain of the MLC, which is the domain where the MLC is expected to behave according to its re- quirement(s) [22], can facilitate the evaluation of the MLC’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, for example, DeepJanus can be useful in distinguishing between the performance of two MLCs that perform the same task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, it cannot solve the issue of identifying the hazard boundary, as the impact of MLCs on safety can only be assessed when evaluating the entire system in a given environmental context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, as illustrated in Figure 3, the interpolated frontier of behavior and the hazard boundary of an MLC are not necessarily the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' More precisely, a member of the frontier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', a pair of safe and unsafe inputs) does not necessarily lie in proximity to the hazard boundary since the violations can occur in probabilistic safe regions (as argued in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, we need a novel method to identify the hazard boundary of an MLC within a system, considering the probabilistic nature of (un)safe inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 Sampling-based Methods Unlike search-based methods, which are guided by fitness functions, sampling-based methods use repeated random samplings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', Monte Carlo methods) or statistical metrics to identify certain system inputs that lead to safety vio- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, Meltz and Guterman [23] proposed SmARTest, which uses Monte Carlo methods to identify a scenario domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', set of system inputs) that lead to safety requirement violations determined by measuring Perfor- mance Assessment Functions (PAFs) defined based on the re- quirements’ Key Performance Indicators (KPIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Sinha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' [24] proposed Neural Bridge Sampling (NBS), a method to measure the probability of rare events, such as accidents, using Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' NBS decomposes the proba- bility of a rare event into chained conditional probabilities, which are tractable to compute using standard Monte Carlo methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This provides a better estimate than the naive Monte Carlo or Adaptive Multi-Splitting (AMS) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Sampling-based methods can efficiently identify safe and unsafe inputs from the system’s input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, they only consider system inputs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', scenarios) and not the effect of different MLC behaviors for the same scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2, both the scenario and the MLC behavior must be taken into account to determine the conditions when MLC behavior leads to safety violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 Formal Methods Formal methods rely on formal representations of the input space, the system (including the MLC), and the output space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Examples of such representations include hybrid system or dynamical system formalisms [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Tools like SMT solvers [26] and Mixed Integer Linear Programming (MILP) [27] can be used to analyze whether the system containing the MLC can reach an unsafe region given its input space [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This is known as reachability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Reachability analysis is used to identify the MLC’s barrier certificate, which is an invariant function that constrains the state space of the system and ensures the satisfaction of a safety property [28] while considering the closed-loop behavior of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Barrier certificates can be seen as an over-approximation of the hazard boundary of the MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' [29] proposed Verisig, which can be ap- plied to Cyber-Physical Systems (CPSs) with DNN-based feed-forward controllers with ReLU activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Verisig transforms the ReLU DNN into a hybrid system representation and combines it with the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This recasts the problem as a hybrid system verification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Given a set of system inputs, the outputs can be approximated using Flow∗ [30], a nonlinear system reacha- bility analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Tuncali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' [31] proposed another method to identify barrier certificates of DNN-based, feed-forward controllers, which is not limited to architectures with ReLU activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This method first identifies candidate barrier certificates using simulations, then evaluates their suitability using the dReal [32], an SMT solver for nonlinear formulas in real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' [33] proposed NNV, a method to perform closed-loop reachability analysis of con- trol systems with Deep Reinforcement Learning (DRL) con- trollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' These controllers have a feed-forward architecture with ReLU/Saturation activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' NNV calculates a low-error over-approximation of the output region, which are reached by the system given its inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Although the aforementioned methods provide guaran- tees for the hazard boundary and cover all possible tra- Unsafe Frontierof Behavior Safe6 jectories of the system, they suffer from practicality and scalability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, over-approximation of the hazard boundary might incorrectly reduce (or even remove in the worst case) the safe operating envelope of the system by incorrectly considering some safe behaviors unsafe, thus limiting the practicality of the methods [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, reachability analysis can only be applied to feed-forward controllers with specific activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, it cannot be used for practical MLCs that perform perception, obstacle tracking, or prediction tasks with different DNN architec- tures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', recurrent neural networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Also, reachability analysis has not yet been applied to closed-loop, industrial Cyber-Physical Systems (CPS) with feedback DNN con- trollers [29, 31, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In such a context, scalability is very likely to become an acute problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 Remark on Differences in Objectives A common goal underlying all the above-mentioned meth- ods is to identify the hazard boundary of a given MLC embedded within its containing system (MLAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' It could be useful when the MLC under test is fixed, but as soon as the MLC changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', via retraining), the previously identified hazard boundary would be invalid, and the whole safety verification exercise would have to be repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' On the other hand, in our research, we aim to identify the combinations of conditions and MLC behaviors, without referring to a specific MLC implementation, that could potentially lead to hazards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Once characterized, such situations could then, independently of a specific MLC implementation, be used to monitor the operation of the system and MLC and warn the user in case it is operating near to the hazard boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In the following section, we propose a novel method that addresses the challenges discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2, and is applicable to various types of MLCs, such as perception, planning, and control, without making any assumptions about their architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5 OUR APPROACH In this section, we provide a solution to the problem de- scribed in Section 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the hazard boundary identification of an MLC in the MLAS under analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Our key idea is to recast the problem as a cooperative co-evolutionary search problem where scenarios and MLC behaviors co- evolve as two separate populations but contribute together to find complete solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the combinations of scenar- ios and MLC behaviors) close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then, we use CCEAs, the algorithms that are well known to be effective at solving search problems such as the one described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In the following subsections, we first describe how scenarios and MLC behaviors can be represented as two separate populations in an search problem (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We then define a novel fitness function of the search problem to assess how close a complete solution is from the boundary (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Finally, we present our novel method based on CCEAs using the representation and the fitness function (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Representations We consider two populations, one for scenarios and another one for MLC behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The individuals of the MLC popu- lation, subjected to evolutionary operators, are only repre- sented as MLC-outputs (𝑜𝑢𝑡𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This is due to the initial MLC- input (𝑖𝑛𝑚) being (indirectly) determined by the scenario whereas the next MLC-inputs are affected by previous MLC- outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, 𝑖𝑛𝑚 is recorded in an archive of complete solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', 𝐴𝑐 in Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 for details) but not included in the representation of the behavior of the MLC that can be manipulated by the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Recording the 𝑖𝑛𝑚 and 𝑜𝑢𝑡𝑚 sequences, along with their corresponding scenarios (𝑢), is indeed crucial as it records unsafe behaviors of an MLC (its input and output sequences) given a set of environmental conditions (its scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This information enables the design of safety monitors that will prevent the MLC from contributing to a systemic hazard via leveraging the recorded information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 MLC behaviors One of the two populations considered for the search is the set of MLC behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The behavior of an MLC can be expressed as a sequence of input and output tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The inputs of an MLC are indirectly controlled by the environ- mental input to the system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', scenario parameters) and the components of the system that process that input before it is passed on to the MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, the parameters that we can directly manipulate during the search are the outputs of the MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We represent an individual in the population of the MLC behaviors as a sequence of MLC outputs where the 𝑡-th element of the sequence denotes an MLC output at time step 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The output of an MLC depends on the task performed by the MLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For instance, in the case of a steering angle estimator, the output is a single real value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Whereas, in the case of an object classifier, the output is a vector of probabilities (real values between 0 and 1), where each ele- ment corresponds to a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Finally, similar to our running example, in the case of obstacle detection, the outputs in an ML component (MLC) are detected obstacles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', their bounding box1, their label (such as pedestrian, vehicle, lamp post, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' ), and their timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, an mlco for a simulation duration 𝑇 can be defined as a sequence of the trajectories of detected obstacles during 𝑇 in the case of obstacle detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, given the maximum number of detectable objects 𝑛 and the simulation duration 𝑇, an mlco can be defined as a sequence of 𝑛 trajectories ⟨trj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' , trj𝑛⟩ where trj𝑖 represents the trajectory of the 𝑖-th object (in terms of the bounding boxes) for 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' By allowing the search algorithm to manipulate individual trajectories, an arbitrary mlco can be generated for obstacle detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, allowing the search algorithm to generate all the bounding boxes for individual time steps will likely yield an unrealistic trajectory randomly moving around without a consistent direction, which we observed during our initial trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, it is better to allow the search 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' A bounding box specifies the area on the image processed by the obstacle detector that contains the detected obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' It can be expressed as (𝑥𝑚𝑖𝑛, 𝑥𝑚𝑎𝑥, 𝑦𝑚𝑖𝑛, 𝑦𝑚𝑎𝑥) corresponding to a specific 2D box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 7 algorithm to generate only the start and end bounding boxes, and then generate the remaining bounding boxes for intermediate time steps using linear interpolation between the start and end boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, trj𝑖 can be defined as a triple (class𝑖, start𝑖, end𝑖) where class𝑖 is the class of the 𝑖-th object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', car, bicycle, pedestrian), start𝑖 is the position and the size of the bounding box of the 𝑖-th object at time step 𝑡 = 𝑡start, and 𝑒𝑛𝑑𝑖 is the position and the size of the bounding box of the 𝑖-th object at time step 𝑡 = 𝑡end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, start or end can be defined as a quintuple (𝑡, 𝑥𝑚𝑖𝑛, 𝑥𝑚𝑎𝑥, 𝑦𝑚𝑖𝑛, 𝑦𝑚𝑎𝑥), which are time and bounding box parameters for the beginning or the end of the trajectory, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then, for a given trajectory trj = (class, start, end), we can easily generate the positions and sizes of bounding boxes for intermediate time steps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', 1 < 𝑡 < 𝑇) based on start and end (using linear interpolation) whenever needed for a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 Scenarios A scenario can be represented as a heterogeneous vector of real and integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For the case of an AV, a scenario consists of the vehicle itself, the weather, the road and other static (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', lamp posts and other obstacles) and dynamic objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', pedestrians and other cars) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Each of them have many attributes of various types, namely float (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', speed) and enumerated types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', line pattern) which can be encoded as integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The size of a scenario individual is determined by the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, a finer-grained level of simulation control implies a larger scenario size as more parameters have to be manipulated by the search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For in- stance, one can manipulate all weather-related parameters separately (10 parameters in the case of CARLA [20]) or manipulate them using the weather preset parameter (1 parameter) which sets the value of all granular weather parameters according to high-level modalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', rainy sunset, clear noon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Figure 4 is the scenario domain model for our running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' A Scenario consists of one or more Vehicles (in- cluding the ego vehicle), zero or more Pedestrians and, Mission and Weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The attributes of the domain model that act as the parameters for a scenario representation are written in bold font in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, a scenario can be defined by the time of day, weather preset, map of the town, start point of the ego vehicle, its target destination and target velocity, the number of Pedestrians, and the number and position of other Vehicles with respect to the ego vehicle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', in front, on the opposite lane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Operational Design Domain (ODD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The Operational De- sign Domain or ODD defines an operational envelope of the AV, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', a set of bounds on the environmental parameters of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For instance, highway driving is an ODD for AVs which determines the type of the road, the average speed of the surrounding vehicles, and the (lack of) pedes- trians in the vicinity [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, within an ODD, many scenarios can still be defined, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', weather, the number of cars, the length and shape of the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, a search can be done within an ODD, which sets the values or the bounds of some parameters, such as the target speed of the ego vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The parameter bounds or values set by the ODD will remain static for the duration of the search, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', a target speed of 90kph in a highway driving ODD, or the angle of the sunlight during a daytime driving ODD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 Fitness Function This section presents our proposed fitness function in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Our aim is to design a fitness function that can effectively guide the search towards the boundary of unsafe regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, as mentioned in Section 3, we cannot assume that a region is either 100% unsafe or safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To address this, we first define the notion of safe and unsafe inputs, followed by probabilistic unsafe regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 (Safe and Unsafe Inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' An input is unsafe if and only if it leads to the violation of a given requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Otherwise, the input is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Recall that an input is a combination of a scenario and an MLC behavior in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 (Probabilistic Unsafe Region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Let 𝑋 be a set of all possible inputs, representing the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Given a threshold probability 𝑝th, a region 𝐺 ⊆ 𝑋 is 𝑝th-unsafe if and only if the proportion of unsafe inputs in 𝐺 is higher than or equal to 𝑝th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, if we randomly draw an input from a 5%- unsafe region, we have more than or equal to 5% chance of leading to a safety violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The value of 𝑝th should be determined by a domain expert within a specific application context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Notice that the shape of a probabilistic unsafe region is unknown, as is its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Nevertheless, we can ap- proximate how far an arbitrary input is from the boundary by sampling its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, for an input 𝑥 ∈ 𝑋, let 𝑝𝑥 be the proportion of unsafe inputs in the neighborhood of 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' If 𝑝𝑥 < 𝑝th, it implies that 𝑥 is not likely to be located in a 𝑝th-unsafe region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Otherwise, if 𝑝𝑥 > 𝑝th, it implies that 𝑥 is likely in a 𝑝th-unsafe region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, if 𝑝𝑥 is close to 𝑝th, it implies that 𝑥 is close to the boundary of a 𝑝th-unsafe region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To leverage this idea, we define the notion of neighborhood as follows: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 (Neighborhood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For an input 𝑥 ∈ 𝑋 and a non-negative real number 𝛿 ∈ R+, a neighborhood of 𝑥 with the radius of 𝛿, denoted by 𝑁(𝑥, 𝛿), is defined as follows: 𝑁(𝑥, 𝛿) = {𝑥′ ∈ 𝑋| dist(𝑥, 𝑥′) ≤ 𝛿} where dist(𝑥, 𝑥′) indicates the distance between 𝑥 and 𝑥′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Notice that various distance functions dist can be adopted depending on the nature of complete solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, if a complete solution can be represented as a heterogeneous vector composed of numerical, ordi- nal, and categorical values, Heterogeneous Distance Metrics (HDMs) [36] are good candidates to measure the distance between two complete solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In Figure 2, a neighbor- hood with a radius of 𝛿 is visualised as a circle between safe and unsafe regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Based on Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3, let 𝑝𝑥,𝛿 be the proportion of unsafe inputs in 𝑁(𝑥, 𝛿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then, as discussed above, we can use the difference between 𝑝𝑥,𝛿 and 𝑝th to approximate the distance between 𝑥 and the boundary of a 𝑝th-unsafe region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, we cannot compute the exact value of 𝑝𝑥,𝛿 since 8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='.* 1 Scenario Weather + preset: WeatherPreset + time_of_day: TimeOfDay Vehicle + id: Integer + drive_control: VehicleControl + bounding_box: BoundingBox Pedestrian + id: Integer + bounding_box: BoundingBox + control: WalkerControl Mission + map: Map + start_point: Waypoint + target_destination: Waypoint + target_velocity: Float 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='.* <> WeatherPreset Clear Cloudy Wet WetCloudy SoftRain MidRainy HardRain <> TimeOfDay Noon Sunset Night Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 4: The scenario domain model for the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The model is based on the concepts provided in the Carla World domain model [10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The scenario parameters are shown on the figure in bold font, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the weather preset, the attributes of Mission and the number of Actors such as vehicles and pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑁(𝑥, 𝛿) has too many complete solutions to exhaustively evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Nevertheless, we can compute an estimate of 𝑝𝑥,𝛿, denoted by ˆ𝑝𝑥, 𝛿, and its confidence interval since the con- secutive trials of checking whether an input 𝑥′ ∈ 𝑁(𝑥, 𝛿) is safe or not are assumed to be independent and can be treated as Bernoulli Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, the probability distribution of 𝑝𝑥,𝛿 can be modelled as a Binomial distribution, and we can compute ˆ𝑝𝑥,𝛿 as follows: ˆ𝑝𝑥, 𝛿 = unsafe(𝑁(𝑥, 𝛿)) evaluated(𝑁(𝑥, 𝛿)) where evaluated(𝑁(𝑥, 𝛿)) is the number of inputs evaluated (sampled) in 𝑁(𝑥, 𝛿) and unsafe(𝑁(𝑥, 𝛿)) is the number of unsafe inputs among those evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, using the Wilson Confidence Intervals [37], we can compute the confidence interval of 𝑝𝑥, 𝛿, denoted by CI(𝑝𝑥, 𝛿), as follows: CI(𝑝𝑥, 𝛿) = ˆ𝑝𝑥, 𝛿 ± 𝑧 × √︄ ˆ𝑝𝑥, 𝛿 × (1 − ˆ𝑝𝑥, 𝛿) evaluated(𝑁(𝑥, 𝛿)) where 𝑧 is determined by the standard normal distribution for a given confidence level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', for a 95% confidence level, 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Based on CI(𝑝𝑥, 𝛿), we can assess the maximum differ- ence2 between 𝑝𝑥, 𝛿 and 𝑝th as follows: diff (𝑝𝑥, 𝛿, 𝑝th) = max �|UL(𝑝𝑥, 𝛿) − 𝑝th|, |LL(𝑝𝑥, 𝛿) − 𝑝th|� where UL(𝑝𝑥, 𝛿) and LL(𝑝𝑥, 𝛿) are the upper and lower limits of CI(𝑝𝑥, 𝛿), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Using diff (𝑝𝑥, 𝛿, 𝑝th), we define our fitness function as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 (Boundary-Seeking Fitness Function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For an input 𝑥, a neighborhood radius 𝛿, and a threshold proba- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We consider the maximum difference to be conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' bility 𝑝th, the fitness value of 𝑥 given 𝛿 and 𝑝𝑡, denoted by fitness(𝑥, 𝛿, 𝑝th), is defined as follows: fitness(𝑥, 𝛿, 𝑝th) = diff (𝑝𝑥,𝛿, 𝑝th) max(𝑝th, (1 − 𝑝th)) where the denominator is a normalisation factor, making the range of the fitness value between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In other words, we compute the fitness value of an input 𝑥 using the difference between 𝑝𝑥,𝛿 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the proportion of unsafe inputs in the neighborhood of 𝑥 with the radius of 𝛿) and 𝑝th (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the probability threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Note that the fitness function is meant to be mini- mized and decreases as the difference between 𝑝th and 𝑝𝑥,𝛿 decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The fitness function also takes the number of observations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', evaluated inputs) within 𝑁(𝑥, 𝛿) into account, as the size of CI(𝑝𝑥,𝛿) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the confidence interval of 𝑝𝑥,𝛿) decreases when the number of observations in the neighborhood increases, thereby also decreasing the value of the fitness function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' A sparsely populated neighborhood therefore tends to yield high fitness values, which is what we would expect as 𝑝𝑥,𝛿 in such neighborhoods comes with much uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To better illustrate how the boundary-seeking fitness function distinguishes between inputs based on their prox- imity to the boundary, let us consider an input space 𝑋 and two inputs 𝑥1 ∈ 𝑋 and 𝑥2 ∈ 𝑋 where CI(𝑝𝑥1,𝛿) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05 and CI(𝑝𝑥2,𝛿) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 for a small 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This means that the proportions of unsafe inputs around 𝑥1 and 𝑥2 are estimated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' If we consider the boundary of a 5%-unsafe region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', 𝑝th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05), we can say that 𝑥1 is closer to the boundary than 𝑥2 since the proportion of unsafe inputs around 𝑥1 is up to 15% while that around 𝑥2 is up to 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This is exactly captured by the fitness function since diff (𝑝𝑥1,𝛿, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 and diff (𝑝𝑥2,𝛿, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55, thus yielding fitness(𝑥1, 𝛿, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='105 and 9 fitness(𝑥2, 𝛿, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='95 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='579, showing that 𝑥1 is closer to the boundary than 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 MLC Systemic Hazard Envelope (MLCSHE) Algo- rithm Based on the representations of scenarios and MLC behav- iors described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 and the boundary-seeking fit- ness function described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2, this section proposes a novel algorithm, MLC Systemic Hazard Envelope (MLCSHE) [/mIlS/], based on CCEA as described at the beginning of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Algorithm 1 shows the pseudocode of MLCSHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' It takes as input a population size 𝑛, a minimum number of joint fitness assessments per individual 𝑘, a threshold probability 𝑝th to define a probabilistic unsafe region, a threshold dis- tance 𝑑𝑎 to ensure the diversity of individuals and complete solutions in archives, a maximum population archive size 𝑙, a distance threshold 𝑑th to filter the complete solutions that are distinct enough, and a boundary fitness threshold 𝑡𝑏 to filter complete solutions close enough to the boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' it returns an archive 𝐴𝑏 of distinct complete solutions, with the pairwise distance of more than 𝑑th, whose fitness values are less than 𝑡𝑏 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', close to the boundary of a 𝑝th-unsafe re- gion), while 𝑘 and 𝑙 are parameters to control the algorithm’s search behavior (detailed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' MLCSHE in essence is a CCEA that uses population archives as described in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, it is different from other similar methods as its goal is to return a set of complete solutions satisfying certain properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', close to the boundary) rather than returning a single-best complete solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Algorithm 1: MLC Hazard Envelope Search algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='(MLCSHE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Input : Population Size 𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Minimum Number of Fitness Assessments per ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Individual 𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Threshold Probability 𝑝th ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Distance Threshold for Population Archives 𝑑𝑎 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Maximum Size of Population Archive 𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Distance Threshold for Post-processing 𝑑th ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Boundary Fitness Threshold for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Post-processing 𝑡𝑏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Output: Archive of Distinct Boundary Complete ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Solutions 𝐴𝑏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Population of MLC Output Sequences 𝑃𝑂 ← ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='initPopulation(𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 Population of Scenarios 𝑃𝑆 ← initPopulation(𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 Archive of MLC Output Sequences 𝐴𝑂 ← 𝑃𝑂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 Archive of Scenarios 𝐴𝑆 ← 𝑃𝑆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 Archive of Complete Solutions 𝐴𝑐 ← ∅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 while not(stopping condition) do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='𝑃𝑂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑃𝑆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐴𝑐 ← assessFitness�𝑃𝑂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑃𝑆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐴𝑂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐴𝑆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐴𝑐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑝th � 8 𝐴𝑂 ← updatePopulationArchive(𝑃𝑂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑑𝑎) 9 𝐴𝑆 ← updatePopulationArchive(𝑃𝑆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑑𝑎) 10 𝑃𝑂 ← Breed(𝑃𝑂) ∪ 𝐴𝑂 11 𝑃𝑆 ← Breed(𝑃𝑆) ∪ 𝐴𝑆 12 Archive of Complete Solutions 𝐴𝑏 ← postProcess(𝐴𝑐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑑th,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑡𝑏) 13 return 𝐴𝑏 The algorithm first randomly initializes the population of MLC Output sequences 𝑃𝑂 (line 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' the population of scenarios 𝑃𝑆 (line 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' and their population archives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐴𝑂 (line 3) and 𝐴𝑆 (line 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The algorithm also initializes the archive of complete solutions 𝐴𝑐 as an empty set (line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The algorithm then co-evolves 𝑃𝑂 and 𝑃𝑆 using 𝐴𝑂 and 𝐴𝑆, until the stopping condition is met (line 6), such that it guides them towards the complete solutions that are close to the boundary of a 𝑝th-unsafe region (lines 6–11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' During the co-evolution, the algorithm repeats the following three steps: 1) assess the fitness values of individuals in both 𝑃𝑂 and 𝑃𝑆 and update 𝐴𝑐 to include complete solutions with their joint fitness values evaluated by the simulator (using function assessFitness at line 7, described in detail in Algorithm 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 2) update 𝐴𝑂 and 𝐴𝑆 based on the individual fitness values, 𝑑, and 𝑙 (using function updatePopulation- Archive at lines 8–9, described in detail in Algorithm 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' and 3) evolve 𝑃𝑂 and 𝑃𝑆 (using the function breed detailed at the end of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3), and merging them with 𝐴𝑂 and 𝐴𝑆, respectively, to make up the next generation of 𝑃𝑂 and 𝑃𝑆 (lines 10–11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' After the co-evolution, the algorithm creates a set of complete solutions 𝐴𝑏 from 𝐴𝑐 such that the distance between two arbitrary, complete solutions in 𝐴𝑏 is at least 𝑑th and the fitness value of every complete solution in 𝐴𝑏 is less than 𝑡𝑏 (using function postProcess at line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The algorithm ends by returning 𝐴𝑏 (line 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Fitness Assessment The function assessFitness is to first calculate the joint fitness values of complete solutions, generated by joining the individuals in 𝑃𝑂 and 𝑃𝑆 (with higher priorities to the individuals in 𝐴𝑂 and 𝐴𝑆, respectively) such that each individual is joined at least 𝑘 times, using the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To reduce the number of computationally intensive simu- lations, complete solutions that are the same as the ones in 𝐴𝐶 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', generated in the previous generations) are not simulated again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then, the function assesses the fitness value of each individual using the joint fitness values of the complete solutions that contain the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, Algorithm 2 shows the pseudocode of as- sessFitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' It takes as input the population of MLC output sequences 𝑃𝑂, the population of scenarios 𝑃𝑆, the popula- tion archive of MLC output sequences 𝐴𝑂, the population archive of scenarios 𝐴𝑆, the minimum number of fitness assessments per individual 𝑘, the archive of previously evaluated complete solutions 𝐴𝐶, the neighborhood radius 𝛿, and the threshold probability 𝑝th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' it then returns 𝑃𝑂 and 𝑃𝑆 updated to include individual fitness values, and 𝐴𝐶 updated to include newly generated complete solutions and their joint fitness values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The algorithm begins by initializing a set of populations 𝑃𝑆 as {𝑃𝑆, 𝑃𝑂} (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' It also initializes a set of complete solutions 𝐶 by selecting and collaborating individuals from 𝑃𝑆 and 𝑃𝑂 using the collaborate function (line 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This func- tion first makes every individual of 𝑃𝑆 and 𝑃𝑂 collaborate with every individual of 𝐴𝑂 and 𝐴𝑆, respectively, and if the number of collaborations for each individual is less than 𝑘 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', when the size of 𝐴𝑂 and 𝐴𝑆 are less than 𝑘), randomly selected individuals of 𝑃𝑂 \\ 𝐴𝑂 and 𝑃𝑆 \\ 𝐴𝑆 are used in addition to 𝐴𝑂 and 𝐴𝑆, respectively, to ensure at least 𝑘 collaborations for each individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then, for each complete solution 𝑐 ∈ 𝐶 (line 3), if 𝑐 ∉ 𝐴𝐶, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑐 has not been pre- viously evaluated (line 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' the algorithm evaluates 𝑐 using ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Algorithm 2: assessFitness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Input : Population of MLC Output Sequences 𝑃𝑂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Population of Scenarios 𝑃𝑆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Archive of MLC Output Sequences 𝐴𝑂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Archive of Scenarios 𝐴𝑆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Minimum Number of Fitness Assessments per ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Individual 𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Archive of Complete Solutions 𝐴𝑐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Complete Solutions Pairwise Distance Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='𝐷𝑐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Neighborhood Radius 𝛿 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Threshold Probability 𝑝th ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Output: Updated Population of MLC Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Sequences 𝑃𝑂 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Updated Population of Scenarios 𝑃𝑆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='Updated Archive of Complete Solutions 𝐴𝑐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Set of Populations 𝑃𝑆 ← {𝑃𝑂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑃𝑆} 2 Set of Complete Solutions 𝐶 ← collaborate(𝑃𝑂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑃𝑆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐴𝑂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐴𝑆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑘) 3 foreach Complete Solution 𝑐 ∈ 𝐶 do 4 if 𝑐 ∉ 𝐴𝑐 then 5 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='isUnsafe ← simulate(c) 6 𝐴𝑐 ← 𝐴𝑐 ∪ {𝑐} 7 foreach Complete Solution 𝑐 ∈ 𝐴𝑐 do 8 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='fitness ← computeBoundaryFitness(𝑐, 𝐴𝑐, 𝛿, 𝑝th) 9 foreach Population 𝑃 ∈ 𝑃𝑆 do 10 foreach Individual 𝑖 ∈ 𝑃 do 11 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='fitness ← assessIndividualFitness(𝑖, 𝐴𝑐) 12 return 𝑃𝑂, 𝑃𝑆, 𝐴𝑐 the high-fidelity simulator to identify if 𝑐 is unsafe (line 5) and adds 𝑐 with its evaluated result into 𝐴𝑐 (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Once 𝐴𝑐 is updated using 𝐶, for each complete solution 𝑐 ∈ 𝐴𝑐 (line 7), the algorithm computes its joint fitness value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the boundary-seeking fitness value) using 𝐴𝑐, 𝛿, and 𝑝th by calculating the proportion of unsafe complete solutions in the neighborhood of 𝑐 and its difference from the threshold probability as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Although computing the neighborhood of 𝑐 requires many distance computations, we can significantly reduce the computations by reusing the distances among the complete solutions that were originally in the input 𝐴𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For each individual 𝑃 ∈ 𝑃𝑆 and for each 𝑖 ∈ 𝑃 (lines 9–10), the algorithm sets the mini- mum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the best since we aim to minimize fitness values) joint fitness value of the complete solutions involving 𝑖 as the individual fitness of 𝑖 (line 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The algorithm ends by returning the updated 𝑃𝑂, 𝑃𝑆, and 𝐴𝑐 (line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 Update Population Archive The function updatePopulationArchive updates the popu- lation archives 𝐴𝑂 and 𝐴𝑆, for the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' They play a key role in guiding the search algorithm since every other individual has to form a complete solution with them, whose joint fitness will be assessed afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' There are many ways to update the population archive such as the ones proposed in iCCEA and pCCEA [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, as mentioned in Section 2, they can be inefficient due to additional fitness evaluations for updating popu- lation archives, making them impractical for our problem involving computationally expensive simulations for fitness evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Instead, we can consider more efficient archive update strategies as follows: selecting individuals with the best fitness values (Best), selecting the best individual plus random individuals (BestRandom), or randomly selecting in- dividuals (Random) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Our preliminary evaluation results on a widely used benchmark problem known as the MTQ (Maximum of Two Quadratics) [38] showed that both Best and BestRandom work similarly well for updating population archives in MLCSHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To ensure the diversity of individuals in each population archive and maximize exploration, we choose BestRandom with a similarity threshold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the distance threshold 𝑑th) that filters out individuals deemed too similar to be included in a population archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The pseudocode for updating a population archive is provided in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The algorithm takes as input a target population 𝑃, a maximum size of a population archive 𝑙, and a threshold distance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the minimum distance between arbitrary two individuals in a population archive) 𝑑th;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' it returns a popu- lation archive 𝐴𝑃 of 𝑃 such that |𝐴𝑃| ≤ 𝑙 and 𝑑(𝑖, 𝑗) ≥ 𝑑th, based on the distance function 𝑑 as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2, for all 𝑖, 𝑗 ∈ 𝐴𝑃 if 𝑖 ≠ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Algorithm 3: updatePopulationArchive Input : Population 𝑃 Maximum Size of Population Archive 𝑙 Threshold Distance 𝑑th Output: Population Archive 𝐴𝑃 1 Archive 𝐴𝑃 ← {popBestFitnessIndividual(𝑃)} 2 while |𝐴𝑃| < 𝑙 or |𝑃| > 0 do 3 Individual 𝑖 ← randomPop(𝑃) 4 if isDistinct(𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐴𝑃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝑑th) then 5 𝐴𝑃 ← 𝐴𝑃 ∪ {𝑖} 6 return 𝐴𝑃 The algorithm starts by initializing a population archive 𝐴𝑃 for 𝑃 using the individual with the best fitness value among all the individuals in 𝑃 (using function popBestFit- nessIndividual at line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' While |𝐴𝑃| < 𝑙 or |𝑃| > 0, the algorithm iteratively pops a random individual 𝑖 from 𝑃 (line 3) and add 𝑖 into 𝐴𝑃 (line 5) if 𝑖 is distinct from all individuals in 𝐴𝑃 with based on the distance threshold of 𝑑th (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The algorithm ends by returning 𝐴𝑃 (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 Evolution As illustrated in Algorithm 1, after updating 𝐴𝑂 and 𝐴𝑆, 𝑃𝑂 and 𝑃𝑆 undergo evolution to generate their next generation using a breed operation, which entails three main steps: 1) Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' MLCSHE selects the candidate individuals for breeding via the standard tournament selection tech- nique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the most widely used selection technique for evolutionary algorithms [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' It is simple yet effective since it only requires the rank ordering of individuals in terms of their fitness values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 2) Crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Selected individuals of each population act as parents to create offspring individuals using a crossover operation [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For our problem, the widely used uni- form crossover technique [11, 39] is used, since there is no preference for specific points in individuals as crossover points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 11 3) Mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Finally, offspring individuals are mutated via the introduction of stochastic noise [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Since the individuals are heterogeneous vectors with both float and integer values, the standard Gaussian and integer randomization mutation techniques are applied to indi- vidual elements, respectively [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Through these mu- tations, all valid individuals can be considered during the search, making it possible to find the global opti- mum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' There is a chance a mutated individual might be invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For example, the x-coordinates (𝑥𝑚𝑖𝑛 and 𝑥𝑚𝑎𝑥) of a bounding box used to define a detected obstacle can be out of the camera frame width bounds (from 0 to 800 pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Such invalid cases are handled by a simple repair function in our implementation, which is available in the replication package (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We want to note that there are hyperparameter values for selection, crossover, and mutation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the tournament size, crossover and mutation rates) that could affect breeding performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' More details on tuning hyperparameter values in our evaluation is provided in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6 EVALUATION In this section, we report on the empirical evaluation of MLCSHE when applied to an open-source MLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifi- cally, we provide answers for the following research ques- tions: RQ1 (Effectiveness) How effectively can MLCSHE find the MLAS hazard envelop boundary compared to baseline boundary search approaches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' RQ2 (Efficiency) How efficiently can MLCSHE find the MLAS hazard envelop boundary compared to baseline boundary search approaches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To answer RQ1, we investigate how many complete solu- tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', combinations of scenarios and MLC behaviors) that are close to the boundaries are found by different boundary search approaches, including MLCSHE, given a same time budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To answer RQ2, we investigate how quickly complete solutions that are close to the boundaries are found by different boundary search approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The re- sults of RQ1 and RQ2 may depend on the distance threshold between complete solutions and the boundaries3 (𝑑th and 𝑡𝑏 in algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, we also consider the effects of the distance thresholds while answering RQ1 and RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Evaluation Subjects We use Pylot [40], one of the highest ranking component- based AV on the CARLA Autonomous Driving Leader- board [41], at the time of the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The leaderboard evaluates AV according to 11 metrics that are designed to assess safe driving performance such as collision, red light infractions and route completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Pylot’s high performance on the leaderboard makes it a good candidate to consider as an evaluation case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, Pylot is one of the only high-ranking AV that is open-source, has been deployed on a real-life vehicle [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Recall that the fitness of a complete solution is defined based on its distance from the hazard boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Throughout the rest of the section boundary distance threshold and fitness threshold are used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We also use CARLA [10], a high-fidelity open-source AV simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' CARLA allows us to control various static and dynamic elements in driving environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Based on the controllable elements, following a previous study using CARLA [42], we consider the following seven scenario ele- ments: road curve and length, start and end points on maps, the density of pedestrians, time of day, and weather condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The detailed explanation for the scenario elements and their values (ranges) are available in the supporting material (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For the ML component under test in Pylot, we tar- get a DNN-based obstacle detection module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The obstacle detection module takes digital images of the front-facing camera and detects obstacles in the images in terms of their location and size (captured as a bounding box), their type, and the uncertainty associated with the predicted label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The output of the obstacle detection module is then used by an obstacle prediction module that predicts the trajectories of the detected obstacles for future timestamps, followed by planning and control modules that generate driving commands considering the obstacles’ predicted trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, we let the boundary search methods manipulate the parameters that define the output sequence of the target MLC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the object detection module) during the execution of a simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, for each obstacle, there are 11 parameters: the label of the detected obstacles (pedestrian or vehicle), the start and end time the obstacles are detected, and the 2D coordinates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', 𝑥𝑚𝑖𝑛, 𝑥𝑚𝑎𝑥, 𝑦𝑚𝑖𝑛 and 𝑦𝑚𝑎𝑥) that define the start and end bounding boxes of the trajecto- ries on the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Additional details regarding the scenario and MLC output are available in the supporting material (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The above-mentioned categorical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', road curve, weather condition, and obstacle’s label) and numeric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', obstacle’s position) parameters defining scenarios and MLC outputs are used to define a distance function dist which measures the distance between two complete solutions as discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Given that these parameters are heterogeneous, we use an HDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, dist is defined as the average of the normalized Hamming distance [43] of categorical values and the normalized City Block dis- tance [43] of numeric values, where the latter are normalized by their maximum range of values that each parameter can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For instance, the 𝑦-coordinates of the MLC outputs range from 0 to 600 due to the height of the camera frame, and thus they are divided by 600 to be normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As we have many pairwise distance calculations during the search, we opted for these distance metrics since they are computationally efficient compared to alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Given its definition above, dist ranges between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' If dist = 1 between two complete solutions, it means all the categorical values of the two are different, and the differences in all the numeric values of the two are the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Among various AV safety requirements used in the literature [35, 44, 45], considering the capability of CARLA and the major functionality of our target MLC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the object detection module), we focus on the following safety requirement: “AV should have a distance no less than 𝑑𝑚𝑖𝑛 from the vehicle in front.” To detect safety violations, if any, during the simulation of a complete solution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the com- bination of a scenario and an MLC behavior), we measure 12 the distance between the ego vehicle and the vehicle in front for each simulation time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Based on feedback from domain experts and values provided in [42, 46], the value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 m was chosen for 𝑑𝑚𝑖𝑛 for the evaluation, which is reasonable as minimum distance behind a stopped car in the front since AVs that have much quicker reaction times than humans [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' If the distance is less then 𝑑𝑚𝑖𝑛 at any time, the violation is detected and the complete solution is marked as unsafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Due to the execution time of individual simulations in CARLA, which is around five minutes on average, the total computing time for the evaluation is more than 1800 hours (75 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To address this issue, we conduct our evaluation on two machines, M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Machine M1 is a desktop computer with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 GHz Intel i7-10750H CPU, NVidia GeForce RTX 2070 with Max-Q Design GPU (with 8 GB memory), and 32 GB RAM, running Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Machine M2 is a g4dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2xlarge node configured as NVIDIA GPU-Optimized AMI (version 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0) in Amazon Elastic Cloud (EC2) with eight virtual cores, NVIDIA T4 GPU (with 16GB memory), and 32 GB RAM, running Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, we use M1 for Random Search (RS) and standard Genetic Algorithm (GA), while MLCSHE is run on M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Note that since we keep the number of simulations, as opposed to the execution time, constant over all the experiments, the experiments on M1 and M2 are comparable (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 RQ1: Effectiveness 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Methodology To answer RQ1, we execute MLCSHE and other comparable methods to generate sets of complete solutions that are close to the boundary and measure their boundary search effectiveness in terms of Distinct Boundary Solutions (DBS) capturing the number of distinct complete solutions close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, given a distance threshold 𝑑th (for the distinctiveness of complete solutions) and a boundary fitness threshold 𝑡𝑏 (for the closeness to the boundary), let 𝐶𝑉 be the set of complete solutions generated by a boundary-seeking method 𝑉, satisfying the following con- ditions4: (1) the pairwise distance between two arbitrary complete solutions in 𝐶𝑉 is more than 𝑑th and (2) the fitness value of every complete solution in 𝐶𝑉 is less than 𝑡𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Then, DBS of 𝑉 is defined as DBS(𝑉) = |𝐶𝑉 | (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', the size of 𝐶𝑉 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To better understand how DBS varies depending on different 𝑑th and 𝑡𝑏 thresholds, we vary 𝑑th from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 and 𝑡𝑏 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20 in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Recall that both distance and fitness values are normalized (𝑑th, 𝑡𝑏 ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For the other methods to compare with MLCSHE, as discussed in section 4, we could not find any other work that has been proposed to address the problem targeted by this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Note that DeepJanus is incomparable to MLCSHE, as discussed in Section 4, because: 1) its goal is to study an MLC’s safety under various conditions, which is different than the goal of this research effort, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', finding the conditions under which an MLC’s behavior can impact the safety of the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 2) the boundary identified by DeepJanus consists of safe-unsafe pairs that can exist in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 𝐶𝑉 is computed via the post-processing function postProcess shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' probabilistic safe or unsafe regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, we compare MLC- SHE against two baseline methods, namely Random Search (RS) and standard Genetic Algorithm (GA) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' RS randomly generates complete solutions, and GA evolves complete solutions without considering two separate populations of scenarios and MLC behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In all the methods (including MLCSHE), the fitness function is the same as defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The results of RS will show how difficult the search problem is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, the comparison between MLCSHE and GA will show how effective our CCEA-based method is compared to a standard search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For all methods, we set the total number of simulations as the search budget to 1,300 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 days to run with two parallel simulations per run), which was a large enough number to see the convergence of the effectiveness metrics on our preliminary evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Since most of the execution cost is dedicated to running simulations, the com- putation budget of the experiments is mainly determined by the number of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Thus, we use the total number of simulations as the search budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Note that, for MLCSHE and GA, the actual number of simulations could be slightly more than the predefined total number since population- based method check if the search budget is exhausted only after the completion of one generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In addition to the search budget, to ensure the comparability, we set the same boundary threshold probability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', 𝑝𝑡) and the same maxi- mum number of obstacle trajectories per mlco to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 and 2, respectively, for all the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This makes a complete solution have 7 (𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜) + 2 × 11 (𝑚𝑙𝑐𝑜) = 29 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' MLCSHE and GA have additional hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For GA, we used recommended values in [49];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' the population size, the mutation rate, and the crossover rate are set to 60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='01, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='85, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, since there are no suggested values for CCEAs, for which there is much less experience, we decided to tune them on two benchmark problems, namely MTQ and Onemax, that are widely used in evaluating CCEAs [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As a result, we used the following hyperparameters for MLCSHE: population size = 10, maxi- mum population archive size = 3, mutation rate = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0, and crossover rate = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The reason for the high mutation rate is to compensate for the individuals in the population archives that are directly passed to the next generation without mutation and crossover in CCEAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Similarly, regarding the distance threshold for population archives (𝑑𝑎) in MLCSHE, we set it to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 based on the two benchmark results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To account for the randomness of the search-based meth- ods, we repeat the experiments for each method 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To evaluate the statistical significance of the difference in effectiveness metrics of different search methods, we use the Mann-Whitney U test [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To measure the effect size of the differences, we measure Vargha and Delaney’s ˆ𝐴𝐴𝐵, where 0 ≤ ˆ𝐴𝐴𝐵 ≤ 1 [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Typically, the value of ˆ𝐴𝐴𝐵 indicates a small, medium, and large difference (effect size) between populations 𝐴 and 𝐵 when it is higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='56, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='64, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='71, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 Results Table 1 reports the DBS achieved by MLCSHE, RS and GA over 10 runs at various distance threshold (𝑑th) and fitness threshold (𝑡𝑏) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 13 TABLE 1: DBS values for different search methods at differ- ent values of 𝑡𝑏 and 𝑑th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Average DBS±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 × 𝐶𝐼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='95 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='10 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='15 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 RS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 GA 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 MLCSHE 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 ± 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 ± 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 ± 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 RS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 GA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 MLCSHE 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 RS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 GA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 MLCSHE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 RS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 GA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 MLCSHE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 RS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 GA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 MLCSHE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 For all three boundary-seeking methods, DBS values increase as fitness threshold (𝑡𝑏) values increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This is expected since increasing the value of 𝑡𝑏 results in more boundary solutions to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Further, DBS values plum- met as the distance threshold (𝑑th) values rise, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5 depicts how the DBS values of the different meth- ods vary with increasing 𝑑th for different 𝑡𝑏 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In each plot, the x-axis is 𝑑th and the y-axis is the average DBS over 10 repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The DBS values for MLCSHE, GA, and RS are marked with circles, triangles, and squares, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The 95% confidence intervals for the average DBS values are also shown as error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' First, RS achieves extremely low DBS values when com- pared to the other two methods in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This implies that the problem of identifying MLAS boundaries is sufficiently challenging for RS not to be able to satisfactorily address it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Regarding MLCSHE and GA, we can see that the DBS values drop rapidly with increasing 𝑑th since identifying complete solutions that are distinct enough with respect to higher distance thresholds becomes quickly more chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Further, given how we normalized, it is unrealistic to expect many complete solutions near the boundary with normalized pairwise distances above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', 𝑑th > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' this is especially true considering that the hazard boundary is expected to span over a rather limited region in the input space of the system under test as the safe input region is of- ten much smaller than the overall input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, for realistic cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', when 𝑑th ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5), MLCSHE significantly outperforms GA in terms of DBS, except when 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 and 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05—that is when the very low thresholds make it infeasible to find many complete solutions that are both distinct enough from each other and close enough to the hazard boundary—for which the average DBS of MLCSHE is only slightly higher than that of GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' In short, this means that MLCSHE is significantly more effective than GA at finding complete solutions, at least when the distance threshold is low enough (𝑑th ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5) to be able to find them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, for the same 𝑑th value, the gap between MLCSHE and GA increases as 𝑡𝑏 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' A plausible explanation for these results is that MLC- SHE uses a cooperative co-evolutionary algorithm which decomposes a high-dimensional problem into two lower- dimensional sub-problems, making the search more effec- tive than GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, MLCSHE takes advantage of population archives that not only carry information re- garding the highest-performing individuals but also enforce diversity among archive members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Our visual observations are supported by the results of the statistical comparisons provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Columns 𝐴 and 𝐵 indicate the search methods being compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Columns 𝑝 and ˆ𝐴𝐴𝐵 indicate statistical significance and effect size, respectively, when comparing A and B in terms of DBS at different 𝑡𝑏 and 𝑑th values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' MLCSHE outperforms both RS and GA in terms of DBS, when 𝑑th ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5, except when 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05 and 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 for the reasons explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Given a significance level of 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='01, the differ- ences between MLCSHE and other methods are significant (𝑝-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='01) when 𝑑th ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5, at all 𝑡𝑏 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Moreover, ˆ𝐴𝐴𝐵 is always greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='71 when 𝐴 = MLCSHE and 𝑑th ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5, indicating that MLCSHE always has a large effect size when compared to other search methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' For realistic distance thresholds (𝑑th ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5), MLC- SHE is significantly more effective than GA and RS with high effect size, meaning that MLCSHE finds significantly more diverse regions near the hazard boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 RQ2: Efficiency 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 Methodology To answer RQ2, we follow the same methodology as for RQ1, including the hyperparameters and 10 repeats for each method, except for the search (simulation) budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Specifically, we measure DBS across different methods while varying the simulation budget from 10% (130 simulations) to 100% (1300 simulations) in steps of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We then report and analyze how the effectiveness values of different meth- ods vary over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 Results Based on the data we collected in our experiment, we analyzed how all different threshold values for 𝑑th and 𝑡𝑏 affect the relationship between the percentage of simulation budget consumed and the average DBS values for 10 runs across MLCSHE, GA, and RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Though the trends remain similar for different 𝑡𝑏 values, 𝑑th affects them significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, it makes sense to focus on 𝑑th ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 as we already found in RQ1 that it is unrealistic to expect many distinct boundary solutions when 𝑑th > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Therefore, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6, we selected three 𝑑th values (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5) that, together, are representative of the overall trends, whereas 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 Distance Threshold (dth) 0 10 20 30 40 50 DBS MLCSHE GA RS (a) 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 Distance Threshold (dth) 0 50 100 150 200 DBS MLCSHE GA RS (b) 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 Distance Threshold (dth) 0 50 100 150 200 250 DBS MLCSHE GA RS (c) 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 Distance Threshold (dth) 0 50 100 150 200 250 300 350 DBS MLCSHE GA RS (d) 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 5: The relationship between 𝑑th (distance threshold) and DBS (distinct boundary solutions) along with their confidence intervals (shown as error bars) for MLCSHE, GA, and RS for different 𝑡𝑏 (fitness threshold) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 0 25 50 75 100 % Simulation Budget 0 50 100 150 200 250 300 Average DBS a) dth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 0 25 50 75 100 % Simulation Budget 0 5 10 15 20 25 Average DBS b) dth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 0 25 50 75 100 % Simulation Budget 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='0 Average DBS c) dth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 MLCSHE GA RS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6: Plots of DBS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' % simulation budget for MLCSHE, GA, and RS 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 and 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 15 TABLE 2: Statistical comparison of DBS values for different search methods at different values of 𝑡𝑏 and 𝑑th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Comparison 𝐷𝐵𝑆 𝐴 𝐵 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='10 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='15 𝑡𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20 𝑝 ˆ𝐴𝐴𝐵 𝑝 ˆ𝐴𝐴𝐵 𝑝 ˆ𝐴𝐴𝐵 𝑝 ˆ𝐴𝐴𝐵 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='46 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='78 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='83 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='51 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='40 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='83 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='83 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 𝑅𝑆 𝐺𝐴 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='46 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='78 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='83 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='44 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='73 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='77 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='88 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='79 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='81 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='73 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 𝑅𝑆 𝐺𝐴 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='44 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='73 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='32 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='14 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='3 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='38 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='63 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='70 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='35 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='74 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='72 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='75 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 𝑅𝑆 𝐺𝐴 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='36 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='62 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='45 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='13 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='22 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='66 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='99 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='79 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='46 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='92 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='99 𝑅𝑆 𝐺𝐴 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='29 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='78 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='01 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='17 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='10 × 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='63 × 10−5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='57 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='99 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='09 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='29 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='77 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='05 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='89 𝑅𝑆 𝐺𝐴 − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='99 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='10 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='94 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='10 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='6 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='47 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='91 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='64 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='87 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='01 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='60 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='29 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='81 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='63 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='41 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='77 𝑅𝑆 𝐺𝐴 − − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='63 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='01 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='95 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='36 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='7 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='79 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='86 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='67 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='65 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='36 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='70 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='68 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='67 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='65 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='51 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='98 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='60 𝑅𝑆 𝐺𝐴 − − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='02 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='67 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='01 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='40 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='8 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='43 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='68 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='67 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='60 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='68 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='68 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='68 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='67 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='60 𝑅𝑆 𝐺𝐴 − − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='02 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 𝑑th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='9 𝑀𝐿𝐶𝑆𝐻𝐸 𝑅𝑆 − − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='02 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 𝑀𝐿𝐶𝑆𝐻𝐸 𝐺𝐴 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 𝑅𝑆 𝐺𝐴 − − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='02 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='50 we fix 𝑡𝑏 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The remaining plots are available in the supporting material (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Overall, MLCSHE leads to significantly higher DBS once the consumed budget is above 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We suspect that the results during the first 10% of the simulation budget is related to the initial overhead of MLCSHE: since MLCSHE simulates all possible complete solutions that can be gener- ated by joining the scenario and MLC output populations in the first generation, it could complete only one search generation while GA could complete two or more gener- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, MLCSHE continues to find new distinct complete solutions near the boundary as the budget in- creases, whereas GA quickly starts to stagnate and reach a plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As a result, after only spending 20% of the total budget, MLCSHE always significantly outperforms GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Note that all the search methods start to converge earlier with higher 𝑑th values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' for example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6(c), all the methods already achieved their best DBS values before consuming 40% of the simulation budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This is because, as mentioned in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2, few or no complete solutions with normalized pairwise distances above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 can be found near the hazard boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We also want to note that, even though we had to set the maximum simulation budget to 1,300 simulations due to the large size of experiments and the unavoidable limitations in computational resources, the DBS values of MLCSHE keep increasing until the budget is exhausted for realistic distance thresholds (𝑑th ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This suggests that MLCSHE is able to find considerably more boundary solutions with more simulation budget, when available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' MLCSHE is significantly more efficient than GA and RS: MLCSHE finds significantly more diverse regions that overlap with the hazard boundary at a faster rate than GA and RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='4 Threats to Validity This section discusses potential threats to the validity of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As mentioned in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1, the actual number of executed simulations is slightly higher than the allocated simulation budget (1,300) for the population-based methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', MLCSHE and GA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Although the same budget should be used for different methods for a fair comparison, the deviations are so small (less than 5% of the allocated budget) that they cannot significantly impact the results in terms of effectiveness and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The hyperparameter values for GA can affect the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' To mitigate this threat, as mentioned in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='1, we relied on the values recommended by Mirjalili [49], which are commonly used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' One notable factor to the generalizability of our results is related to the fact that we have relied only on a specific ADS (Pylot) and simulator (Carla).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, Carla is a widely used open-source, high-fidelity simulator, and Pylot was the only component-based AV among those high-ranking 16 in the Carla leaderboard [41] at the time of our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Moreover, running the experiments on Pylot and Carla took more than 75 days of execution, even with paralleliza- tion, making it infeasible to consider additional evaluation subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Nonetheless, further studies involving other ML- enabled Autonomous Systems in autonomous driving as well as other domains, such as aerospace, agriculture, and manufacturing, are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The specific encoding of scenarios and MLC outputs would be another generalizability factor since it determines the search space, which could significantly affect the effec- tiveness and efficiency of each search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' However, for the large search space problems that are common in practice, we expect MLCSHE to fare increasingly better than GA and RS since MLCSHE is designed to decom- pose high-dimensional problems into lower-dimensional subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='5 Data Availability The search algorithms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=', MLCSHE, GA, RS), the parallel simulation execution module, and the postprocess script are all implemented in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The replication package, includ- ing the aforementioned implementations, the instructions to set up and configure Pylot and CARLA, the detailed de- scriptions of the initial conditions used in the experiments, and the detailed results, is available at [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' 7 CONCLUSION AND FUTURE WORK In this paper, we presented MLCSHE, a cooperative co- evolutionary search algorithm to effectively and efficiently approximate the systemic hazard boundary of a machine learning component embedded in an ML-enabled au- tonomous system, given a system-level safety requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We address the challenge of the high-dimensional search space and expensive high-fidelity simulations by using cooperative coevolutionary search, which decomposes the problem into two smaller subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We rely on a prob- abilistic fitness function that guides the search towards the boundary of probabilistic unsafe regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' We apply the method to an AV case study, where we run large- scale experiments with parallel simulations to evaluate the effectiveness and efficiency of MLCSHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' The evaluation results show that MLCSHE is significantly more effective and efficient than random search and a standard genetic algorithm in identifying diverse boundary regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' As part of the future work, we plan to apply MLCSHE to other AVs as well as other ML-enabled autonomous sys- tems in various domains such as agriculture or aerospace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' Furthermore, we plan to use the hazard boundary approxi- mated using MLCSHE in developing and evaluating safety monitors, and guiding the testing of ML components being integrated in ML-enabled autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors are very grateful to Auxon Corporation for their financial support and to Zachary Pierce for his insightful feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFST4oBgHgl3EQfdjiO/content/2301.13807v1.pdf'} +page_content=' This work was also supported through the Nat- ural Sciences and Research Council of Canada 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0000000000000000000000000000000000000000..3c687770497219a03916145f3a2e0325861e1b03 --- /dev/null +++ b/adFQT4oBgHgl3EQffzab/content/tmp_files/2301.13341v1.pdf.txt @@ -0,0 +1,2469 @@ +1 +Neural Target Speech Extraction: An Overview +Katerina Zmolikova, Marc Delcroix, Tsubasa Ochiai, Keisuke Kinoshita, Jan ˇCernock´y, Dong Yu +Abstract—Humans can listen to a target speaker even in +challenging acoustic conditions that have noise, reverberation, +and interfering speakers. This phenomenon is known as the +cocktail-party effect. For decades, researchers have focused on +approaching the listening ability of humans. One critical issue is +handling interfering speakers because the target and non-target +speech signals share similar characteristics, complicating their +discrimination. Target speech/speaker extraction (TSE) isolates +the speech signal of a target speaker from a mixture of several +speakers with or without noises and reverberations using clues +that identify the speaker in the mixture. Such clues might be a +spatial clue indicating the direction of the target speaker, a video +of the speaker’s lips, or a pre-recorded enrollment utterance +from which their voice characteristics can be derived. TSE is an +emerging field of research that has received increased attention in +recent years because it offers a practical approach to the cocktail- +party problem and involves such aspects of signal processing +as audio, visual, array processing, and deep learning. This +paper focuses on recent neural-based approaches and presents +an in-depth overview of TSE. We guide readers through the +different major approaches, emphasizing the similarities among +frameworks and discussing potential future directions. +Index Terms—Speech processing, target speech extraction, +speech enhancement, multi-modal, deep learning +I. INTRODUCTION +In everyday life, we are constantly immersed in complex +acoustic scenes consisting of multiple sounds, such as a mix- +ture of speech signals from multiple speakers and background +noise from air-conditioners or music. Humans naturally extract +relevant information from such noisy signals as they enter +our ears. The cocktail-party problem is a typical example +[1], where we can follow the conversation of a speaker +of interest (target speaker) in a noisy room with multiple +interfering speakers. Humans can manage this complex task +due to selective attention or a selective hearing mechanism +that allows us to focus our attention on a target speaker’s +voice and ignore others. Although the mechanisms of human +selective hearing are not fully understood yet, many studies +have identified essential cues exploited by humans to attend to +a target speaker in a speech mixture: spatial, spectral (audio), +visual, or semantic cues [1]. One long-lasting goal of speech +processing research is designing machines that can achieve +similar listening abilities as humans, i.e., selectively extracting +the speech of a desired speaker based on auxiliary cues. +In this paper, we present an overview of recent devel- +opments in target speech/speaker extraction (TSE), which +estimates the speech signal of a target speaker in a mixture +Katerina Zmolikova and Jan ˇCernock´y are with Brno University of Tech- +nology, Speech@FIT. Marc Delcroix, Tsubasa Ochiai and Keisuke Kinoshita +are with NTT Corporation. Dong Yu is with Tencent, AI Lab. +Target speech +extraction +Spatial +Target speaker clues +Visualal +ua +Audio +Fig. 1. TSE problem and examples of clues +of several speakers, given auxiliary cues to identify the tar- +get1. In the following, we call auxiliary cues, clues, since +they represent hints for identifying the target speaker in the +mixture. Fig. 1 illustrates the TSE problem and shows that by +exploiting the clues, TSE can focus on the voice of the target +speaker while ignoring other speakers or noise. Inspired by +psychoacoustic studies [1], several clues have been explored +to tackle the TSE problem, such as spatial clues that provide +the direction of the target speaker [2], [3], visual clues from +video of their face [4]–[9], or audio clues extracted from pre- +recorded enrollment recording of their voice [10]–[12]. +The TSE problem is directly related to human selective +hearing, although we approach it from an engineering point of +view and do not try to precisely mimic human mechanisms. +TSE is related to other speech and audio-processing tasks +such as noise reduction and blind source separation (BSS) +that do not use clues about the target speaker. Although +noise reduction does suppress the background noise, it cannot +handle well interfering speakers. BSS estimates each speech +source signal in a mixture, which usually requires estimating +the number of sources, a step that is often challenging. +Moreover, it estimates the source signals without identifying +them, which leads to global permutation ambiguity at its +output; it remains ambiguous which of the estimated source +signals corresponds to the target speaker. In contrast, TSE +focuses on the target speaker’s speech by exploiting clues +without assuming knowledge of the number of speakers in +the mixture and avoids global permutation ambiguity. It thus +offers a practical alternative to noise reduction or BSS when +the use case requires focusing on a desired speaker’s voice. +Solving the TSE problem promises real implications for +the development of many applications: (1) robust voice user +interfaces or voice-controlled smart devices that only respond +to a specific user; (2) teleconferencing systems that can remove +1Alternative terms in the literature for TSE include informed source sepa- +ration, personalized speech enhancement, or audio-visual speech separation, +depending on the context and the modalities involved. +Copyright ©2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any +copyrighted component of this work in other works. +arXiv:2301.13341v1 [eess.AS] 31 Jan 2023 + +12 +interfering speakers close by; (3) hearing aids/hearables that +can emphasize the voice of a desired interlocutor. +TSE ideas can be traced back to early works on beam- +formers [2]. Several works also extended BSS approaches to +exploit clues about the target speaker [4], [5], [12]. Most of +these approaches required a microphone array [5] or models +trained on a relatively large amount of speech data from the +target speaker [4]. The introduction of neural networks (NNs) +enabled the building of powerful models that learn to perform +complex conditioning on various clues by leveraging large +amounts of speech data of various speakers. This evolution re- +sulted in impressive extraction performance. Moreover, neural +TSE systems can operate with a single microphone and with +speakers unseen during the training of the models, allowing +more flexibility. +This overview paper covers recent TSE development and +focuses on neural approaches. Its remaining sections are +organized as follows. In Section II, we formalize the TSE +problem and its relation to noise reduction and BSS and +introduce its historical context. We then present a taxonomy of +TSE approaches and motivate the focus of this overview paper +in Section III. We describe a general neural TSE framework +in Section IV. The later sections (V, VI, and VII) introduce +implementations of TSE with different clues, such as audio, +visual, and spatial clues. We discuss extensions to other tasks +in Section VIII. Finally, we conclude by describing the outlook +on remaining issues in Section IX and provide pointers to +available resources for experimenting with TSE in Section X. +II. PROBLEM DEFINITION +A. Speech recorded with a distant microphone +Imagine recording a target speaker’s voice in a living room +using a microphone placed on a table. This scenario represents +a typical use case of a voice-controlled smart device or a +video-conferencing device in a remote-work situation. Many +sounds may co-occur while the speaker is speaking, e.g., +a vacuum cleaner, music, children screaming, voices from +another conversation, or from a TV. The speech signal captured +at a microphone thus consists of a mixture of the target +speaker’s speech and interference from the speech of other +speakers and background noise2. We can express the mixture +signal recorded at a microphone as +ym = xm +s + +� +k̸=s +xm +k + vm +� +�� +� +≜im +, +(1) +where ym = [ym[0], . . . , ym[T]] ∈ RT , xm +s ∈ RT , xm +k ∈ RT , +and vm ∈ RT are the time-domain signal of the mixture, +the target speech, the interference speech, and noise signals, +respectively. Variable T represents the duration (number of +samples) of the signals, m is the index of the microphone +in an array of microphones, s represents the index of the +target speaker and k is the index for the other speech sources. +2In this paper, we do not explicitly consider the effect of reverberation +caused by the reflection of sounds on the walls and surfaces in a room, +which also corrupt the recorded signal. Some of the approaches we discussed +implicitly handle reverberation. +We drop microphone index m whenever we deal with single +microphone approaches. In the TSE problem, we are interested +in only recovering the target speech of speaker s, xm +s , and view +all the other sources as undesired signals to be suppressed. +We can thus define the interference signal as im ∈ RT . Note +that we make no explicit hypotheses about the number of +interfering speakers. +B. TSE problem and its relation to BSS and noise reduction +The TSE problem is to estimate the target speech, given a +clue, Cs, as +ˆxs = TSE(y, Cs; θTSE), +(2) +where ˆxs is the estimate of the target speech, TSE(·; θTSE) +represents a TSE system with parameters θTSE. The clue, Cs, +allows identifying the target speaker in the mixture. It can be +of various types, such as a pre-recorded enrollment utterance, +C(a) +s , a video signal capturing the face or lips movements of +the target speaker, C(v) +s , or such spatial information as the +direction of arrival (DOA) of the speech of the target speaker, +C(d) +s . +In the later sections, we expand on how to design TSE +systems. Here, we first emphasize the key difference between +TSE and BSS and noise reduction. Fig. 2 compares these three +problems. +BSS [13], [14] estimates all the source signals in a mixture +without requiring clues: +{ˆx1, . . . , ˆxK} = BSS(y; θBSS), +(3) +where BSS(·; θBSS) represents a separation system with pa- +rameters θBSS, ˆxk are the estimates of the speech sources, and +K is the number of sources in the mixture. As seen in Eq. (3), +BSS does not and cannot differentiate the target speech from +other speech sources. Therefore, we cannot know in advance +which output corresponds to the target speech, i.e., there is +a global permutation ambiguity problem between the outputs +and the speakers. Besides, since the number of outputs is given +by the number of sources, the number of sources K must be +known or estimated. Comparing Eqs. (2) and (3) emphasizes +the fundamental difference between TSE and BSS: (1) TSE +estimates only the target speech signal, while BSS estimates +all the signals, and (2) TSE is conditioned on speaker clue +Cs, while BSS only relies on the observed mixture3. Typical +use cases for BSS include applications that require estimating +speech signals of every speaker, such as automatic meeting +transcription systems. +Noise reduction is another related problem. It assumes that +the interference only consists of background noise, i.e., i = v, +and can thus enhance the target speech without requiring clues: +ˆxs = Denoise(y; θDenoise), +(4) +where Denoise(·; θDenoise) represents a noise reduction sys- +tem with parameters θDenoise. Unlike BSS, a noise reduction +system’s output only consists of target speech ˆxs, and there +3Another setup sitting between TSE and BSS is a task that extracts multiple +target speakers, e.g., extracting the speech of all the meeting attendees given +such information about them as enrollment or videos of all the speakers. + +3 +Fig. 2. Comparison of TSE with BSS and noise reduction +is thus no global permutation ambiguity. This is possible if +the background noise and speech have distinct characteristics. +For example, we can assume that ambient noise and speech +signals exhibit different spectro-temporal characteristics that +enable their discrimination. However, noise reduction cannot +suppress interfering speakers because it cannot discriminate +among different speakers in a mixture without clues4. Noise +reduction is often used, e.g., in video-conferencing systems or +hearing aids. +TSE is an alternative to BSS and noise reduction, which +uses a clue to simplify the problem. Like BSS, it can handle +speech mixtures. Like noise reduction, it only estimates the +target speaker, thus avoiding global permutation ambiguity +and the need to estimate the number of sources. However, +TSE requires access to clues, unlike BSS and noise reduction. +Moreover, it must internally perform two sub-tasks: (1) iden- +tifying the target speaker and (2) estimating the speech of that +speaker in the mixture. TSE is thus a challenging problem that +introduces specific issues and requires dedicated solutions. +A straightforward way to achieve TSE using BSS methods is +to first apply BSS and next select the target speaker among the +estimated sources. Such a cascade system allows the separate +development of BSS and speaker identification modules. How- +ever, this scheme is usually computationally more expensive +and imports some disadvantages of BSS, such as the need to +estimate the number of speakers in the mixture. Therefore, +we focus on approaches that directly exploit the clues in the +extraction process. Nevertheless, most TSE research is rooted +in BSS, as argued in the following discussion on the historical +context. +C. Historical context +The first studies related to TSE were performed in the +1980s. Flanagan et al. [2] explored enhancing a target +speaker’s voice in a speech mixture, assuming that the target +speech originated from a fixed and known direction. They +employed a microphone array to record speech and designed +a fixed beamformer that enhanced the signals from the target +direction [2], [16]. We consider that this work represents an +early TSE system that relies on spatial clues. +In the mid-1990s, the BSS problem gained attention with +pioneering works on independent component analysis (ICA). +4Some works propose to exploit clues for noise reduction and apply +similar ideas of TSE to reduce background noise (and sometimes interfering +speakers). In the literature, this is called personalized speech enhancement, +which in this paper, we view as a special case of the TSE problem, where +only the target speaker is actively speaking [15]. +ICA estimates spatial filters that separate the sources by +relying on the assumption of the independence of the sources +in the mixture and the fact that speech signals are non- +Gaussian [13]. A frequency-domain ICA suffers from a fre- +quency permutation problem because it treats each frequency +independently. In the mid-2000s, independent vector analysis +(IVA) addressed the frequency-permutation problem by work- +ing on vectors spanning all frequency bins, which allowed +modeling dependency among frequencies [13]. Several works +have extended ICA and IVA to perform TSE, which simplifies +inference by focusing on a single target source. For example, in +the late 2000s, TSE systems were designed by incorporating +the voice activity information of the target speaker derived +from video signals to the ICA criterion, allowing identification +and extraction of only the target source [5]. In the late 2010s, +independent vector extraction (IVE) extended IVA to extract +a single source out of the mixture. In particular, IVE exploits +clues to guide the extraction process, such as the enrollment of +the target speaker to achieve TSE [12]. All these approaches +require a microphone array to capture speech. +In the first decade of the 2000s, single-channel approaches +for BSS emerged, such as factorial hidden Markov model +(F-HMM) [17] and non-negative matrix factorization (NMF) +[18]. These approaches relied on pre-trained spectral models +of speech signals learned on clean speech data. An F-HMM is +a model of speech mixtures, where the speech of each speaker +in the mixture is explicitly modeled using a separate hidden +Markov model (HMM). The parameters of each speaker-HMM +are learned on the clean speech data of that speaker. The sep- +aration process involves inferring the most likely HMM state +sequence associated with each speaker-HMM, which requires +approximations to make inference tractable. This approach +was the first to achieve super-human performance using only +single-channel speech [17]. In the early 2000s, F-HMM was +also among the first approaches to exploit visual clues [4]5. In +NMF, the spectrogram of each source is modeled as a multi- +plication of pre-learned bases, representing the basic spectral +patterns and their time-varying activations. NMF methods have +also been extended to multi-channel signals [13] and used to +extract a target speaker [19] with a flexible multi-source model +of the background. The main shortcoming of the F-HMM +and NMF methods is that they require pre-trained source +models and thus struggle with unseen speakers. Furthermore, +5This framework needs having clues for all of the speakers, a requirement +that negates some of the advantages of TSE, e.g., the number of speakers +must be known beforehand. Despite that, the method does not suffer from +global permutation ambiguity, since visual clues identify the target speaker, +and we thus include this work in the broader view of TSE methods. + +4 +the inference employs a computationally expensive iterative +optimization. +In the mid-2010s, deep NNs (DNNs) were first introduced +to address the BSS problem. These approaches rapidly gained +attention with the success of deep-clustering and permutation +invariant training (PIT) [20], [21], which showed that single- +channel speaker-open6 BSS was possible. In particular, the +introduction of DNNs enabled more accurate and flexible spec- +trum modeling and computationally efficient inference. These +advances were facilitated by supervised training methods that +can exploit a large amount of data. +Neural BSS rapidly influenced TSE research. For example, +Du et al. [22] trained a speaker-close NN to extract the +speech of a target speaker using training data with mixed +various interfering speakers. This work is an initial neural +TSE system using audio clues. However, using speaker-close +models requires a significant amount of data from the target +speaker and cannot be extended to speakers unseen during +training. Subsequently, the introduction of TSE systems con- +ditioned on speaker characteristics derived from an enrollment +utterance significantly mitigated this requirement [10], [11], +[23]. Enrollment consists of a recording of a target speaker’s +voice, which amounts to a few seconds of speech. With +these approaches, audio clue-based TSE became possible for +speakers unseen during training as long as an enrollment +utterance was available. Furthermore, the flexibility of NNs to +integrate different modalities combined with the high modeling +capability of face recognition or lip-reading systems offered +new possibilities for speaker-open visual clue-based TSE [7], +[8]. More recently, neural approaches have also been intro- +duced for spatial-clue-based TSE [3], [24]. +TSE has gained increased attention. For example, dedi- +cated tasks were part of such recent evaluation campaigns as +the deep noise suppression (DNS)7 and Clarity8 challenges. +Many works have extended TSE to other tasks, such as a +direct automatic speech recognition (ASR) of a target speaker +from a mixture, which is called target speaker ASR (TS- +ASR) [25], [26], or personalized voice activity detection +(VAD)/diarization [27], [28]. Notably, target speaker VAD +(TS-VAD)-based diarization [28] has been very successful +in such evaluation campaigns as CHiME-69 or DIHARD- +310, outperforming state-of-the-art diarization approaches in +challenging conditions. +III. TSE TAXONOMY +TSE is a vast research area spanning a multitude of ap- +proaches. This section organizes them to emphasize their +relations and differences. We categorized the techniques using +four criteria: 1) type of clues, 2) number of channels, 3) +speaker-close vs. open, and 4) generative vs. discriminative. +Table I summarizes the taxonomy; the works in the scope of +this overview paper are emphasized in red. +6BSS is possible for speakers unseen during training, i.e., not present in +the training data. +7https://www.microsoft.com/en-us/research/academic-program/ +deep-noise-suppression-challenge-icassp-2022/ +8https://claritychallenge.github.io/clarity CC doc +9https://chimechallenge.github.io/chime6/results.html +10https://dihardchallenge.github.io/dihard3/results +A. Type of clue +The type of clue used to determine the target speaker is +an important factor in distinguishing among TSE approaches. +The most prominent types are audio, visual, and spatial clues. +This classification also defines the main organization of this +article, which covers such approaches in Sections V, VI, and +VII. Other types have and could be proposed, as we briefly +discuss in Section IX. +An audio clue consists of a recording of a speech signal +of the target speaker. Such a clue can be helpful, e.g., in the +use case of personal devices, where the user can pre-record +an example of their voice. Alternatively, for long recordings, +such as meetings, clues can be obtained directly from part of +the recording. The interest in audio clues sharply increased +recently with the usage of neural models for TSE [10]–[12]. +Audio clues are perhaps the most universal, because they do +not require using any additional devices, such as multiple +microphones or a camera. However, the performance may be +limited compared to other clues, since discriminating speakers +based only on their voice characteristics is prone to errors due +to inter- and intra-speaker variability. For example, the voice +characteristics of different speakers, such as family members, +often closely resemble each other. On the other hand, the voice +characteristics of one speaker may change depending on such +factors as emotions, health, or age. +A visual clue consists of a video of the target speaker +talking. This type is often constrained to the speaker’s face, +sometimes just to the lip area. Unlike audio clues, visual +clues are typically synchronized with audio signals that are +processed, i.e., not pre-recorded. A few works also explored +just using a photo of the speaker [37]. Visual clues have been +employed to infer the activity pattern and location of the target +speaker [5] or to jointly model audio and visual signals [4], [5]. +Recent works usually use visual clues to guide discriminative +models toward extracting the target speaker [7]–[9]. Visual +clues are especially useful when speakers in the recording +have similar voices [8]. However, they might be sensitive to +physical obstructions of the speaker in the video. +A spatial clue refers to the target speaker’s location, e.g., +the angle from the recording devices. The location can be +inferred in practice from a video of the room or a recording +of a speaker in the same position. Extracting the speaker +based on their location has been researched from mid 1980’s, +with beamforming techniques that pioneered this topic [2], +[16]. More recent IVE models use location for initialization +[12]. Finally, several works have shown that NNs informed +by location can also achieve promising performance [3], [24]. +Spatial clues are inherently applicable only when a recording +from multiple microphones is available. However, they can +identify the target speaker in the mixture rather reliably, +especially when the speakers are stationary. +Different clues may work better in different situations. For +example, the performance with audio clues might depend on +the similarity of voices of the present speakers, and obstruc- +tions in the video may influence visual clues. As such, it is +advantageous to use multiple clues simultaneously to combine +their strengths. Many works have combined audio and visual + +5 +TABLE I +TAXONOMY OF TSE WORKS: APPROACHES WITHIN SCOPE OF THIS OVERVIEW PAPER ARE EMPHASIZED IN RED. +Representative approaches +References +Year +Type of clues +Number of mic. +Speaker-close/open +Audio +Visual +Spatial +Single +Multi +Close +Open +Fixed beamforming +[2], [16]11 +1985 +- +- +✓ +- +✓ +- +✓ +Generative +Audio-visual F-HMM +[4] +2001 +✓12 +✓ +- +✓ +- +✓ +- +ICA with visual voice activity +[5] +2007 +- +✓ +- +- +✓ +- +✓ +Multi-channel NMF +[19] +2011 +✓12 +- +- +- +✓ +✓ +- +IVE with x-vectors +[12] +2020 +✓ +- +- +- +✓ +- +✓ +Audio-visual VAE +[29] +2020 +- +✓ +- +✓ +- +- +✓ +Discriminative +Speaker-specific network +[22] +2014 +✓12 +- +- +✓ +- +✓ +- +Multi-channel SpeakerBeam +[10], [30] +2017 +✓ +- +- +- +✓ +- +✓ +SpeakerBeam +[10] +2019 +✓ +- +- +✓ +- +- +✓ +VoiceFilter +[11] +2019 +✓ +- +- +✓ +- +- +✓ +SpEx +[31] +2020 +✓ +- +- +✓ +- +- +✓ +The conversation +[7] +2018 +- +✓ +- +✓ +- +- +✓ +Looking-to-listen +[8] +2018 +- +✓ +- +✓ +- +- +✓ +On/off-screen audio-visual separation +[9] +2018 +- +✓ +- +✓ +- +- +✓ +Landmark-based AV speech enh. +[32] +2019 +- +✓ +- +✓ +- +- +✓ +Multi-modal SpeakerBeam +[33], [34] +2019 +✓ +✓ +- +✓ +- +- +✓ +AV speech enh. through obstructions +[35] +2019 +✓ +✓ +- +✓ +- +- +✓ +Neural spatial filter +[3] +2019 +✓ +- +✓ +- +✓ +- +✓ +Spatial speaker extractor +[24] +2019 +✓ +- +✓ +- +✓ +- +✓ +Multi-channel multi-modal TSE +[36] +2020 +✓ +✓ +✓ +- +✓ +- +✓ +clues [4], [33], and some have even added spatial clues [36]. +B. Number of microphones +Another way to categorize the TSE approaches is based +on the number of microphones (channels) they use. Mul- +tiple channels allow the spatial diversity of the sources to +be exploited to help discriminate the target speaker from +interference. Such an approach also closely follows human +audition, where binaural signals are crucial for solving the +cocktail-party problem. +All approaches with spatial clues require using a micro- +phone array to capture the direction information of the sources +in the mixture [2], [3], [16], [24], [36]. Some TSE approaches +that exploit audio or visual clues also assume multi-channel +recordings, such as the extensions of ICA/IVA approaches [5], +[12]. +Multi-channel approaches generally generate extracted sig- +nals with better quality and are thus preferable when record- +ings from a microphone array are available. However, some- +times they might fail when the sources are located in the +same direction from the viewpoint of the recording device. +Moreover, adopting a microphone array is not always an option +when developing applications due to cost restrictions. In such +cases, single-channel approaches are requested. They rely on +spectral models of speech mixture using either F-HMM or +recently NNs and exploit audio [10], [11] or visual clues [7], +[8] to identify the target speech. +Recent single-channel neural TSE systems have achieved re- +markable performance. Interestingly, such approaches can also +be easily extended to multi-channel processing by augmenting +the input with spatial features [3] or combining the processing +11Since the first works that proposed beamforming were not model-based, +we consider them neither generative nor discriminative. +12In speaker-close cases, the models are trained on target speaker’s audio. +We consider this an audio clue in Table I. +with beamforming [24], [30], as discussed in Section IV-C. +For example, using a beamformer usually extracts a higher +quality signal due to employing a spatial linear filter to perform +extraction, which can benefit ASR applications [10]. +C. Speaker-open vs speaker-close methods +We usually understand the clues used by TSE as short +evidence about the target speaker obtained at the time of +executing the method, e.g., one utterance spoken by the target +speaker, a video of him/her speaking, or their current location. +There are, however, also methods that use a more significant +amount of data from the target speaker (e.g., several hours of +their speech) to build a model specific to that person. These +methods can also be seen as TSE except that the clues involve +much more data. +We refer to these two categories as the speaker-open and +speaker-close methods13. In speaker-open methods, the data +of the target speaker are available only during the test time, +i.e., the model is trained on the data of different speakers. +In contrast, the target speaker is part of the training data +in speaker-close methods. Many methods in the past were +speaker-close, e.g., [4] or [19], where the models were trained +on the clean utterances of the target speaker. Also, the first +neural models for TSE used a speaker-specific network [22]. +Most recent works on neural methods, which use a clue as +an additional input, are speaker-open methods [3], [7], [8], +[10], [11]. Recent IVE methods [12] are also speaker-open, +i.e., they guide the inference of IVE using the embedding of +a previously unseen speaker. +13Speaker-open and speaker-close categories are sometimes referred to +as speaker-independent and speaker-dependent, respectively. We avoid this +terminology, as in TSE, all systems are informed about the target speaker, +and therefore the term speaker-independent might be misleading. + +6 +D. Generative vs discriminative +We can classify TSE into approaches using generative or +discriminative models. +Generative approaches model the joint distribution of the +observations, target signals, and clues. The estimated target +speech is obtained by maximizing the likelihood. In contrast, +discriminative approaches directly estimate the target speech +signal given observations and clues. +In the TSE literature, generative models were the dominant +choice in the pioneering works, including one [4] that used +HMMs to jointly model audio and visual modalities. IVE [12] +is also based on a generative model of the mixtures. +The popularity of discriminative models, in particular NNs, +has increased since mid-2010’s, and such models today are +the choice for many problems, including TSE. With dis- +criminative models, TSE is treated as a supervised problem, +where the parameters of a TSE model are learned using +artificially generated training data. The modeling power of +NNs enables us to exploit large amounts of such data to +build strong speech models. Moreover, the versatility of NNs +enables complex dependencies to be learned between different +types of observations (e.g., speech mixture and video/speaker +embeddings), which allows the successful conditioning of the +extraction process on various clues. However, NNs also bring +new challenges, such as generalization to unseen conditions +or high computational requirements [38]. +Some recent works have also explored using generative +NNs, such as variational autoencoders (VAEs) [29], which +might represent a middle-ground between the traditional gen- +erative approaches and those using discriminative NNs. +E. Scope of overview paper +In the remainder of our paper, we focus on the neu- +ral methods for TSE emphasized in Table I. Recent neural +TSE approaches opened the possibility of achieving high- +performance extraction with various clues. They can be op- +erated with a single microphone and applied for speaker- +open conditions, which are very challenging constraints for +other schemes. Consequently, these approaches have received +increased attention from both academia and industry. +In the next section, we introduce a general framework to +provide a uniformized view of the various NN-based TSE +approaches, for both single- and multi-channel approaches, +and independently of the type of clues. We then respectively +review the approaches relying on audio, visual, and spatial +clues in Sections V, VI, and VII. +IV. GENERAL FRAMEWORK FOR NEURAL TSE +In the previous section, we introduced a taxonomy that de- +scribed the diversity of approaches to tackle the TSE problem. +However, recent neural TSE systems have much in common. +In this section, we introduce a general framework that provides +a unified view of a neural TSE system, which shares the same +processing flow independently of the type of clue used. By +organizing the existing approaches into a common framework, +we hope to illuminate their similarities and differences and +establish a firm foundation for future research. +A neural TSE system consists of an NN that estimates the +target speech conditioned on a clue. Fig. 3 is a schematic +diagram of a generic neural TSE system that consists of two +main modules: a clue encoder and a speech extraction module, +described in more detail below. +A. Clue encoder +The clue encoder pulls out (from the clue, Cs) information +that allows the speech extraction module to identify and extract +the target speech in the mixture. We can express the processing +as +Es = ClueEncoder(Cs; θClue), +(5) +where ClueEncoder(·; θClue) represents the clue encoder, +which can be an NN with learnable parameters θClue, and Es +are the clue embeddings. Naturally, the specific implementa- +tion of the clue encoder and the information carried within +Es largely depend on the type of clues. For example, when +the clue is an enrollment utterance, Es = E(a) +s +∈ RDEmb +will be a speaker embedding vector of dimension DEmb that +represents the voice characteristics of the target speaker. When +dealing with visual clues, Es = E(v) +s +∈ RDEmb×N can be a +sequence of the embeddings of length N, representing, e.g., +the lip movements of the target speaker. Here N represents +the number of time frames of the mixture signal. +Interestingly, the implementation of the speech extraction +module does not depend on the type of clues used. To provide +a description that is independent of the type of clues, hereafter, +we consider that Es ∈ RDEmb×N consists of a sequence of +embedding vectors of dimension DEmb of length N. Note that +we can generate a sequence of embedding vectors for audio +clue-based TSE systems by repeating the speaker embedding +vector for each time frame. +B. Speech extraction module +The speech extraction module estimates the target speech +from the mixture, given the target speaker embeddings. We +can use the same configuration independently of the type of +clue. Its process can be decomposed into three main parts: a +mixture encoder, a fusion layer, and a target extractor: +Zy = MixEncoder(y; θMix), +(6) +Zs = Fusion(Zy, Es; θFusion), +(7) +ˆxs = TgtExtractor(Zs, y; θTgtExtractor), +(8) +where +MixEncoder(·; θMix), +Fusion(·; θFusion), +and +TgtExtractor(·; θTgtExtractor) +respectively +represent +the +mixture encoder, the fusion layer, and the target extractor +with parameters θMix, θFusion, and θTgtExtractor. Zy ∈ RDy×N +and Zs ∈ RDs×N are the internal representations of the +mixture before and after conditioning on embedding Es. +The mixture encoder performs the following: +Y = FE(y; θFE), +(9) +Zy = MixNet(Y; θMixNet), +(10) +where FE(·) and MixNet(·) respectively represent the feature +extraction process and an NN with parameters θFE and θMixNet. + +7 +Mixture +encoder +Clue +encoder +������������ +������������������������ +Mixture +Clue +Target +extractor +Fusion +layer +������������������������� +Target speech +������������������������ +������������������������ +������������������������ +Feature +extraction +NN +(MixNet) +Extraction +process +(Mask/Beamformer) +NN +(MaskNet) +Signal +reconstruction +Speech extraction module +Mixture encoder +Target extractor +Fig. 3. General framework for neural TSE +TABLE II +TYPE OF FUSION LAYERS: L, L1, AND L2 ARE LINEAR TRANSFORMATIONS FOR MAPPING THE DIMENSION OF THE CLUE EMBEDDINGS, DEMB, TO THE +DIMENSION OF Zy, DZ. ⊙ REPRESENTS THE ELEMENT-WISE HADAMARD MULTIPLICATION OPERATION OF MATRICES. ei IS A VECTOR CONTAINING +THE ELEMENTS OF THE i-TH ROW OF Es AND diag(·) IS AN OPERATOR THAT CONVERTS A VECTOR INTO A DIAGONAL MATRIX. +Fusion type +Equation +Parameters (θFusion) +Concatenation +Zs = [Zy, Es] +- +Addition +Zs = Zy + LEs +L ∈ RDZ×DEmb +Multiplication +Zs = Zy ⊙ (LEs) +L ∈ RDZ×DEmb +Feature-wise Linear Modulation (FiLM) +Zs = Zy ⊙ (L1Es) + L2Es, +L1 ∈ RDZ×DEmb, L2 ∈ RDZ×DEmb +Factorized layer +Zs = �DEmb +i=1 LiZy diag(ei), +Li ∈ RDZ×DZ +The feature extractor computes the features from the observed +mixture signal, Y ∈ RD×N. These can be such spectral +features as magnitude spectrum coefficients derived from the +short-time Fourier transform (STFT) of the input mixture [7], +[8], [10], [11]. When using a microphone array, spatial features +like interaural phase difference (IPD) defined in Eq. (21) in +Section VII can also be appended. Alternatively, the feature +extraction process can be implemented by an NN such as a +1-D convolutional layer that operates directly on the raw input +waveform of the microphone signal [23], [39]. This enables +learning of a feature representation optimized for TSE tasks. +The features are then processed with an NN, MixNet(·), +which performs a non-linear transformation and captures the +time context, i.e., several past and future frames of the signal. +The resulting representation, Zy, of the mixture is (at this +point) agnostic of the target. +The fusion layer, sometimes denoted as an adaptation layer, +is a key component of a TSE system and allows conditioning +of the process on the clue. It combines Zy with the clue +embeddings, Es. Conditioning an NN on auxiliary information +is a general problem that has been studied for multi-modal +processing or the speaker adaptation of ASR systems. TSE +systems have borrowed fusion layers from these fields. Table +II lists several options for the fusion layer. Some widely used +fusion layers include: (1) the concatenation of Zy with the +clue embeddings Es [7], [8]; (2) addition14 after transforming +the embeddings with linear transformation L to match the +dimension of Zy; (3) multiplication [10]; (4) a combination of +14Concatenation is similar to addition if a linear transformation follows it. +addition and multiplication denoted as FiLM; (5) a factorized +layer [10], [30], i.e., the combination of different transfor- +mations of the mixture representation weighted by the clue +embedding values. Other alternatives have also been proposed, +including attention-based fusion [40]. Note that the fusion +operations described here assume just one clue. It is also +possible to use multiple clues, as discussed in Section VI-B. +Some works also employ the fusion repeatedly at multiple +positions in the model [31]. +The last part of the speech extraction module is the target +extractor, which estimates the target signal. We explain below +the time-frequency masking-based extractor, which has been +widely used [3], [7], [8], [41]. Recent approaches also perform +a similar masking operation in the learned feature domain [23], +[39]. +The time-frequency masking approach was inspired by early +BSS studies that relied on the sparseness assumption of speech +signals, an idea based on the observation that the energy +of a speech signal is concentrated in a few time-frequency +bins of a speech spectrum. Accordingly, the speech signals of +different speakers rarely overlap in the time-frequency domain +in a speech mixture. We can thus extract the target speech +by applying a time-frequency mask on the observed speech +mixture, where the mask indicates the time-frequency bins +where the target speech is dominant over other signals. Fig. 4 +shows an example of an ideal binary mask for extracting a +target speech in a mixture of two speakers. Such an ideal +binary mask assumes that all the energy in each TF bin belongs +to one speaker. In recent mask-based approaches that use real- + +8 +Mixture +TF-mask +Extracted signal +Time +Frequency +Time +Frequency +Time +Frequency +Fig. 4. Example of time-frequency mask for speech extraction: Time-frequency mask shows spectrogram regions where target source is dominant. By applying +this mask to the mixture, we obtain an extracted speech signal that estimates the target speech. +valued (or complex) masks, this assumption or observation is +not needed. +The processing of the masking-based extractor can be +summarized as +Ms = MaskNet(Zs; θMask), +(11) +ˆXs = Ms ⊙ Y, +(12) +ˆxs = Reconstruct( ˆXs; θReconst), +(13) +where MaskNet(·) is an NN that estimates the time-frequency +mask for the target speech, Ms ∈ RD×N, θMask are the net- +work parameters, and ⊙ denotes the element-wise Hadamard +multiplication. Y and ˆXs are the mixture and the estimated +target speech signals in the feature domain. Eq. (12) shows the +actual extraction process. Reconstruct(·) is an operation to +reconstruct the time-domain signal by performing the inverse +operation of the feature extraction of the mixture encoder, +i.e., either inverse STFT (iSTFT) or a transpose convolution +if using a learnable feature extraction. In the latter case, the +reconstruction layer has learnable parameters, θReconst. +There are other possibilities to perform the extraction pro- +cess. For example, we can modify the MaskNet(·) NN to +directly infer the target speech signal in the feature domain. +Alternatively, as discussed in Section IV-C, we can replace +the mask-based extraction process with beamforming when a +microphone array is available. +C. Integration with microphone array processing +If we have access to a microphone array to record the speech +mixture, we can exploit the spatial information to extract the +target speech. One approach is to use spatial clues to identify +the speaker in the mixture by informing the system about the +target speaker’s direction, as discussed in Section VII. Another +approach combines TSE with beamforming and uses the latter +to perform the extraction process instead of Eq. (12). For +example, we can use the output of a TSE system to estimate +the spatial statistics needed to compute the coefficients of a +beamformer steering in the direction of the target speaker. This +approach can also be used with audio or visual clue-based TSE +systems and requires no explicit use of spatial clues to identify +the target speaker in the mixture. +We briefly review the mask-based beamforming approach, +which was introduced initially for noise reduction and BSS +[42], [43]. A beamformer performs the linear spatial filtering +of the observed microphone signals: +ˆXs[n, f] =WH[f]Y[n, f], +(14) +where +ˆXs[n, f] ∈ C is the STFT coefficient of the esti- +mated target signal at time frame n and frequency bin f, +W[f] ∈ CM is a vector of the beamformer coefficients, +Y[n, f] = +� +Y 1[n, f], . . . , Y M[n, f] +� T ∈ CM is a vector of +the STFT coefficients of the microphone signals, M is the +number of microphones, and H is the conjugate transpose. +We can derive the beamformer coefficients from the spatial +correlation matrices of the target speech and the interference. +These correlation matrices can be computed from the observed +signal and the time-frequency mask estimated by the TSE +system [30]. +This way of combining a TSE system with beamforming +replaces the time-frequency masking operation of Eq. (12) +with the spatial linear filtering operation of Eq. (14). It allows +distortionless extraction, which is often advantageous when +using TSE as a front-end for ASR [10]. +D. Training a TSE system +Before using a TSE model, we first need to learn its param- +eters: θTSE = {θMix, θClue, θFusion, θTgtExtractor}. Most existing +studies use fully supervised training, which requires a large +amount of training data consisting of the triplets of speech +mixture y, target speech signal xs, and corresponding clue Cs +to learn parameters θTSE. Since this requires access to a clean +target speech signal, such training data are usually simulated +by artificially mixing clean speech signals and noise following +the signal model of Eq. (1). +Figure 5 illustrates the data generation process using a +multi-speaker audio-visual speech corpus containing multiple +videos for each speaker. First, we generate a mixture using +randomly selected speech signals from the target speaker, the +interference speaker, and the background noise. We obtain +an audio clue by selecting another speech signal from the +target speaker as well as a visual clue from the video signal +associated with the target speech. +The training of a neural TSE framework follows the training +scheme of NNs with error back-propagation. The parameters +are estimated by minimizing a training loss function: +θTSE = arg min +θ +L (xs, ˆxs) , +(15) +where L(·) is a training loss, which measures how close +estimated target speech ˆxs = TSE (y, Cs; θ) is to the target +source signal xs. We can use a similar loss as that employed +for training noise reduction or BSS systems [14], [39]. + +9 +Audio clue +(Enrollment) +Visual clue +(Video) +Speech/Video +(target speaker) +Speech +(interfering speaker) +- Audio clue: Speech sample from the target speaker different from the target speech +- Visual clue: Video signal associated with the target speech +Noise +samples +Mixture +Target speech +Interfering speech +Noise +(1) Sample source signals +(2) Get the clues +(1) Sample target speech, interfering speech and noise signal from the dataset +(2) Get the clues as: +- +Audio clue: Sample from the dataset an utterance from the target speaker different +from the target speech +Fig. 5. +Example of generating simulation data for training or testing: This example assumes videos are available so that audio and visual clues can be +generated. No video is needed for audio clue-based TSE. For visual clue-based TSE, we do not necessarily need multiple videos from the same speaker. +Several variants of the losses operating on different domains +exist, such as the cross-entropy between the oracle and the +estimated time-frequency masks and the mean squared error +(MSE) loss between the magnitude spectra of the source and +the estimated target speech. Recently, a negative signal-to- +noise ratio (SNR) measured in the time-domain has been +widely used [6], [23], [39]: +LSNR(xs, ˆxs) = −10 log10 +� +∥xs∥2 +∥xs − ˆxs∥2 +� +. +(16) +The SNR loss is computed directly in the time-domain, which +forces the TSE system to learn to correctly estimate the +magnitude and the phase of the target speech signal. This loss +has improved extraction performance [23]. Many works also +employ versions of the loss which are invariant to arbitrary +scaling, i.e., scale-invariant SNR (SI-SNR) [39] or linear +filtering of the estimated signal, often calledsignal-to-distortion +ratio (SDR) +[44]. Besides training losses operating on the +signal or mask levels, it is also possible to train a TSE system +end-to-end with a loss defined on the output of an ASR system +[45]. Such a loss can be particularly effective when targeting +ASR applications, as discussed in Section VIII. +The clue encoder can be an NN trained jointly with a speech +extraction module [10] or pre-trained on a different task, such +as speaker identification for audio clue-based TSE [11] or lip- +reading for visual clue-based TSE [7]. Using a pre-trained +clue encoder enables the leveraging of large amounts of data +to learn robust and highly discriminative embeddings. On the +other hand, jointly optimizing the clue encoder allows learning +embeddings to be optimized directly for TSE. These two trends +can also be combined by fine-tuning the pre-trained encoder +or using multi-task training schemes, which add a loss to the +output of the clue embeddings [46]. +E. Considerations when designing a TSE system +We conclude this section with some considerations about the +different options for designing a TSE system. In the above +description, we intentionally ignored the details of the NN +architecture used in the speech extraction module, such as the +type of layers. Indeed, novel architectures have been and will +probably continue to be proposed regularly, leading to gradual +performance improvement. For concrete examples, we refer to +some public implementations of TSE frameworks presented in +Section X. +Most TSE approaches borrow a network configuration from +architectures proven effective for BSS or noise reduction. One +important aspect is that an NN must be able to see enough +context in the mixture to identify the target speaker. This has +been achieved using such recurrent neural network (RNN)- +based architectures as a stack of bidirectional long short-term +memory (BLSTM) layers [10], convolutional neural network +(CNN)-based architectures with a stack of convolutional layers +that gradually increases the receptive field over the time axis to +cover a large context [7], [23] or attention-based architectures +[47]. +The networks in the mixture encoder and the extraction pro- +cess generally use a similar architecture. The best performance +was reported when using a shallow mixture encoder (typically +a single layer/block) and a much deeper extraction network, +i.e., where a fusion layer is placed on the lower part of the +extraction module. Furthermore, we found in our experiments +that the multiplication or FiLM layers usually perform well. + +10 +Fig. 6. Illustration of i-vector, NN-based vector, and jointly-trained embeddings: Orange parts are included only in training stage. +However, the impact of the choice of the fusion layer seems +rather insignificant. +For the feature extraction, early studies used spectral fea- +tures computed with STFT [7], [8], [10]. However, most +recent approaches employ a learned feature extraction module +following its success for separation [23], [39]. This approach +allows direct optimization of the features for the given task. +However, the choice of input features may depend on the +acoustic conditions, and some have reported superior perfor- +mance using STFT under challenging reverberant conditions +[48] or using handcrafted filterbanks [49]. +Except for such general considerations, it is difficult to make +solid arguments for a specific network configuration since +performance may depend on many factors, such as the task, +the type of clue, the training data generation, and the network +and training hyper-parameters. +V. AUDIO-BASED TSE +In this section, we explain how the general framework +introduced in Section IV can be applied in the case of audio +clues. In particular, we discuss different options to implement +the clue encoder, summarize the development of the audio- +based TSE, and present some representative experimental +results. +A. Audio clue encoder +An audio clue is an utterance spoken by the target speaker +from which we derive the characteristics of their voice, al- +lowing identification in a mixture. This enrollment utterance +can be obtained by pre-recording the user of a personal device +or with a part of a recording where a wake-up keyword was +uttered. The clue encoder is usually used to extract a single +vector that summarizes the entire enrollment utterance. +Since the clue encoder’s goal is to extract information +that defines the voice characteristics of the target speaker, +embeddings from the speaker verification field are often used, +such as i-vectors or NN-based embeddings (e.g., d-vectors or +x-vectors). Clue encoders trained directly for TSE tasks are +also used. Fig. 6 describes these three options. +1) I-vectors: From their introduction around 2010, i-vectors +[50] were the ruling speaker verification paradigm until the rise +of NN speaker embeddings. The main idea behind i-vectors +is modeling the features of an utterance using a Gaussian +mixture model (GMM), whose means are constrained to a +subspace and depend on the speaker and the channel effects. +The subspace is defined by the Universal Background model +(UBM), i.e., GMM trained on a large amount of data from +many speakers, and a total variability subspace matrix. The +super-vector of the means of utterance GMM µ is decom- +posed: +µ = m + Tw, +(17) +where m is a super-vector of the means of the UBM, T is a +low-rank rectangular matrix representing the bases spanning +the subspace, and w is a random variable with standard +normal prior distribution. Since an i-vector is the maximum a +posteriori estimate of w, it thus consists of values that enable +the adaptation of the parameters of the generic UBM speaker +model (m) to a specific recording. As a result, it captures the +speaker’s voice characteristics in the recording. +An important characteristic of i-vectors is that they capture +both the speaker and channel variability. This case may be +desired in some TSE applications, where we obtain enrollment +utterances in identical conditions as the mixed speech. In such +a situation, the channel information might also help distinguish +the speakers. I-vectors have also been used in several TSE +works [10]. +2) Neural network-based embeddings: The state-of-the- +art speaker verification systems predominantly use NN-based +speaker embeddings, which were adopted later for TSE. The +common idea is to train an NN for the task of speaker +classification. Such an NN contains a “pooling layer” which +converts a sequence of features into one vector. The pooling +layer computes the mean and optionally the standard deviation +of the sequence of features over the time dimension. The +pooled vector is then classified into speaker classes or used + +oro11 +in other loss functions that encourage speaker discrimination. +For TSE, the speaker embedding is then the vector of the +activation coefficients of one of the last network layers. The +most common of such NN-based speaker embeddings are d- +vectors and x-vectors [51]. Many TSE works employ d-vectors +[11]. +Since NNs are trained for speaker classification or a related +task, embeddings are usually highly speaker-discriminative. +Most other sources of variability are discarded, such as the +channel or content. Another advantage of this class of em- +beddings is that they are usually trained on large corpora +with many speakers, noises, and other variations, resulting in +very robust embedding extractors. Trained models are often +publicly available, and the embeddings can be readily used +for TSE tasks. +3) Jointly-learned embeddings: +NN-based embeddings, +such as x-vectors, are designed and trained for the task of +speaker classification. Although this causes them to contain +speaker information, it is questionable whether the same +representation is optimal for TSE tasks. An alternative is to +train the neural embedding extractor jointly with a speech +extraction module. The resulting embeddings are thus directly +optimized for TSE tasks. This approach has been used for TSE +in several works [10], [31]. +The NN performing the speaker embedding extraction takes +an enrollment utterance C(a) +s +as input and generally contains +a pooling layer converting the frame-level features into one +vector, similar to the embedding extractors discussed above. +This NN is trained with the main NN using a common ob- +jective function. A second objective function can also be used +on the embeddings to improve their speaker discriminability +[46]. +As mentioned above, the advantage of such embeddings is +that they are trained directly for TSE and thus collect essential +information for this task. On the other hand, the pre-trained +embedding extractors are often trained on larger corpora and +may be more robust. A possible middle ground might take a +pre-trained embedding extractor and fine-tune it jointly with +the TSE task. However, this has, to the best of our knowledge, +not been done yet. +B. Existing approaches +The first neural TSE methods were developed around 2017. +One of the first published works, SpeakerBeam [10], ex- +plored both the single-channel approach, where the target +extractor was implemented by time-frequency masking, and +the multi-channel approach using beamforming. This work +also compared different variants of fusion layers and clue +encoders. This was followed by VoiceFilter [11], which put +more emphasis on ASR applications using TSE as a front-end +and also investigated streaming variants with minimal latency. +A slightly modified variant of the task was presented in works +on speaker inventory [40], where not one but multiple speakers +can be enrolled. Such a setting might be suitable for meeting +scenarios. Recently, many works, such as SpEx [31], have +started to use time-domain approaches, following their success +in BSS [39]. +7 +9 +11 +13 +15 +17 +WSJ0-2mix +WHAM +WHAMr +SI-SNR improvement [dB] +BSS (oracle) +BSS (x-vector) +TSE +Fig. 7. Comparison of TSE and cascade BSS systems when using an audio +clue in terms of SI-SNR improvement (higher is better) [52]. +C. Experiments +An audio clue is a simple way to condition the system for +extracting the target speaker. Many works have shown that +the speaker information extracted from audio clues is suffi- +cient for satisfactory performance. Demonstrations of many +works are available online15. We present here some results +to demonstrate the potential of audio clue-based approaches. +The experiments were done with time-domain SpeakerBeam16, +which uses a convolutional architecture, a multiplicative fusion +layer, and a jointly-learned clue encoder. +The experiments were done on three different datasets +(WSJ0-2mix, WHAM!, and WHAMR!) to show the perfor- +mance in different conditions (clean, noisy, and reverberant, +respectively). We describe these datasets in more detail in +Section X. All the experiments were evaluated with the SI- +SNR metric and measured the improvements over the SI-SNR +of the observed mixture. More details about the experiments +can be found in [52]. +Figure 7 compares the TSE results with a cascade system, +first doing BSS and then independent speaker identification. +Speaker identification is done either in an oracle way (selecting +the output closest to the reference) or with x-vectors (ex- +tracting the x-vectors from all the outputs and the enrollment +utterances and selecting the output with the smallest cosine +distance as the target). The BSS system uses the same con- +volutional architecture as TSE, differing only in that it does +not have a clue encoder and the output layer is twice larger +as it outputs two separated speech signals. The direct TSE +scheme outperformed the cascade system, especially in more +difficult conditions such as WHAMR!. This difference reflects +a couple of causes: 1) the TSE model is directly optimized for +the TSE task and does not spend any capacity on extracting +other speakers or 2) the TSE model has additional speaker +information. +Figure 8 shows an example of spectrograms obtained using +TSE on a recording of two speakers from the WHAMR! +database, including noise and reverberation. TSE correctly +15Demonstrations of audio clues approaches: VoiceFilter [11] https://google. +github.io/speaker-id/publications/VoiceFilter/, SpeakerBeam [10] https://www. +youtube.com/watch?v=7FSHgKip6vI. +16https://github.com/butspeechfit/speakerbeam + +12 +2 +4 +6 +8 +10 +0 +2000 +4000 +Frequency [Hz] +Mixture +2 +4 +6 +8 +10 +0 +2000 +4000 +Frequency [Hz] +Reference +2 +4 +6 +8 +10 +Time [s] +0 +2000 +4000 +Frequency [Hz] +Extracted (SI-SNR 11.56 dB) +100 +50 +100 +50 +100 +50 +Fig. 8. Example of spectrograms of mixed, reference, and extracted speech: Example is taken from WHAMR! database. +identifies the target speaker and removes all the interference, +including the second speaker, noise, and reverberation. +D. Limitations and outlook +Using TSE systems conditioned on audio clues is particu- +larly practical due to the simplicity of obtaining the clues, i.e., +no additional hardware is needed, such as cameras or multi- +ple microphones. Considering the good performance demon- +strated in the literature, these systems are widely applicable. +Nowadays, the methods are rapidly evolving and achieving +increasingly higher accuracy. +The main challenge in audio-clue-based systems is correct +identification of the target speaker. The speech signal of the +same speaker might have highly different characteristics in +different conditions due to such factors as emotional state, +channel effects, or the Lombard effect. TSE systems must +be robust enough to such intra-speaker variability. On the +other hand, different speakers might have very similar voices, +leading to erroneous identification if the TSE system lacks +sufficient accuracy. +Resolving both issues requires precise speaker modeling. +In this regard, the TSE methods may draw inspiration from +the latest advances in the speaker verification field, including +advanced model architectures, realistic datasets with a huge +number of speakers for training, or using pre-trained features +from self-supervised models. +VI. VISUAL/MULTI-MODAL CLUE-BASED TSE +Visual clue-based TSE assumes that a video camera captures +the face of the target speaker who is talking in the mixture [7], +[8]. Using visual clues is motivated by psycho-acoustic studies +(see the references in a previous work [6]) that revealed that +humans look at lip movements to understand speech better. +Similarly, the visual clues of TSE systems derive hints about +the state of the target speech from the lip movements, such +as whether the target speaker is speaking or silent or more +refined information about the phoneme being uttered. +A visual clue, which presents different characteristics than +audio clues because it captures information from another +modality, is time-synchronized with the target speech in the +mixture without being corrupted by the interference speakers. +Therefore, a visual clue-based TSE can better handle mixtures +of speakers with similar voices, such as same-gender mixtures, +than audio clue-based systems because the extraction process +is not based on the speaker’s voice characteristics17. Another +potential advantage is that the users may not need to pre-enroll +their voice. Video signals are also readily available for many +applications such as video-conferencing. +Figure 9 shows a diagram of a visual TSE system that +follows the same structure as the general TSE framework +introduced in Section IV. Only the visual clue encoder part +is specific to the task. We describe it in more detail below and +then introduce a multi-modal clue extension. We conclude this +section with some experimental results and discussions. +A. Visual clue encoder +The visual clue encoder computes from the video signal +a representation that allows the speech extraction module to +identify and extract the target speech in the mixture. This +processing involves the steps described below: +E(v) +s += Upsample(NN(VFE(C(v) +s ), θv-clue)), +(18) +where E(v) +s +∈ +RDEmb×N represents the sequence of the +visual embedding vectors, C(v) +s +is the video signal obtained +after pre-processing, VFE(·) is the visual feature extraction +module, NN(·, θv-clue) is an NN with parameters θv-clue, and +Upsample(·) represents the up-sampling operation. The latter +up-sampling step is required because the sampling rates of the +audio and video devices are usually different. Up-sampling +17Some works can even perform extraction from a mixture of the same +speaker’s speech [8]. + +13 +Pre- +processing +Feature +extraction +Up- +sampling +Speech extraction module +Visual clue encoder +NN +Fig. 9. Visual clue-based TSE system. +matches the number of frames of the mixture and visual clue +encoders. +1) Pre-processing: First, the video signal captured by the +camera requires pre-processing to isolate the face of the +target speaker. Depending on the application, this may require +detecting and tracking the target speaker’s face and cropping +the video. These pre-processing steps can be performed using +previously well-established video processing algorithms [6]. +2) Visual feature extraction: Similar to an audio-clue-based +TSE, the visual clue encoder can directly extract embeddings +from raw video data or visual features. With the first option, +the raw video is processed with a CNN whose parameters are +jointly-learned with the speech extraction module to enable +direct optimization of the features for the extraction task +without any loss of information. However, since the video +signals are high-dimensional data, achieving joint optimization +can be complex. This approach has been used successfully +with speaker-close conditions [53]. Extending it to speaker- +open conditions might require a considerable amount of data +or careful design of the training loss using, e.g., multi-task +training to help the visual encoder capture relevant informa- +tion. +Most visual TSE works use instead a visual feature extractor +pre-trained on another task to reduce the dimensionality of the +data. Such feature extractors can leverage a large amount of +image or video data (that do not need to be speech mixtures) +to learn representation robust to variations, such as resolution, +luminosity, or head orientation. The first option is to use +facial landmark points as features. Facial landmarks are the +key points on a face that indicate the mouth, eyes, or nose +positions and offer a very low-dimension representation of a +face, which is interpretable. Moreover, face landmarks can be +easily computed with efficient off-the-shelf algorithms [32]. +The other option is to use neural embeddings derived +from an image/video processing NN trained on a different +task, which was proposed in three concurrent works [7]–[9]. +Ephrat et al. [8] used visual embeddings obtained from an +intermediate layer of a face recognition system called FaceNet. +This face recognition system is trained so that embeddings +derived from photographs of the same person are close and +embeddings from different persons are far from each other. It +thus requires only a corpus of still images with person identity +labels for training the system. However, the embeddings do +not capture the lip movement dynamics and are not explicitly +related to the acoustic content. +Alternatively, Afouras et al. [7] proposed using embeddings +obtained from a network trained to perform lip-reading, i.e., +where a network is trained to estimate the phoneme or word +uttered from the video of the speaker’s lips. The resulting +embeddings are thus directly related to the acoustic content. +However, the training requires video with the associated +phoneme or word transcriptions, which are more demanding +and costly to obtain. +The third option introduced by Owens et al. [9] exploits +embeddings derived from an NN trained to predict whether +the audio and visual tracks of a video are synchronized. This +approach enables self-supervised training, where the training +data are simply created by randomly shifting the audio track +by a few seconds. The embeddings capture information on +the association between the lip motions and the timing of the +sounds in the audio. All three options [7]–[9] can successfully +perform a visual TSE. +3) Transformation and up-sampling: Except with joint- +training approaches, the visual features are (pre-)trained on +different tasks and thus do not provide a representation optimal +for TSE. Besides, since some of the visual features are ex- +tracted from the individual frames of a video, the dynamics of +lip movements are not captured. Therefore, the visual features +are further transformed with an NN, which is jointly trained +with the speech extraction module. The NN, which allows +learning a representation optimal for TSE, can be implemented +with long short-term memory (LSTM) or convolutional layers +across the time dimension to model the time series of the visual +features, enabling the lip movement dynamics to be captured. +Finally, the visual embeddings are up-sampled to match the +sampling rate of audio features Zy. +B. Audio-visual clue-based TSE +Audio and visual clue-based TSE systems have complemen- +tary properties. An audio clue-based TSE is not affected by +speaker movements and visual occlusions. In contrast, a visual +clue-based TSE is less affected by the voice characteristics of +the speakers in the mixture. By combining these approaches, +we can build TSE systems that exploit the strengths of both +clues for improving the robustness to various conditions [33], +[36]. +Figure 10 shows a diagram of an audio-visual TSE system, +which assumes access to the pre-recorded enrollment of the +target speaker to provide an audio clue and a video camera +for a visual clue. The system uses the audio and visual clue +encoders described in Sections V-A and VI-A and combines +these clues into an audio-visual embedding, which is given to +the speech extraction module. Audio-visual embeddings can be +simply the concatenation [35] or the summation of the audio +and visual embeddings, or obtained as a weighted sum [33], + +14 +Speech extraction module +Enrollment +Video +Mixture +Visual clue +encoder +Audio clue +encoder +Multi-modal +clue fusion +Fig. 10. Audio-visual clue-based TSE system +[34], where the weights can vary depending on the reliability +of each clue. The weighted sum approach can be implemented +with an attention layer widely used in machine learning, which +enables dynamic weighting of the contribution of each clue. +C. Experimental results and discussion +Several visual TSE systems have been proposed, which dif- +fer mostly by the type of visual features used and the network +configuration. These systems have demonstrated astonishing +results, which can be attested by the demonstrations available +online18. Here we briefly describe experiments using the audio, +visual, and audio-visual time-domain SpeakerBeam systems +[34], which use a similar configuration as the system in Section +V-C. The speech extraction module employs a stack of time- +convolutional blocks and a multiplicative fusion layer. The +audio clue encoder consists of the jointly-learned embeddings +described in Section V-A3. The visual clue encoder uses visual +features derived from face recognition like a previous work [8]. +The audio-visual system combines the visual and audio clues +with an attention layer [34]. +The experiments used mixtures of utterances from the +LRS3-TED corpus19, which consists of single speaker ut- +terances with associated videos. We analyzed the behavior +under various conditions by looking at results from same +and different gender mixtures and two examples of clue +corruptions (enrollment corrupted with white noise at SNR +of 0 dB and video with a mask on the speaker’s mouth). The +details of the experimental setup are available in [34]. +Figure 11 compares the extraction performance measured +in terms of SDR improvement for audio, visual, and audio- +visual TSE under various mixture and clue conditions. We +confirmed that a visual clue-based TSE is less sensitive to +the characteristics of the speakers in the mixture since the +performance gap between different- and same-gender mixtures +is smaller than with an audio clue-based TSE. When using a +single clue, performance can be degraded when this clue is +corrupted. However, the audio-visual system that exploits both +clues can achieve superior extraction performance and is more +robust to clue corruption. +18Demo +samples +for +several +approaches +are +available, +e.g., +for +[9]: https://andrewowens.com/multisensory, for [8]: https://looking-to-listen. +github.io, for [7]: https://www.robots.ox.ac.uk/∼vgg/demo/theconversation, +and for [34]: http://www.kecl.ntt.co.jp/icl/signal/member/demo/audio visual +speakerbeam.html +19https://www.robots.ox.ac.uk/∼vgg/data/lip reading/lrs3.html +D. Discussions and outlook +Visual clue-based TSE approaches offer an alternative to +audio-clue-based ones when a camera is available. The idea +of using visual clues for TSE is not new [4], [5], although +recent neural systems have achieved an impressive level of +performance. This is probably because NNs can effectively +model the relationship between the different modalities learned +from a large amount of training data. +Issues and research opportunities remain with the current +visual clue-based TSE systems. First, most approaches do not +consider the speaker tracking problem and assume that the +audio and video signals are synchronized. These aspects must +be considered when designing and evaluating future TSE sys- +tems. Second, video processing involves high computational +costs, and more research is needed to develop efficient online +systems. +VII. SPATIAL CLUE-BASED TSE +When using a microphone array to record a signal, spatial +information can be used to discriminate among sources. In +particular, access to multi-channel recordings opens the way +to extract target speakers based on their location, i.e., using +spatial clues (as indicated in Fig. 1). This section explains how +such spatial clues can be obtained and used in TSE systems. +While enhancing speakers from a given direction has a long +research history [2], we focus here on neural methods that +follow the scope of our overview paper. +Note that multi-channel signals can also be utilized in the +extraction process using beamforming. Such an extraction +process can be used in systems with any type of clue, only +requiring that the mixed speech be recorded with multiple +microphones. This beamforming process was reviewed in +Section IV-C. In this section, we focus specifically on the +processing of spatial clues. +A. Obtaining spatial clues +In some situations, the target speaker’s location is approxi- +mately known in advance. For example, for an in-car ASR, the +driver’s position is limited to a certain region in a car. In other +scenarios, we might have access to a multi-channel enrollment +utterance of the speaker recorded in the same position as the +final mixed speech. In such a case, audio source localization +methods can be applied. Conventionally, this can be done +by methods based on generalized cross-correlation or steered- +response power, but recently, deep learning methods have also +shown success in this task. An alternative is to skip the explicit +estimation of the location and directly extract features in which +the location is encoded when a multi-channel enrollment is +available. We will detail this approach further in the next +section. +Spatial clues can also be obtained from a video using +face detection and tracking systems. A previous work [36] +demonstrated this possibility with a 180-degree wide-angle +camera positioned parallel to a linear microphone array20. By +identifying the target speaker in the video, the azimuth with +20https://yongxuustc.github.io/grnnbf + +15 +12 +12.5 +13 +13.5 +14 +14.5 +15 +15.5 +16 +16.5 +Different gender +Same gender +Corrupted audio clue Corrupted visual clue +SDR improvement (dB) +Audio +Visual +Audio-visual +Clues: +Visual +clean +no mask +Audio +Effect of gender in mixtures +Effect of clue corruption +no mask +0dB +Visual +Audio +mask +clean +Visual +Audio +Visual +clean +no mask +Audio +Fig. 11. SDR Improvement of TSE with audio, visual, and audio-visual clues for mixtures of same/different gender and for corruptions of audio and visual +clues: Audio clues were corrupted by adding white noise at SNR of 0 dB to enrollment utterance. Video clues were corrupted by masking mouth region in +video. +Fig. 12. Illustration of usage of spatial clue encoder and directional features +respect to the microphone array was roughly approximated. +Depth cameras can also be used to estimate not only the +azimuth but also the elevation and distance of the speaker. +B. Spatial clue encoder +The left part of Fig. 12 shows the overall structure and the +usage of a spatial clue encoder, which usually consists of two +parts: the extraction of directional features and an NN post- +processing of them. Two possible forms of spatial clues are +dominant in the literature: the angle of the target speaker with +respect to the microphone array or a multi-channel enrollment +utterance recorded in the target location. Both can be encoded +into directional features. +When the spatial clue is DOA, the most commonly used +directional features are the angle features, which are computed +as the cosine of the difference between the IPD and the target +phase difference (TPD): +AF[n, f] = +� +m1,m2∈M +cos +� +TPD (m1, m2, φs, f) +− IPD (m1, m2, n, f) +� +(19) +TPD(m1, m2, φs, f) = 2πfFs +F +cos φs ∆m1,m2 +c +(20) +IPD(m1, m2, n, f) = ∠Y m2[n, f] − ∠Y m1[n, f], +(21) +where M is a set of pairs of microphones used to compute +the feature, Fs is the sampling frequency, φs is the target +direction, c is the sound’s velocity, and ∆m1,m2 is the distance +from microphone m1 to microphone m2. An example of angle +features is shown on the right of Fig. 12. For time-frequency +bins dominated by the source from direction φs, the value of +the angle feature should be close to 1 or -1. Other directional +features have been proposed that exploit a grid of fixed +beamformers. A directional power ratio measures the ratio +between the power of the response of a beamformer steered +into the target direction and the power of the beamformer +responses steered into all the directions in the grid. In a +similar fashion, a directional signal-to-noise ratio can also be +computed, which compares the response of a beamformer in +the target direction with the response of a beamformer in the +direction with the strongest interference. +If the spatial clue consists of a multi-channel enrollment +utterance, the directional feature can be formed as a vector of +IPDs computed from the enrollment. Alternatively, the DOA +can be estimated from the enrollment, and the spatial features +derived from it can be used. +Note that when using a spatial clue to determine the target +speaker, the multi-channel input of the speech extraction +module must also be used. This enables the identification of +the speaker coming from the target location in the mixture. +Furthermore, a target extractor is often implemented as beam- +forming, as explained in Section IV-C. + +5 +-10-10 +15-5 +10 +1516 +0 +2 +4 +6 +8 +10 +12 +<15° +15-45° +45-90° +>90° +SI-SNR improvement +[dB] +Audio +Visual +Spatial +Combined +Fig. 13. SI-SNR improvement of TSE with audio, visual, and spatial clues +in four conditions based on angle separation between speakers [36] +C. Combination with other clues +Although a spatial clue is very informative and generally +can lead the TSE system to a correct extraction of the target, +it does fail in some instances. Estimation errors of DOA +are harmful to proper extraction. Furthermore, if the spatial +separation of the speakers with respect to the microphone array +is not significant enough, the spatial clue may not discriminate +between them. Combining a spatial clue with audio or visual +clues is an option to combat such failure cases. +D. Experimental results +We next report the results from an experiment with spatial +clues [36] that compared the effectiveness of using audio, +visual, and spatial clues. The audio-clue encoder was trained +jointly with the extraction module, and the visual encoder +was a pre-trained lip-reading network. The target speaker’s +direction was encoded in the angle feature. The spatial and +visual embeddings were fused with the extraction network by +concatenation and the audio embedding with a factorized layer. +The extraction module employed a neural network consisting +of temporal convolutional layers. +The experiments were performed on a Mandarin audio- +visual dataset containing mixtures of two and three speakers. +The results in Fig. 13 were divided into several conditions, +based on the angle separation between the closest speakers. +The spatial clue is very effective, although the performance +declines when speakers are near each other (< 15°). A combi- +nation with other modalities outperformed any individual type +of clue in all the conditions. +E. Discussion +Using spatial clues is a powerful way of conditioning a TSE +system to extract the target speaker. It relies on the availability +of signals from a microphone array and a way to determine the +location of the target speaker. Unfortunately, these restrictions +limit the applications to some extent. Neural TSE methods +with spatial clues follow a long history of research on the +topic, such as beamforming techniques, and extend them with +non-linear processing. This approach unifies the methods with +those using other clues and allows a straightforward combi- +nation of different clues into one system. Such combinations +can alleviate the shortcomings of spatial clues, including the +failures when the speakers are located in the same direction +from the point of view of the microphones. +In most current neural TSE works, the target speaker’s +location is assumed to be fixed. Although the methods should +be easily extended to a dynamic case, investigations of such +settings remain relatively rare [24]. +VIII. EXTENSION TO OTHER TASKS +The ideas of TSE can be applied to other speech processing +tasks, such as ASR and diarization. +A. Target-speaker ASR +An important application of TSE is TS-ASR, where the +goal is to transcribe the target speaker’s speech and ignore all +the interference speakers. The TSE approaches we described +can be naturally used as a front-end to an ASR system to +achieve TS-ASR. Such a cascade combination allows for +a modular system, which offers ease of development and +interpretability. However, the TSE system is often optimized +with a signal loss, as in Eq. (16). Such a TSE system inevitably +introduces artifacts caused by the remaining interferences, +over-suppression, and other non-linear processing distortions. +These artifacts limit the expected performance improvement +from a TSE front-end. +One approach to mitigate the effect of such artifacts is to +optimize the TSE front-end with an ASR criterion [10]. The +TSE front-end and the ASR back-end are NNs and can be +interconnected with differentiable operations, such as beam- +forming and feature extraction. Therefore, a cascade system +can be represented with a single computational graph, allowing +all parameters to be jointly trained. Such joint-training can +significantly improve the TS-ASR performance. +Another approach inserts a fusion layer into an ASR system +[26], [45] to directly perform clue conditioning. These inte- +grated TS-ASR systems avoid any explicit signal extraction +step, a decision that reduces the computational cost, although +such systems may be less interpretable than cascade systems. +TS-ASR can use the audio clues provided by pre-recorded +enrollment utterances [10], [26], [45] or from a keyword +(anchor) for a smart-device scenario [54], for example. Some +works have also exploited visual clues, which can be used for +the extraction process and to implement an audio-visual ASR +back-end, since lip-reading also improves ASR performance +[55]. +B. Target-speaker VAD and diarization +The problem of speech diarization consists of detecting who +spoke when in a multi-speaker recording. This technology is +essential for achieving, e.g., meeting recognition and analysis +systems that can transcribe a discussion between multiple +participants. Several works have explored using speaker clues +to perform this task [27], [28]. +For example, a personalized VAD [27] exploits a speaker +embedding vector derived from an enrollment utterance of the +target speaker to predict its activity, i.e., whether they are +speaking at a given time. In principle, this can be done with +a system like that presented in Section IV, where the output +layer performs the binary classification of the speaker activity + +17 +instead of estimating the target speech signal. Similar systems +have also been proposed using visual clues, called audio-visual +VAD [56]. Predicting the target speaker’s activity is arguably +a more straightforward task than estimating its speech signal. +Consequently, TS-VAD can use simpler network architectures, +leading to more lightweight processing. +The above TS-VAD systems, which estimate the speech +activity of a single target speaker, have been extended to simul- +taneously output the activity of multiple target speakers [28]. +The resulting system achieved the top diarization performance +in the CHiME 6 evaluation campaign21. +IX. REMAINING ISSUES AND OUTLOOK +Research toward computational selective hearing has been +a long endeavor. Recent developments in TSE have enabled +identifying and extracting a target speaker’s voice in a mixture +by exploiting audio, visual, or spatial clues, which is one +step closer to solving the cocktail-party problem. Progress in +speech processing (speech enhancement, speaker recognition) +and image processing (face recognition, lip-reading), com- +bined with deep learning technologies to learn models that can +effectively condition processing on auxiliary clues, triggered +the progress in the TSE field. Some of the works we presented +have achieved levels of performance that seemed out-of-reach +just a few years ago and are already being deployed in +products22. +Despite substantial achievements, many opportunities re- +main for further research, some of which we list below. +A. Deployment of TSE systems +Most of the systems we described operate offline and are +computationally expensive. They are also evaluated under +controlled (mostly simulated mixture) settings. Deploying such +systems introduces engineering and research challenges to re- +duce computational costs while maintaining high performance +under less controlled recording conditions. We next discuss +some of these aspects. +1) Inactive target speaker: Most TSE systems have been +evaluated assuming that the target speaker is actively speaking +in the mixture. In practice, we may not know beforehand +whether the target speaker will be active. We expect that a +TSE system can output no signal when the target speaker is +inactive, which may not actually be the case with most current +systems that are not explicitly trained to do so. The inactive +target speaker problem is specific to TSE. The type of clue +used may also greatly impact the difficulty of tackling this +problem. For instance, visual voice activity detection [5] might +alleviate this issue. However, it is more challenging with audio +clues [57], and further research may be required. +21The results of the CHiME 6 challenge can be found at: https:// +chimechallenge.github.io/chime6/results.html. The top system used TS-VAD +among other technologies. DiHARD 3 performed a diarization evaluation +on the CHiME 6 challenge data. Here the top system also used TS-VAD: +https://dihardchallenge.github.io/dihard3/results +22The following blog details the effort for deploying a visual clue- +based TSE system for on-device processing: https://ai.googleblog.com/2020/ +10/audiovisual-speech-enhancement-in.html. +2) Training and evaluation criteria: Most TSE systems are +trained and evaluated using such signal level metrics as SNR +or SDR. Although these metrics are indicative of the extraction +performance, their use presents two issues. +First, they may not always be correlated with human per- +ception and intelligibility or with ASR performance. This +issue is not specific to TSE; it is common to BSS and +noise reduction methods. For ASR we can train a system +end-to-end, as discussed in Section VIII-A. When targeting +applications for human listeners, the problem can be partly +addressed using other metrics for training or evaluation that +correlate better with human perception, such as short-time +objective intelligibility (STOI) or perceptual evaluation of +speech quality (PESQ) [6]. However, controlled listening tests +must be conducted to confirm the impact of a TSE on human +listeners [6]. +Second, unlike BSS and noise reduction, a TSE system +needs to identify the target speech, implying other sources +of errors. Indeed, failing to identify the target may lead to +incorrectly estimating an interference speaker or inaccurately +outputting the mixture. Although these errors directly impact +the SDR scores, it would be fruitful to agree on the evaluation +metrics that separate extraction and identification performance +to better reveal the behavior of TSE systems. Signal level +metrics might not satisfactorily represent the extraction perfor- +mance for inactive speaker cases. A better understanding of the +failures might help develop TSE systems that can recognize +when they cannot identify the target speech, which is appealing +for practical applications. +Consequently, developing better training and evaluation +criteria are critical research directions. +3) Robustness to recording conditions: Training neural TSE +systems requires simulated mixtures, as discussed in Section +IV-D. Applying these systems to real conditions (multi-speaker +mixtures recorded directly with a microphone) requires that +the training data match the application scenario relatively +well. For example, the type of noise and reverberation may +vary significantly depending on where a system is deployed. +This raises questions about the robustness of TSE systems to +various recording conditions. +Neural TSE systems trained with a large amount of simu- +lated data have been shown to generalize to real recording +conditions [8]. However, exploiting real recordings where +no reference target speech signal is available could further +improve performance. Real recordings might augment the +training data or be used to adapt a TSE system to a new +environment. The issue is defining unsupervised training losses +correlated with the extraction performance of the target speech +without requiring access to the reference target signal. +Another interesting research direction is combining neural +TSE systems, which are powerful under matched conditions, +with such generative-based approaches as IVE [12], which are +adaptive to recording conditions. +4) Lightweight and low-latency systems: +Research on +lightweight and low-latency TSE systems is gaining mo- +mentum as the use of teleconferencing systems in noisy +environments has risen in response to the Covid pandemic. +Other important use cases for TSE are hearing aids and + +18 +hearables, both of which impose very severe constraints in +terms of computation costs and latency. The recent DNS23 and +Clarity24 challenges that target teleconferencing and hearing +aid application scenarios include tracks where target speaker +clues (enrollment data) can be exploited. This demonstrates +the growing interest in practical solutions for TSE. +Since TSE is related to BSS and noise reduction, the +development of online and low-latency TSE systems can be +inspired from the progress of BSS/noise reduction in that +direction. However, TSE must also identify the target speech, +which may need specific solutions that exploit the long-context +of the mixture to reliably and efficiently capture a speaker’s +identity. +5) Spatial rendering: For applications of TSE to hearing +aids or hearables, sounds must be localized in space after +the TSE processing. Therefore, a TSE system must not only +extract the target speech but also estimate its direction to allow +rendering it so that a listener feels the correct direction of the +source. +B. Self-supervised and cross-modal learning +A TSE system identifies the target speech in a mixture +based on the intermediate representation of the mixture and +the clue. Naturally, TSE benefits from better intermediate +representations. For example, speech models learned with self- +supervised learning criteria have gained attention as a way +to obtain robust speech representations. They have shown +potential for pre-training many speech processing downstream +tasks, such as ASR, speaker identification, and BSS. Such +self-supervised models could also reveal advantages for TSE +since they could improve robustness by allowing efficient pre- +training on various acoustic conditions. Moreover, for audio- +based TSE, using the same self-supervised pre-trained model +for the audio clue encoder and the speech extraction module +will help to learn the common embedding space between the +enrollment and speech signals in the mixture. Similarly, the +progress in cross-modal learning, which aims to learn the joint +representation of data across modalities, could benefit such +multi-modal approaches as visual clue-based TSE. +C. Exploring other clues +We presented three types of clues that have been widely +used for TSE. However, other clues can also be considered. +For example, recent works have explored other types of spatial +clues such as the distance [58]. Moreover, humans do not only +rely on physical clues to perform selective hearing. We also +use more abstract clues, such as semantic ones. Indeed, we +can rapidly focus our attention on a speaker when we hear our +name or a topic we are interested in. Reproducing a similar +mechanism would require TSE systems that operate with +semantic clues, which introduces novel challenges concerning +how to represent semantic information and exploit it within a +TSE system. Some works have started to explore this direction, +23https://www.microsoft.com/en-us/research/academic-program/ +deep-noise-suppression-challenge-icassp-2022/ +24https://claritychallenge.github.io/clarity CC doc/ +such as conditioning on languages [59] or more abstract +concepts [60]. +Other interesting clues consist of signals that measure a +listener’s brain activity to guide the extraction process. Indeed, +the electroencephalogram (EEG) signal of a listener focusing +on a speaker correlates with the envelope of that speaker’s +speech signal. Ceolini et al. identified the possibility of using +EEG as clues for TSE with a system similar to the one +described in Section IV [61]. An EEG-guided TSE might +open the door for futuristic hearing aids controlled by the +user’s brain activity, which might automatically emphasize the +speaker a user wants to hear. However, research is still needed +because developing a system that requires marginal tuning to +the listener is especially challenging. Moreover, collecting a +large amount of training data is very complicated since it is +more difficult to control the quality of such clues. Compared +to audio and visual TSE clues, EEG signals are very noisy +and affected by changes in the attention of the listener, body +movements, and other factors. +D. Beyond speech +Human selective listening abilities go beyond speech sig- +nals. For example, we can focus on listening to the part of +an instrument in an orchestra or switch our attention to a +siren or a barking dog. In this paper, we focused on TSE, +but similar extraction problems have also been explored for +other audio-processing tasks. For example, much research has +been performed on extracting the track of an instrument in +a piece of music conditioned on, e.g., the type of instrument +[62], video of the musician playing [63], or EEG signal of the +listener [64]. These approaches may be important to realize, +e.g., audio-visual music analysis [65]. +Recently, the problem was extended to the extraction of +arbitrary sounds from a mixture [66], [67], e.g., extracting the +sound of a siren or a klaxon from a recording of a mixture +of street sounds. We can use such systems as that introduced +in Section IV to tackle these problems, where the clue can +be a class label indicating the type of target sound [66], the +enrollment audio of a similar target sound +[67], a video +of the sound source [9] or a text description of the target +sound [68]. Target sound extraction may become an important +technology to design, e.g., hearables or hearing aids that could +filter out nuisances and emphasize important sounds in our +surroundings, or audio visual scene analysis [9]. +Psycho-acoustic studies suggest that humans process speech +and music partly using shared auditory mechanisms and that +exposure to music can lead to better discrimination of speech +sounds +[69]. It would be interesting to explore whether, +similarly to humans, TSE systems could benefit from exposure +to other acoustic signals by training a system to extract target +speech, music, or arbitrary sounds. +X. RESOURCES +We conclude by providing pointers to selected datasets and +toolkits available for those motivated to experiment with TSE. +TSE works mostly use datasets designed for BSS. These +datasets consist generally of artificial mixtures generated from + +19 +TABLE III +SOME DATASETS AND TOOLKITS +Name +Description +Link +Dataset +WSJ0-mix +Mixtures of two or three speakers +www.merl.com/demos/deep-clustering +WHAM(R)! +Noisy and reverberant versions of WSJ0-mix +wham.whisper.ai +Librimix +Larger dataset of mixtures of two or three speakers +github.com/JorisCos/LibriMix +LibriCSS +Meeting-like mixtures recorded in a room +github.com/chenzhuo1011/libri css +MC-WSJ0-mix Spatialized version of WSJ0-2mix +www.merl.com/demos/deep-clustering +SMS-WSJ +Multi-channel corpus based on WSJ +github.com/fgnt/sms wsj +LRS +Audio-visual corpus from TED or BBC videos +www.robots.ox.ac.uk/∼vgg/data/lip reading +AVSpeech +Very large audio-visual corpus from YouTube videos +looking-to-listen.github.io/avspeech +Tools +SpeakerBeam +Time-domain audio-based TSE system +github.com/butspeechfit/speakerbeam +SpEx+ +Time-domain audio-based TSE system [31] +github.com/xuchenglin28/speaker extraction SpEx +VoiceFilter +Time-domain audio-based TSE system (Unofficial) [11] github.com/mindslab-ai/voicefilter +Multisensory +Visual clue-based TSE [9] +github.com/andrewowens/multisensory +AV Speech enh. Face landmark-based visual clue-based TSE [32] +github.com/dr-pato/audio visual speech enhancement +FaceNet +Visual feature extractor used in [8], [33], [34] +github.com/davidsandberg/facenet +the isolated signals of the individual speakers and background. +This allows evaluation of the performance by comparing the +estimated signals to the original references. Additionally, TSE +methods also require a clue, i.e., an enrollment utterance for +the target speaker or video signal. We can obtain enrollment +utterances by choosing a random utterance of the target +speaker from the same database, provided that the utterance +is different from the one in the mixture. For a video clue, it +requires using an audio-visual dataset. The top of Table III +lists some of the most commonly used datasets for audio and +visual TSE. +Several implementations of TSE systems are openly avail- +able and listed in the lower part of Table III. Although there +are no public implementations for some of the visual TSE +systems, they can be re-implemented following the audio TSE +toolkits and using openly available visual feature extractors +such as FaceNet, which was used in some previous works [8], +[33], [34]. +XI. ACKNOWLEDGMENTS +This work was partly supported by the Czech Ministry of +Education, Youth and Sports from project no. LTAIN19087 +”Multi-linguality in speech technologies.” Computing on IT4I +supercomputer was supported by the Czech Ministry of Ed- +ucation, Youth and Sports from the Large Infrastructures for +Research, Experimental Development and Innovations project +”e-Infrastructure CZ – LM2018140”. The figures contain +elements designed by pikisuperstar/Freepik. +REFERENCES +[1] A. W. Bronkhorst, “The cocktail-party problem revisited: early pro- +cessing and selection of multi-talker speech,” Attention, Perception, & +Psychophysics, vol. 77, no. 5, pp. 1465–1487, 2015. +[2] J. L. Flanagan, J. D. Johnston, R. 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Interspeech 2022, pp. 1801–1805, 2022. +[69] S. S. Asaridou and J. M. McQueen, “Speech and music shape the listen- +ing brain: evidence for shared domain-general mechanisms,” Frontiers +in psychology, vol. 4, p. 321, 2013. + diff --git a/bNA0T4oBgHgl3EQfGf8S/content/tmp_files/2301.02046v1.pdf.txt b/bNA0T4oBgHgl3EQfGf8S/content/tmp_files/2301.02046v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ea4b9539def4909712367d3c6267756ee0ee8ee --- /dev/null +++ b/bNA0T4oBgHgl3EQfGf8S/content/tmp_files/2301.02046v1.pdf.txt @@ -0,0 +1,823 @@ +ON THE INFLUENCE OF GRADIENT RECONSTRUCTION +PROCEDURES OVER THE ACCURACY OF FINITE VOLUME BASED +SCHEMES +Frederico Bolsoni Oliveira1 & João Luiz F. Azevedo2 +1Instituto Tecnológico de Aeronáutica, DCTA/ITA, Pr. Mal. Eduardo Gomes, São José dos Campos, 12228-900, SP, Brazil +2Instituto de Aeronáutica e Espaço, DCTA/IAE, Pr. Mal. Eduardo Gomes, São José dos Campos, 12228-904, SP, Brazil +Abstract +In the context of the cell centered finite volume approach, care must be taken when performing the recon- +struction of property gradients at cell interfaces. The present work analyzes three different gradient reconstruc- +tion procedures, using three different turbulent simulation test cases, namely the zero-gradient flat plate, the +subsonic NACA 0012 airfoil and the transonic OAT15A airfoil. The analysis is concerned mainly with the usage +of quadrilateral meshes. The gas dynamics equations are solved using an implicit implementation of Roe’s +second-order upwind scheme. The RANS closure problem is solved by using the negative Spalart-Allmaras +turbulence model. The solution quality of each gradient discretization procedure is analyzed and compared +to experimental data and other numerical solutions available in the literature. For the cases considered here, +excellent agreement is obtained between the computed solutions and the expected results, regardless of which +gradient reconstruction scheme is used. +Keywords: Finite Volume, Gradient Reconstruction, Flat Plate, NACA 0012, OAT15A +1. Introduction +With the advent of the Industry 4.0, the demand for high-fidelity numerical simulations has been +steadily increasing over the past few years among all fields of application [1]. Therefore, any modifi- +cation to well established numerical methodologies that yields improvements over simulation results +are of great interest to the industry. In the realm of computational fluid dynamics (CFD), an approach +that has been greatly used throughout the decades and that has been demonstrated to be capable +of achieving great results is the finite volume method (FV). In the aerospace industry, FV is com- +monly used to solve systems of conservation laws, among others, the compressible Navier-Stokes +equations. +Multiple numerical schemes have been developed over the years in the context of the FV approach +[2]. Those that rely on a cell-centered formulation, though, have one particular trait in common: they +require the evaluation of flow properties, and also flow property gradients, at discrete cell interfaces, +where those values are not readily available. On a cell-centered formulation, the known discrete +properties are taken to be volumetric averages inside each cell. Hence, a reconstruction procedure +must be used in order to define suitable values for the unknown properties at each cell interface. +Depending on the property in question, the definition of a reconstruction procedure is not necessarily +straightforward. This is especially troublesome when the reconstruction of property gradients are +considered. On the subject of fluid dynamics, some of the properties that fall into this category are, +for instance, the gradients of the velocity components, required in the calculation of the viscous forces. +Furthermore, if compressibility effects are taken into account, then the gradient of the fluid internal +energy must also be reconstructed during the evaluation of Fourier’s law in the energy conservation +equation. +It has been previously reported in the literature that the use of different gradient reconstruction +techniques can drastically change the outcome of viscous fluid simulations [3]. The effects vary from +changing the overall robustness of the CFD algorithm being employed, to modifying or dissipating +fluid structures that are present in the solution field. Unfortunately, though, no single gradient recon- +struction procedure has been found so far to be well suited for all situations. The present study is +inserted exactly in this context, and aims to provide numerical data for better understanding the ef- +fects of different gradient reconstruction techniques over the solution of compressible turbulent flows +when applied to quadrilateral meshes. Thus, CFD users can make a more informative decision re- +garding what gradient reconstruction procedure to use for a given problem configuration. +arXiv:2301.02046v1 [physics.flu-dyn] 5 Jan 2023 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +In the current work, the three-dimensional flow dynamics are modeled using the compressible +Reynolds-averaged Navier-Stokes (RANS) equations [2]. These equations are discretized in a cell- +centered FV framework by using Roe’s flux-difference splitting scheme for the reconstruction of the +convective fluxes [4, 5, 6]. A second-order, total-variation diminishing (TVD), version of the scheme is +implemented by using a piece-wise linear reconstruction of the solution [7], coupled with Venkatakr- +ishnan’s limiter [8]. In order to solve the closure problem, inherent to the RANS equations, the nega- +tive Spallart-Allmaras (SA-neg) turbulence model is employed [9, 10, 11]. Discrete cell gradients are +computed using a volume-weighted Green-Gauss approach. Property gradients at cell interfaces, +however, are computed using three different reconstruction procedures: A00, A0E and AJ0, whose +naming conventions follow Ref. [3], and that will be described in the forthcoming sections. +Simulations are performed using an in-house code, BRU3D [5, 6], for three different cases. The +first one is a two-dimensional turbulent flat plate [11], which is mainly used as a sanity test case. The +second one is the subsonic NACA 0012 airfoil with a 15 deg. angle of attack [11]. Its main purpose +is to observe the influence that the different procedures have over the value of the aerodynamic +coefficients when a separation bubble is present in the solution. +The final case is the transonic +OAT15A airfoil [12, 5, 6]. It illustrates the performance of each scheme when a shock wave is present +in the domain. +This introduction section is followed by a presentation of the numerical formulation used in the +present work. Then, a brief description of the test cases is made, accompanied by the obtained +results the concluding remarks. +2. Numerical Formulation +2.1 General Formulation of the Method +The system of conservation laws used here, which for now on will be referred to as the RANS +equations, can be written as +∂ ⃗Q +∂t +⃗∇· ⃗ +F(⃗Q,−→ +∇Q) ≡ ∂ ⃗Q +∂t +⃗∇· +� +⃗ +Fe(⃗Q)− ⃗ +Fv(⃗Q,−→ +∇Q) +� += 0 , +(1) +where ⃗Q is the vector of conserved variables. Furthermore, ⃗ +F is a geometric vector of algebraic +vectors, such that +⃗ +Fe(⃗Q) ≡ ⃗Ee(⃗Q) ˆi+⃗Fe(⃗Q) ˆj + ⃗Ge(⃗Q) ˆk , +(2) +and +⃗ +Fv(⃗Q,−→ +∇Q) ≡ ⃗Ev(⃗Q,−→ +∇Q) ˆi+⃗Fv(⃗Q,−→ +∇Q) ˆj + ⃗Gv(⃗Q,−→ +∇Q) ˆk , +(3) +with +⃗ +F ≡ ⃗ +Fe − ⃗ +Fv . +(4) +Vectors ⃗E ≡ ⃗Ee − ⃗Ev, ⃗F ≡ ⃗Fe − ⃗Fv and ⃗G ≡ ⃗Ge − ⃗Gv are the flux vectors associated with the Cartesian +coordinate triad x, y and z, respectively. In the same manner, ˆi, ˆj and ˆk are unit vectors aligned with +the same triad, respectively. The subscripts e and v refer to the inviscid and viscous components of +each flux vector. Notice that the functional relation that exists between each vector and ⃗Q and −→ +∇Q +is explicitly written. This is done in order to emphasize that only the viscous part of this formulation +requires the evaluation of −→ +∇Q. Lastly, t is the time coordinate. The mathematical definitions of each +one of these vectors are well-known in the CFD literature and, therefore, are not repeated here. The +authors refer the interested reader to Ref. [13] for a complete description of the formulation. +In order to discretize Eq. (1), the FV framework is adopted. Thus, Eq. (1) is integrated over an +arbitrary Eulerian domain of constant volume V, and outer surface S, as follows: +� +V +� +∂ ⃗Q +∂t +� +dV+ +� +V +� +∇· ⃗ +F +� +dV = 0 +=⇒ +∂ +∂t +� +V +⃗QdV+ +� +S +⃗ +F ·−→ +dS = 0 . +(5) +2 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +In Eq. (5), the Divergence Theorem, also known as the Green-Gauss Theorem, has been applied +in conjunction with the general form of the Leibniz rule. Moreover, −→ +dS ≡ ˆn dS, where ˆn is the unitary +normal vector that points in the outward direction of S. +The computational domain is assumed to be divided into multiple discrete cells of polyhedral +shape, composing an unstructured grid. The discrete conserved variables vector, ⃗Qi, associated with +the i-th cell of finite volume Vi, is defined as +⃗Qi ≡ 1 +Vi +� +Vi +⃗QdV . +(6) +If each cell has nf faces, then Eq. (5) becomes +Vi +∂ ⃗Qi +∂t + +n f +∑ +k=1 +� +⃗ +Fk ·⃗Sk +� += 0 +=⇒ +∂ ⃗Qi +∂t = − 1 +Vi +n f +∑ +k=1 +� +⃗ +Fk ·⃗Sk +� +, +(7) +after applying a 1-point Gaussian quadrature rule. In the present case, this is a valid construct, since +the resulting formulation is second-order accurate in space. +If higher-order schemes were used +instead, especially compact ones, then the surface integral might need to be numerically performed +by means of higher-order quadrature rules, to which the present formulation would need to be further +enhanced. +Equation (7) is the finite volume discrete form of the RANS equations and must be true for all +cells in the domain. For a mesh of constant geometry, the face area vectors, ⃗Sk, are known at all +times. Consequently, only two procedures are yet to be established: the reconstruction scheme used +for the evaluation of the face flux vectors, ⃗ +Fk, as well as a procedure for integrating ∂ ⃗Qi +∂t over time. +Here, Roe’s second-order TVD scheme, coupled with Venkatakrishnan’s limiter [8], is employed in the +discretization of all inviscid fluxes [5, 6], including the ones related to the turbulence model. The inte- +gration of the temporal derivatives is performed by using an implicit time-march scheme, as described +in Refs. [5, 6]. These schemes were chosen as part of an effort to improve the overall robustness of +the solution process. Thus, the only remaining issue is to define a scheme for computing the viscous +fluxes. +The calculation of the viscous components of −→ +F k requires the reconstruction of both ⃗Q and −→ +∇Q at +the k-th cell face. In the present work, a standard centered approach is followed. Therefore, if i and j +are the indexes of two adjacent cells, then: +⃗Qk = +⃗Qki + ⃗Qkj +2 +, +(8) +in which ⃗Qki and ⃗Qkj are the piece-wise reconstructed properties of i and j, respectively, evaluated at +the centroid of k [7]. +Based on the same idea, (−→ +∇Q)k is also reconstructed as a function of the directly adjacent cell +discrete properties. The definition of this function is what sets the gradient reconstruction proce- +dures apart from each other. In the next subsection, the three gradient reconstruction procedures +considered here are briefly presented. +2.2 Gradient Reconstruction Procedures +2.2.1 Weighted Green-Gauss Gradient Computation +As previously mentioned, the evaluation of a property gradient at a cell interface usually revolves +around the definition of a function with local stencil, responsible for reconstructing the gradient value +at the desired location. The selection of a suitable reconstruction scheme can depend on the problem +configuration, mesh geometry and even on the available computational resources. In the present +work, three different gradient reconstruction procedures are considered: A00, A0E and AJ0, following +the naming conventions from Ref. [3]. It must be made clear that other schemes do exist [3, 14, 15], +but only these three are analyzed here due to their simplicity and overall efficiency. +Before proceeding with a proper description of each scheme, it is important to define a method for +computing the discrete cell property gradient, of which all three distinct schemes herein considered +3 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +are a function of. If A is a property whose discrete values, Ai, are known at each cell, then its gradient, +(∇A)i, can be computed as +(−→ +∇A)i ≡ 1 +Vi +� +Vi +−→ +∇AdV = 1 +Vi +� +S A−→ +dS = 1 +Vi +nf +∑ +k=1 +Ak⃗Sk , +(9) +which is referred to as the Green-Gauss approach for defining discrete cell gradients [16]. The term +Ak can, then, be computed by using some sort of average between the adjacent known values, since +it is related to the diffusive components of the original partial differential equation. Here, a volume- +weighted average is used, as follows: +Ak = ViAi +V jA j +Vi +V j +. +(10) +More robust, but more computationally expensive, schemes for computing cell-averaged property +gradients are also available in the literature, such as the Linear Preserving Gradient (LPG) and the +Least Squares (LS) methods [17]. +2.2.2 Procedure A00 +The first gradient reconstruction procedure presented here, A00, is perhaps the simplest formula- +tion possible. It consists of a simple average between the two directly adjacent cell values: +(−→ +∇A)k = (−→ +∇A)i +(−→ +∇A)j +2 +≡ (∇A)k . +(11) +This reconstruction can also be improved by, instead, using a weighted average [3], without loss of +computational efficiency. However, only the formulation shown in Eq. (11) is considered here. +Although extremely cheap to compute, the usage of this scheme results in a stencil that effectively +does not utilize information from the i and j cells [16, 18]. In turn, high-frequency errors can develop in +the solution [3]. To solve this problem, the formulation from Eq. (11) is augmented by the introduction +of extra terms that ensure dependency on cell-averaged data of the two cells that share the interface. +Schemes A0E and AJ0 are inserted in this category. +2.2.3 Procedure A0E +The A0E scheme, also known as the edge-normal scheme, is one of the possible solutions for the +previously mentioned A00 problem. It consists in exchanging the gradient component in the direction +that connects the i and j cell centroids with a finite difference construct [3, 18]. Following Fig 1, the +A0E formulation can be written as +(−→ +∇A)k = (∇A)k + +� +A j −Ai +��⃗ri j +�� +−(∇A)k · ⃗ri j +��⃗ri j +�� +� +⃗ri j +��⃗ri j +�� . +(12) +Hence, cells i and j are effectively reintroduced to the stencil of (−→ +∇A)k. +2.2.4 Procedure AJ0 +Another approach is to use a jump term construct, AJ0, in which information from the discontinu- +ous solution at the face center is introduced to the face gradient reconstruction [3, 14]. The equation, +then, becomes +(−→ +∇A)k = (∇A)k + +α +��⃗ri j · ˆnk +�� +� +Ak j −Aki +� +ˆnk , +(13) +where Aki and Akj are the piece-wise linear reconstructed A properties of cells i and j, respectively, +evaluated at the face centroid. Furthermore, ˆnk is the face normal unitary vector pointing outwards +from the current cell. Lastly,⃗rik is a vector that points from the centroid of cell i to the centroid of face +k, as seen in Fig. 1. For the A0E scheme, the jump coefficient, α, is taken to be α = 4/3. +4 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +Figure 1 – Mesh schematic diagram picturing cells i and j. Focus is given to the k-th face of the i-th +cell. Face normal unitary vector, ˆnk, as well as relevant distance vectors,⃗r, are also shown. The dots +are used to represent the cell centroid locations. +It can be shown that multiple gradient reconstruction techniques can be cast into the form of +Eq. (13) [14]. In fact, the A0E scheme can be written by using Eq. (13) with the following α: +α = (ˆnk · ˆei j) +��ˆnk · ˆei j +�� , +(14) +where +ˆei j ≡ ⃗ri j +��⃗ri j +�� . +(15) +The above expression for the A0E scheme is the one that is effectively implemented here. +3. Description of Test Cases +In this section, a brief description of each test case is presented. +3.1 Zero-Pressure Gradient Flat Plate +The flat plate case follows NASA Langley’s Turbulence Modeling Resource (TMR) setup [11]. +Hence, it is an incompressible case solved by using a compressible fluid formulation. The problem +consists of a simple rectangular domain with a length of 2.33 m and a height of 1 m. The first 0.33 +m of the bottom boundary is a symmetry plane. An infinitely thin flat plate, which is modeled as an +adiabatic no-slip wall, lies in the other 2 m. The origin of the domain is located at the leading edge of +the plate, with the X axis parallel to the plate surface, pointing towards the right side of the domain. +Moreover, the Z axis points upwards. The top boundary is a non-reflective farfield, implemented +using Riemann invariants. The freestream Mach number is set to M∞ = 0.2, at a static temperature +of T∞ = 300 K and Reynolds number Re∞ = 5×106, computed based on the reference length ℓref = 1 +m. The right boundary is a simple back-pressure output, which is set to enforce the freestream static +pressure p∞ = 114.47 kPa. The left boundary is a non-reflective subsonic intake, with a total pressure +of pt = 1.02828 p∞, and a total temperature of Tt = 1.008T∞. A diagram that illustrates the problem is +shown in Fig. 2. +Since BRU3D is a 3-D code, the quadrilateral mesh is obtained by using hexahedral meshes with +a single cell depth-wise. The mesh employed here is the finest hexahedral mesh available in the TMR +website [11], and is composed of 544 cells in the X direction and 384 cells in the Y direction. Cells are +clustered in the region near the leading edge of the flat plate, as seen in Fig. 3. +5 + +k +rik jk +: +1INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +Figure 2 – Boundary condition placement for the two-dimensional zero-pressure gradient flat plate +case. +Figure 3 – Mesh used in the flat plate case, containing 544 x 384 cells. +6 + +1 +T +0.9 +Farfield +0.8 +0.7 +0.6 +Non-Reflective +Z (m) 0.5 +Subsonic Intake +Back-Pressure Output +0.4 +Pt = 1.02828 Po +p=Po +Tt = 1.008 To +0.3 +0.2 +0. 1 +10 +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +(w)x +A +Symmetry Plane +No-Slip Adiabatic WallINFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +3.2 Subsonic NACA 0012 Airfoil +The NACA 0012 Airfoil case follows, once again, NASA Langley’s Turbulence Modeling Resource +setup [11] for an angle of attack, αAoA, of 15 degrees. The domain has two boundary conditions: +no-slip adiabatic wall and non-reflective farfield, as shown in Fig. 4. +The freestream conditions, +which includes the Reynolds number, Re∞, Mach number, M∞, reference chord, c, and reference +temperature, T∞, are shown in Tab. 1. The mesh used is the finest hexahedral “C”-shaped mesh +available in Ref. [11]. It contains 917504 cells, mainly clustered around the airfoil surface, as illustrated +in Fig. 4. +Figure 4 – Boundary condition placement and overall view of the computational mesh for the +subsonic NACA 0012 airfoil case. +Table 1 – Freestream conditions for the subsonic NACA0012 airfoil case. +Re∞ +M∞ +c +T∞ +αAoA +6×106 +0.15 +1 m +300 K +15 deg. +3.3 Transonic OAT15A Airfoil +The final case is the transonic OAT15A airfoil, described in Ref. [12]. Here, the boundaries are +laid out in a similar manner to the previous case. That is, two continuous surfaces are employed. +The innermost one is the airfoil surface, where a no-slip adiabatic boundary condition is imposed. +Mereover, the outermost one is the freestream, where a non-reflective farfield is enforced. +The +freestream conditions are presented in Tab. 2. The mesh is constructed with 410 cells distributed +along the airfoil chord. The farfield is located at 240 chords away from the airfoil surface. Cells are +clustered around the airfoil surface, in such a way that y+ ≈ 1 at the no-slip wall, as despicted in +Figs. 5 and 6. +4. Results and Discussion +In this section, the obtained results are presented, followed by a brief discussion. In all cases, the +solution is considered converged when a decrease of 10 orders of magnitude is obtained in the L∞ +7 + +500- +450- +400 +350- +Non-Reflective +300 +Non-Reflective +Farfield +250 +Farfield +Adiabatic +Non-Slip Wall +150 +100 +50 +Z (m) 0 +50 +-100- +-150 +-200 +-250- +-300- +-350- +-400 +-450 +-500 +X (m)INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +Table 2 – Freestream conditions for the transonic OAT15A airfoil case. +Re∞ +M∞ +c +T∞ +αAoA +3×106 +0.724 +1 m +246.66 K +1.15 deg. +Figure 5 – Overview of the mesh used in the +transonic OAT15A airfoil case. +Figure 6 – Zommed-in view of the mesh used in +the transonic OAT15A airfoil case. +norm of the residue related to the continuity equation. +4.1 Zero-Pressure Gradient Flat Plate +Values of skin-friction coefficient, cf , plotted along the length of the flat plate are shown in Fig. 7. +The skin friction coefficient is defined as +cf ≡ +τw +1 +2ρ∞u2∞ +, +(16) +in which τw is the fluid shear stress measured at the wall. Experimental data from Ref. [19], along +with von Kármán’s empirical curve [20], are also shown for comparison. The von Kármán empirical +curve is defined as +CfvonKrmn = 0.027 +(Rex) +1 +7 . +(17) +Simulation data from Ref. [11] are also plotted. Such data was obtained with NASA’s CFL3D and +FUN3D codes using the Spalart-Allmaras turbulence model. In spite of the fact that slight changes +can be seen between experimental data and most of the simulation data, it is clear that the results +obtained by the three different gradient reconstruction schemes are virtually identical in the context +of the current case setup. Furthermore, when comparing the current data with simulation results +from CFL3D and FUN3D, it is also clear that they are extremely close to each other. Figure 8 shows +a zommed-in view of Fig. 7, which highlights the fact that the computed c f values differ from each +other by a maximum of, approximately, 0.1%. Therefore, it is safe to say that the differences observed +between the simulation data and the experimental results come from the quality of the turbulence +model itself, and not from the discretization schemes used. +8 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +1 +2 +3 +4 +5 +6 +Rex +106 +2.5 +3 +3.5 +4 +4.5 +Cf +10-3 +CFL3D +FUN3D +V00 +AJ0 +V0E +Cfvon Kármán +Exp. +Figure 7 – Skin friction coefficient, cf , distribution as a function of Rex, along the first 1.2 m of the flat +plate. Experimental data from Ref. [19] are added for comparison. +2.32 +2.33 +2.34 +2.35 +2.36 +Rex +106 +3.005 +3.01 +3.015 +Cf +10-3 +CFL3D +FUN3D +V00 +AJ0 +V0E +Cfvon Kármán +Exp. +Figure 8 – Zoomed-in view of Fig. 7, which highlights the small differences between all numerical +results. +9 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +0 +0.2 +0.4 +0.6 +0.8 +1 +x/c +-12 +-10 +-8 +-6 +-4 +-2 +0 +2 +Cp + = 15 deg. +V00 +AJ0 +A0E +CFL3D, SA +Gregory & O'Reilly +Ladson et. al. +Figure 9 – Pressure coefficient, cp, distribution along the airfoil surface for the subsonic NACA 0012 +case. +4.2 Subsonic NACA 0012 Airfoil +Figure 9 shows the pressure coefficient, cp, plotted along the wall surface for the NACA 0012 +airfoil case. Here, the pressure coefficient is computed as: +cp ≡ (p− p∞) +1 +2ρ∞u2∞ +. +(18) +Additional simulation data from Ref. [11], using the CFL3D code, as well as experimental data from +Gregory & O’Reilly [21] and Ladson [21] are added for comparison. As it can be seen, the same +behavior previously described also repeats here. That is, no meaningful changes are captured be- +tween the schemes for the current case configuration and the present mesh topology. Furthermore, +excellent agreement is observed with the experimental cp data. +In order to spot the differences between the obtained results, a zommed-in view of Fig. 9 is shown +in Fig. 10. Focus is given to the region surrounding the suction peak. The maximum difference +between the predicted cp values is observed when the results obtained by the V00 scheme are com- +pared with the ones from CFL3D. Even then, the relative difference is of only 0.27%, approximately. +Obtained lift and drag coefficients, cL and cD, are compared in Tab. 3. Experimental results are +interpolated from Refs. [21] and [22] and presented in the same table. It is clear that all numerical +results are consistent with each other. Therefore, identical flow behavior is being captured by all +numerical schemes with the SA-neg turbulence model. Any differences between the predicted co- +efficients and the experimental data are likely due to the turbulence model itself, and not due to the +numerical discretization. +4.3 Transonic OAT15A Airfoil +This is a transonic case and, therefore, shock waves are expected to develop in the numerical +solution. Distribution of cp along the chord of the OAT15A airfoil is shown in Fig. 11, compared to +experimental data from Ref. [12]. Once again, no difference is seen from the results obtained by each +scheme throughout the entire length of the airfoil. This is the case even in the region surrounding +the shock wave, as seen from Fig. 12, where an extremely zoomed-in view is presented in order to +visualize separate curves. +Computed aerodynamic coefficients, in the form of cL and cD, are shown in Tab. 4. +Interpo- +lated data is extracted from the plots available in Ref. [12]. There is a significant disparity between +10 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +1 +1.2 +1.4 +1.6 +1.8 +2 +x/c +10-3 +-11.19 +-11.18 +-11.17 +-11.16 +-11.15 +-11.14 +-11.13 +-11.12 +-11.11 +Cp + = 15 deg. +V00 +AJ0 +A0E +CFL3D, SA +Gregory & O'Reilly +Ladson et. al. +Figure 10 – Zoomed-in view of Fig. 9 in the region surrounding the suction peak. +0 +0.2 +0.4 +0.6 +0.8 +1 +x/c +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +Cp +V00 +AJ0 +A0E +Exp. +Figure 11 – Pressure coefficient, cp, distribution along the airfoil surface for the transonic OAT15A +case. +11 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +Table 3 – Comparison between computed aerodynamic coefficients and experimental data for the +subsonic NACA 0012 airfoil case. +Source +cL +cD +CFL3D +1.5461 +0.02124 +BRU3D V00 +1.5498 +0.021234 +BRU3D AJ0 +1.5496 +0.021257 +BRU3D A0E +1.5496 +0.021254 +Exp. Gregory +1.5052 +− +Exp. Ladson +1.4993 +0.0180 +Exp. Abbott +1.4976 +− +0.381 0.38120.38140.38160.3818 0.382 +x/c +-0.8975 +-0.897 +-0.8965 +-0.896 +-0.8955 +-0.895 +-0.8945 +Cp +V00 +AJ0 +A0E +Exp. +Figure 12 – Zoomed-in view of Fig. 11 in the region surrounding the shock wave. +the computed coefficients and the experimental values. This is, however, a known limitation of the +Spalart-Allmaras turbulence model, due to its inability to correctly solve the shock wave location for +this case [5, 6]. +Table 4 – Comparison between computed aerodynamic coefficients and interpolated experimental +data for the transonic OAT15A airfoil case. +Source +cL +cD +BRU3D V00 +0.69178 +0.013152 +BRU3D AJ0 +0.69163 +0.013197 +BRU3D A0E +0.69183 +0.013183 +Interp. Exp. Data +0.5949 +0.0106 +5. Concluding Remarks +Reconstruction of property gradients at cell interfaces is a small portion of the complete discretiza- +tion scheme. However, it can have a profound impact on the quality of finite volume-based schemes. +In the present work, essentially no differences are observed in the solutions obtained with the three +different schemes. It is quite likely that such behavior is a result of the use of “well behaved” quadrilat- +12 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +eral meshes for the three test-cases addressed here. Disparities between the computed values and +experimental data are likely due to limitations of the turbulence model employed, namely, the nega- +tive Spalart-Allmaras model, and not due to discretization errors. This argument is further enhanced +by comparisons with other codes that implement the same turbulence model. In these comparisons, +virtually identical results are obtained. Therefore, any of the three gradient reconstruction schemes +can, theoretically, be used in the simulation of cases similar to the ones investigated here. +It is important to stress, however, that these conclusions are only valid for the type of mesh con- +sidered here. Highly stretched hybrid meshes can lead to wildly different results, and the differences +between each reconstruction scheme might become more obvious. This is precisely what the authors +will address in future work. +6. Acknowledgments +The authors wish to express their gratitude to the São Paulo Research Foundation, FAPESP, +which has supported the present research under the Research Grants No. 2021/00147-8 and No. +2013/07375-0. The authors also gratefully acknowledge the support for the present research provided +by Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, under the Research +Grant No. 309985/2013-7. The work is further supported by the computational resources of the +Center for Mathematical Sciences Applied to Industry (CeMEAI), also funded by FAPESP under the +Research Grant No. 2013/07375-0. +7. Contact Author Email Address +Frederico Bolsoni Oliveira: fredericobolsoni@gmail.com, Tel.: +55 (12) 3947-6488. +João Luiz F. Azevedo: joaoluiz.azevedo@gmail.com, Tel.: +55 (12) 3947-6488. +References +[1] “Digital Twin: +Definition & Value,” an AIAA and AIA Position Paper, available in https://www. +aia-aerospace.org/report/digital-twin-paper/, last accessed in 06 feb. 2022. +[2] Hirsch, C., Numerical Computation of Internal and External Flows - Volume 2: Computational Methods +for Inviscid and Viscous Flows, Wiley, Chichester, 1st ed., 1988, p. 691. +[3] Jalali, A., Sharbatdar, M., and Ollivier-Gooch, C., “Accuracy analysis of unstructured finite volume dis- +cretization schemes for diffusive fluxes,” Computers & Fluids, Vol. 101, 2014, pp. 220–232. +[4] Roe, P. L., “Approximate Riemann Solvers, Parameter Vectors and Difference Schemes,” Journal of Com- +putational Physics, Vol. 43, No. 2, 1981, pp. 357–372. +[5] Bigarella, E. D. V. and Azevedo, J. L. F., “A Unified Implicit CFD Approach for Turbulent-Flow Aerospace- +Configuration Simulations,” AIAA Paper No. 2009-1473, 47th AIAA Aerospace Sciences Meeting including +The New Horizons Forum and Aerospace Exposition, Orlando, Florida, USA, Jan. 2009. +[6] Bigarella, E. D. V. and Azevedo, J. L. F., “A Study of Convective Flux Schemes for Aerospace Flows,” +Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 34, No. 3, 2012, pp. 314– +329. +[7] Barth, T. J. and Jespersen, D. C., “The Design and Application of Upwind Schemes on Unstructured +Meshes,” AIAA Paper No. AIAA-89-0366, 27th AIAA Aerospace Sciences Meeting, Reno, Nevada, USA, +Jan. 1989. +[8] Venkatakrishnan, V., “Convergence to Steady State Solutions of the Euler Equations on Unstructured +Grids with Limiters,” Journal of Computational Physics, Vol. 118, 1995, pp. 120–130. +[9] Spalart, P. R. and Allmaras, S. R., “A One-Equation Turbulence Model for Aerodynamic Flows,” AIAA +Paper No. 92-0439, 30th Aerospace Sciences Meeting & Exhibit, Reno, NV, USA, Jan. 1992. +[10] S. R. Allmaras, F. T. J. and Spalart, P. R., “Modifications and Clarifications for the Implementation of +the Spalart-Allmaras Turbulence Model,” AIAA Paper No. 2007-4079, 7th International Conference on +Computational Fluid Dynamics (ICCFD7), Big Island, Hawaii, USA, jul 2012. +[11] “NASA’s Langley Research Center - Turbulence Modeling Resource Website,” available in https:// +turbmodels.larc.nasa.gov/, last accessed in 25 jan. 2022. +[12] Roddle, A. M. and Archambaud, J. P., “OAT15A Airfoil Data,” A Selection of Experimental Test Cases +for the Validation of CFD Codes, AGARD-AR-303, NATO Advisory Group for Aerospace Research & +Developtment, Case A11, aug. 1994. +13 + +INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES +[13] Hirsch, C., Numerical Computation of Internal and External Flows - Volume 1: Fundamentals of Numerical +Discretization, Wiley, Chichester, 1st ed., 1988, p. 515. +[14] Nishikawa, H., “Beyond Interface Gradient: A General Principle for Constructing Diffusion Schemes,” +AIAA Paper No. 2010-5093, 40th Fluid Dynamics Conference and Exhibit, jun 2010. +[15] Nishikawa, H., “Robust and accurate viscous discretization via upwind scheme – I: Basic principle,” Com- +puters & Fluids, Vol. 49, No. 1, 2011, pp. 62–86. +[16] Blazek, J., Computational Fluid Dynamics: Principles and Applications, Butterworth-Heinemann, Elsevier, +Oxford, UK, 3rd ed., 2015. +[17] Cary, A., Dorgan, A., and Mani, M., “Towards Accurate Flow Predictions Using Unstructured Meshes,” +AIAA Paper No. 2009-3650, 19th AIAA Computational Fluid Dynamics, jun 2009. +[18] Weiss, J. M., Maruszewski, J. P., and Smith, W. A., “Implicit Solution of Preconditioned Navier-Stokes +Equations Using Algebraic Multigrid,” AIAA Journal, Vol. 37, No. 1, 1998, pp. 29–36. +[19] Coles, D. and Hirst, E. A., “Computation of Turbulent Boundary Layers,” Proceedings of the 1968 U.S. Air +Force Office of Scientific Research - Internal Flow Program - Stanford Conference, Stanford University +Press. +[20] White, F. M., Viscous Fluid Flow, McGraw Hill, New York, USA, 3rd ed., 2006, p. 629. +[21] McCroskey, W. J., “A Critical Assessment of Wind Tunnel Results for the NACA 0012 Airfoil,” AGARD +CP-429, 1998. +[22] Abbott, I. and von Doenhoff, A., Theory of Wing Sections, Including a Summary of Airfoil Data, Dover +Books on Aeronautical Engineering Series, Dover Publications, 1959. +14 + diff --git a/bNA0T4oBgHgl3EQfGf8S/content/tmp_files/load_file.txt b/bNA0T4oBgHgl3EQfGf8S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1540dfac5bbb2010656522f0bc56523dd4f77ae --- /dev/null +++ b/bNA0T4oBgHgl3EQfGf8S/content/tmp_files/load_file.txt @@ -0,0 +1,528 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf,len=527 +page_content='ON THE INFLUENCE OF GRADIENT RECONSTRUCTION PROCEDURES OVER THE ACCURACY OF FINITE VOLUME BASED SCHEMES Frederico Bolsoni Oliveira1 & João Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Azevedo2 1Instituto Tecnológico de Aeronáutica, DCTA/ITA, Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Eduardo Gomes, São José dos Campos, 12228-900, SP, Brazil 2Instituto de Aeronáutica e Espaço, DCTA/IAE, Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Eduardo Gomes, São José dos Campos, 12228-904, SP, Brazil Abstract In the context of the cell centered finite volume approach, care must be taken when performing the recon- struction of property gradients at cell interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The present work analyzes three different gradient reconstruc- tion procedures, using three different turbulent simulation test cases, namely the zero-gradient flat plate, the subsonic NACA 0012 airfoil and the transonic OAT15A airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The analysis is concerned mainly with the usage of quadrilateral meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The gas dynamics equations are solved using an implicit implementation of Roe’s second-order upwind scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The RANS closure problem is solved by using the negative Spalart-Allmaras turbulence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The solution quality of each gradient discretization procedure is analyzed and compared to experimental data and other numerical solutions available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' For the cases considered here, excellent agreement is obtained between the computed solutions and the expected results, regardless of which gradient reconstruction scheme is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Keywords: Finite Volume, Gradient Reconstruction, Flat Plate, NACA 0012, OAT15A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Introduction With the advent of the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='0, the demand for high-fidelity numerical simulations has been steadily increasing over the past few years among all fields of application [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Therefore, any modifi- cation to well established numerical methodologies that yields improvements over simulation results are of great interest to the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In the realm of computational fluid dynamics (CFD), an approach that has been greatly used throughout the decades and that has been demonstrated to be capable of achieving great results is the finite volume method (FV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In the aerospace industry, FV is com- monly used to solve systems of conservation laws, among others, the compressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Multiple numerical schemes have been developed over the years in the context of the FV approach [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Those that rely on a cell-centered formulation, though, have one particular trait in common: they require the evaluation of flow properties, and also flow property gradients, at discrete cell interfaces, where those values are not readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' On a cell-centered formulation, the known discrete properties are taken to be volumetric averages inside each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Hence, a reconstruction procedure must be used in order to define suitable values for the unknown properties at each cell interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Depending on the property in question, the definition of a reconstruction procedure is not necessarily straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' This is especially troublesome when the reconstruction of property gradients are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' On the subject of fluid dynamics, some of the properties that fall into this category are, for instance, the gradients of the velocity components, required in the calculation of the viscous forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Furthermore, if compressibility effects are taken into account, then the gradient of the fluid internal energy must also be reconstructed during the evaluation of Fourier’s law in the energy conservation equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It has been previously reported in the literature that the use of different gradient reconstruction techniques can drastically change the outcome of viscous fluid simulations [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The effects vary from changing the overall robustness of the CFD algorithm being employed, to modifying or dissipating fluid structures that are present in the solution field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Unfortunately, though, no single gradient recon- struction procedure has been found so far to be well suited for all situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The present study is inserted exactly in this context, and aims to provide numerical data for better understanding the ef- fects of different gradient reconstruction techniques over the solution of compressible turbulent flows when applied to quadrilateral meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Thus, CFD users can make a more informative decision re- garding what gradient reconstruction procedure to use for a given problem configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='02046v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='flu-dyn] 5 Jan 2023 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES In the current work, the three-dimensional flow dynamics are modeled using the compressible Reynolds-averaged Navier-Stokes (RANS) equations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' These equations are discretized in a cell- centered FV framework by using Roe’s flux-difference splitting scheme for the reconstruction of the convective fluxes [4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' A second-order, total-variation diminishing (TVD), version of the scheme is implemented by using a piece-wise linear reconstruction of the solution [7], coupled with Venkatakr- ishnan’s limiter [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In order to solve the closure problem, inherent to the RANS equations, the nega- tive Spallart-Allmaras (SA-neg) turbulence model is employed [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Discrete cell gradients are computed using a volume-weighted Green-Gauss approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Property gradients at cell interfaces, however, are computed using three different reconstruction procedures: A00, A0E and AJ0, whose naming conventions follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [3], and that will be described in the forthcoming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Simulations are performed using an in-house code, BRU3D [5, 6], for three different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The first one is a two-dimensional turbulent flat plate [11], which is mainly used as a sanity test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The second one is the subsonic NACA 0012 airfoil with a 15 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' angle of attack [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Its main purpose is to observe the influence that the different procedures have over the value of the aerodynamic coefficients when a separation bubble is present in the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The final case is the transonic OAT15A airfoil [12, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It illustrates the performance of each scheme when a shock wave is present in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' This introduction section is followed by a presentation of the numerical formulation used in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Then, a brief description of the test cases is made, accompanied by the obtained results the concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Numerical Formulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='1 General Formulation of the Method The system of conservation laws used here, which for now on will be referred to as the RANS equations, can be written as ∂ ⃗Q ∂t +⃗∇· ⃗ F(⃗Q,−→ ∇Q) ≡ ∂ ⃗Q ∂t +⃗∇· � ⃗ Fe(⃗Q)− ⃗ Fv(⃗Q,−→ ∇Q) � = 0 , (1) where ⃗Q is the vector of conserved variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Furthermore, ⃗ F is a geometric vector of algebraic vectors, such that ⃗ Fe(⃗Q) ≡ ⃗Ee(⃗Q) ˆi+⃗Fe(⃗Q) ˆj + ⃗Ge(⃗Q) ˆk , (2) and ⃗ Fv(⃗Q,−→ ∇Q) ≡ ⃗Ev(⃗Q,−→ ∇Q) ˆi+⃗Fv(⃗Q,−→ ∇Q) ˆj + ⃗Gv(⃗Q,−→ ∇Q) ˆk , (3) with ⃗ F ≡ ⃗ Fe − ⃗ Fv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (4) Vectors ⃗E ≡ ⃗Ee − ⃗Ev, ⃗F ≡ ⃗Fe − ⃗Fv and ⃗G ≡ ⃗Ge − ⃗Gv are the flux vectors associated with the Cartesian coordinate triad x, y and z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In the same manner, ˆi, ˆj and ˆk are unit vectors aligned with the same triad, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The subscripts e and v refer to the inviscid and viscous components of each flux vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Notice that the functional relation that exists between each vector and ⃗Q and −→ ∇Q is explicitly written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' This is done in order to emphasize that only the viscous part of this formulation requires the evaluation of −→ ∇Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Lastly, t is the time coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The mathematical definitions of each one of these vectors are well-known in the CFD literature and, therefore, are not repeated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The authors refer the interested reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [13] for a complete description of the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In order to discretize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (1), the FV framework is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (1) is integrated over an arbitrary Eulerian domain of constant volume V, and outer surface S, as follows: � V � ∂ ⃗Q ∂t � dV+ � V � ∇· ⃗ F � dV = 0 =⇒ ∂ ∂t � V ⃗QdV+ � S ⃗ F ·−→ dS = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (5) 2 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (5), the Divergence Theorem, also known as the Green-Gauss Theorem, has been applied in conjunction with the general form of the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Moreover, −→ dS ≡ ˆn dS, where ˆn is the unitary normal vector that points in the outward direction of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The computational domain is assumed to be divided into multiple discrete cells of polyhedral shape, composing an unstructured grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The discrete conserved variables vector, ⃗Qi, associated with the i-th cell of finite volume Vi, is defined as ⃗Qi ≡ 1 Vi � Vi ⃗QdV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (6) If each cell has nf faces, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (5) becomes Vi ∂ ⃗Qi ∂t + n f ∑ k=1 � ⃗ Fk ·⃗Sk � = 0 =⇒ ∂ ⃗Qi ∂t = − 1 Vi n f ∑ k=1 � ⃗ Fk ·⃗Sk � , (7) after applying a 1-point Gaussian quadrature rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In the present case, this is a valid construct, since the resulting formulation is second-order accurate in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' If higher-order schemes were used instead, especially compact ones, then the surface integral might need to be numerically performed by means of higher-order quadrature rules, to which the present formulation would need to be further enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Equation (7) is the finite volume discrete form of the RANS equations and must be true for all cells in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' For a mesh of constant geometry, the face area vectors, ⃗Sk, are known at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Consequently, only two procedures are yet to be established: the reconstruction scheme used for the evaluation of the face flux vectors, ⃗ Fk, as well as a procedure for integrating ∂ ⃗Qi ∂t over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Here, Roe’s second-order TVD scheme, coupled with Venkatakrishnan’s limiter [8], is employed in the discretization of all inviscid fluxes [5, 6], including the ones related to the turbulence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The inte- gration of the temporal derivatives is performed by using an implicit time-march scheme, as described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' These schemes were chosen as part of an effort to improve the overall robustness of the solution process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Thus, the only remaining issue is to define a scheme for computing the viscous fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The calculation of the viscous components of −→ F k requires the reconstruction of both ⃗Q and −→ ∇Q at the k-th cell face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In the present work, a standard centered approach is followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Therefore, if i and j are the indexes of two adjacent cells, then: ⃗Qk = ⃗Qki + ⃗Qkj 2 , (8) in which ⃗Qki and ⃗Qkj are the piece-wise reconstructed properties of i and j, respectively, evaluated at the centroid of k [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Based on the same idea, (−→ ∇Q)k is also reconstructed as a function of the directly adjacent cell discrete properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The definition of this function is what sets the gradient reconstruction proce- dures apart from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In the next subsection, the three gradient reconstruction procedures considered here are briefly presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 Gradient Reconstruction Procedures 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='1 Weighted Green-Gauss Gradient Computation As previously mentioned, the evaluation of a property gradient at a cell interface usually revolves around the definition of a function with local stencil, responsible for reconstructing the gradient value at the desired location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The selection of a suitable reconstruction scheme can depend on the problem configuration, mesh geometry and even on the available computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In the present work, three different gradient reconstruction procedures are considered: A00, A0E and AJ0, following the naming conventions from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It must be made clear that other schemes do exist [3, 14, 15], but only these three are analyzed here due to their simplicity and overall efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Before proceeding with a proper description of each scheme, it is important to define a method for computing the discrete cell property gradient, of which all three distinct schemes herein considered 3 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES are a function of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' If A is a property whose discrete values, Ai, are known at each cell, then its gradient, (∇A)i, can be computed as (−→ ∇A)i ≡ 1 Vi � Vi −→ ∇AdV = 1 Vi � S A−→ dS = 1 Vi nf ∑ k=1 Ak⃗Sk , (9) which is referred to as the Green-Gauss approach for defining discrete cell gradients [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The term Ak can, then, be computed by using some sort of average between the adjacent known values, since it is related to the diffusive components of the original partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Here, a volume- weighted average is used, as follows: Ak = ViAi +V jA j Vi +V j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (10) More robust, but more computationally expensive, schemes for computing cell-averaged property gradients are also available in the literature, such as the Linear Preserving Gradient (LPG) and the Least Squares (LS) methods [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 Procedure A00 The first gradient reconstruction procedure presented here, A00, is perhaps the simplest formula- tion possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It consists of a simple average between the two directly adjacent cell values: (−→ ∇A)k = (−→ ∇A)i +(−→ ∇A)j 2 ≡ (∇A)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (11) This reconstruction can also be improved by, instead, using a weighted average [3], without loss of computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' However, only the formulation shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (11) is considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Although extremely cheap to compute, the usage of this scheme results in a stencil that effectively does not utilize information from the i and j cells [16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In turn, high-frequency errors can develop in the solution [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' To solve this problem, the formulation from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (11) is augmented by the introduction of extra terms that ensure dependency on cell-averaged data of the two cells that share the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Schemes A0E and AJ0 are inserted in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='3 Procedure A0E The A0E scheme, also known as the edge-normal scheme, is one of the possible solutions for the previously mentioned A00 problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It consists in exchanging the gradient component in the direction that connects the i and j cell centroids with a finite difference construct [3, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Following Fig 1, the A0E formulation can be written as (−→ ∇A)k = (∇A)k + � A j −Ai ��⃗ri j �� −(∇A)k · ⃗ri j ��⃗ri j �� � ⃗ri j ��⃗ri j �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (12) Hence, cells i and j are effectively reintroduced to the stencil of (−→ ∇A)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='4 Procedure AJ0 Another approach is to use a jump term construct, AJ0, in which information from the discontinu- ous solution at the face center is introduced to the face gradient reconstruction [3, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The equation, then, becomes (−→ ∇A)k = (∇A)k + α ��⃗ri j · ˆnk �� � Ak j −Aki � ˆnk , (13) where Aki and Akj are the piece-wise linear reconstructed A properties of cells i and j, respectively, evaluated at the face centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Furthermore, ˆnk is the face normal unitary vector pointing outwards from the current cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Lastly,⃗rik is a vector that points from the centroid of cell i to the centroid of face k, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' For the A0E scheme, the jump coefficient, α, is taken to be α = 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 4 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES Figure 1 – Mesh schematic diagram picturing cells i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Focus is given to the k-th face of the i-th cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Face normal unitary vector, ˆnk, as well as relevant distance vectors,⃗r, are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The dots are used to represent the cell centroid locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It can be shown that multiple gradient reconstruction techniques can be cast into the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (13) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In fact, the A0E scheme can be written by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (13) with the following α: α = (ˆnk · ˆei j) ��ˆnk · ˆei j �� , (14) where ˆei j ≡ ⃗ri j ��⃗ri j �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (15) The above expression for the A0E scheme is the one that is effectively implemented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Description of Test Cases In this section, a brief description of each test case is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='1 Zero-Pressure Gradient Flat Plate The flat plate case follows NASA Langley’s Turbulence Modeling Resource (TMR) setup [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Hence, it is an incompressible case solved by using a compressible fluid formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The problem consists of a simple rectangular domain with a length of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='33 m and a height of 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The first 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='33 m of the bottom boundary is a symmetry plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' An infinitely thin flat plate, which is modeled as an adiabatic no-slip wall, lies in the other 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The origin of the domain is located at the leading edge of the plate, with the X axis parallel to the plate surface, pointing towards the right side of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Moreover, the Z axis points upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The top boundary is a non-reflective farfield, implemented using Riemann invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The freestream Mach number is set to M∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2, at a static temperature of T∞ = 300 K and Reynolds number Re∞ = 5×106, computed based on the reference length ℓref = 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The right boundary is a simple back-pressure output, which is set to enforce the freestream static pressure p∞ = 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='47 kPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The left boundary is a non-reflective subsonic intake, with a total pressure of pt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='02828 p∞, and a total temperature of Tt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='008T∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' A diagram that illustrates the problem is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Since BRU3D is a 3-D code, the quadrilateral mesh is obtained by using hexahedral meshes with a single cell depth-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The mesh employed here is the finest hexahedral mesh available in the TMR website [11], and is composed of 544 cells in the X direction and 384 cells in the Y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Cells are clustered in the region near the leading edge of the flat plate, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 5 k rik jk : 1INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES Figure 2 – Boundary condition placement for the two-dimensional zero-pressure gradient flat plate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 3 – Mesh used in the flat plate case, containing 544 x 384 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 6 1 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='9 Farfield 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='6 Non-Reflective Z (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5 Subsonic Intake Back-Pressure Output 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='4 Pt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='02828 Po p=Po Tt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='008 To 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8 2 (w)x A Symmetry Plane No-Slip Adiabatic WallINFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 Subsonic NACA 0012 Airfoil The NACA 0012 Airfoil case follows, once again, NASA Langley’s Turbulence Modeling Resource setup [11] for an angle of attack, αAoA, of 15 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The domain has two boundary conditions: no-slip adiabatic wall and non-reflective farfield, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The freestream conditions, which includes the Reynolds number, Re∞, Mach number, M∞, reference chord, c, and reference temperature, T∞, are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The mesh used is the finest hexahedral “C”-shaped mesh available in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It contains 917504 cells, mainly clustered around the airfoil surface, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 4 – Boundary condition placement and overall view of the computational mesh for the subsonic NACA 0012 airfoil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Table 1 – Freestream conditions for the subsonic NACA0012 airfoil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Re∞ M∞ c T∞ αAoA 6×106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='15 1 m 300 K 15 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='3 Transonic OAT15A Airfoil The final case is the transonic OAT15A airfoil, described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Here, the boundaries are laid out in a similar manner to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' That is, two continuous surfaces are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The innermost one is the airfoil surface, where a no-slip adiabatic boundary condition is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Mereover, the outermost one is the freestream, where a non-reflective farfield is enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The freestream conditions are presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The mesh is constructed with 410 cells distributed along the airfoil chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The farfield is located at 240 chords away from the airfoil surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Cells are clustered around the airfoil surface, in such a way that y+ ≈ 1 at the no-slip wall, as despicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Results and Discussion In this section, the obtained results are presented, followed by a brief discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In all cases, the solution is considered converged when a decrease of 10 orders of magnitude is obtained in the L∞ 7 500- 450- 400 350- Non-Reflective 300 Non-Reflective Farfield 250 Farfield Adiabatic Non-Slip Wall 150 100 50 Z (m) 0 50 100- 150 200 250- 300- 350- 400 450 500 X (m)INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES Table 2 – Freestream conditions for the transonic OAT15A airfoil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Re∞ M∞ c T∞ αAoA 3×106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='724 1 m 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='66 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='15 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 5 – Overview of the mesh used in the transonic OAT15A airfoil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 6 – Zommed-in view of the mesh used in the transonic OAT15A airfoil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' norm of the residue related to the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='1 Zero-Pressure Gradient Flat Plate Values of skin-friction coefficient, cf , plotted along the length of the flat plate are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The skin friction coefficient is defined as cf ≡ τw 1 2ρ∞u2∞ , (16) in which τw is the fluid shear stress measured at the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [19], along with von Kármán’s empirical curve [20], are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The von Kármán empirical curve is defined as CfvonKrmn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='027 (Rex) 1 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (17) Simulation data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [11] are also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Such data was obtained with NASA’s CFL3D and FUN3D codes using the Spalart-Allmaras turbulence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In spite of the fact that slight changes can be seen between experimental data and most of the simulation data, it is clear that the results obtained by the three different gradient reconstruction schemes are virtually identical in the context of the current case setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Furthermore, when comparing the current data with simulation results from CFL3D and FUN3D, it is also clear that they are extremely close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 8 shows a zommed-in view of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 7, which highlights the fact that the computed c f values differ from each other by a maximum of, approximately, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Therefore, it is safe to say that the differences observed between the simulation data and the experimental results come from the quality of the turbulence model itself, and not from the discretization schemes used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 8 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES 1 2 3 4 5 6 Rex 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5 Cf 10-3 CFL3D FUN3D V00 AJ0 V0E Cfvon Kármán Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 7 – Skin friction coefficient, cf , distribution as a function of Rex, along the first 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 m of the flat plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [19] are added for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='36 Rex 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='005 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='015 Cf 10-3 CFL3D FUN3D V00 AJ0 V0E Cfvon Kármán Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 8 – Zoomed-in view of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 7, which highlights the small differences between all numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 9 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8 1 x/c 12 10 8 6 4 2 0 2 Cp = 15 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=" V00 AJ0 A0E CFL3D, SA Gregory & O'Reilly Ladson et." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 9 – Pressure coefficient, cp, distribution along the airfoil surface for the subsonic NACA 0012 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 Subsonic NACA 0012 Airfoil Figure 9 shows the pressure coefficient, cp, plotted along the wall surface for the NACA 0012 airfoil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Here, the pressure coefficient is computed as: cp ≡ (p− p∞) 1 2ρ∞u2∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' (18) Additional simulation data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [11], using the CFL3D code, as well as experimental data from Gregory & O’Reilly [21] and Ladson [21] are added for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' As it can be seen, the same behavior previously described also repeats here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' That is, no meaningful changes are captured be- tween the schemes for the current case configuration and the present mesh topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Furthermore, excellent agreement is observed with the experimental cp data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In order to spot the differences between the obtained results, a zommed-in view of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 9 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Focus is given to the region surrounding the suction peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The maximum difference between the predicted cp values is observed when the results obtained by the V00 scheme are com- pared with the ones from CFL3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Even then, the relative difference is of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='27%, approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Obtained lift and drag coefficients, cL and cD, are compared in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Experimental results are interpolated from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [21] and [22] and presented in the same table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It is clear that all numerical results are consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Therefore, identical flow behavior is being captured by all numerical schemes with the SA-neg turbulence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Any differences between the predicted co- efficients and the experimental data are likely due to the turbulence model itself, and not due to the numerical discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='3 Transonic OAT15A Airfoil This is a transonic case and, therefore, shock waves are expected to develop in the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Distribution of cp along the chord of the OAT15A airfoil is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 11, compared to experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Once again, no difference is seen from the results obtained by each scheme throughout the entire length of the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' This is the case even in the region surrounding the shock wave, as seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 12, where an extremely zoomed-in view is presented in order to visualize separate curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Computed aerodynamic coefficients, in the form of cL and cD, are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Interpo- lated data is extracted from the plots available in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' There is a significant disparity between 10 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8 2 x/c 10-3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='19 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='18 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='17 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='16 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='15 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='14 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='13 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='12 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='11 Cp = 15 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=" V00 AJ0 A0E CFL3D, SA Gregory & O'Reilly Ladson et." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 10 – Zoomed-in view of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 9 in the region surrounding the suction peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8 1 x/c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5 Cp V00 AJ0 A0E Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 11 – Pressure coefficient, cp, distribution along the airfoil surface for the transonic OAT15A case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 11 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES Table 3 – Comparison between computed aerodynamic coefficients and experimental data for the subsonic NACA 0012 airfoil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Source cL cD CFL3D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='02124 BRU3D V00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='021234 BRU3D AJ0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='021257 BRU3D A0E 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Abbott 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='4976 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='38120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='38140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='38160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='3818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='382 x/c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='8945 Cp V00 AJ0 A0E Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Figure 12 – Zoomed-in view of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 11 in the region surrounding the shock wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' the computed coefficients and the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' This is, however, a known limitation of the Spalart-Allmaras turbulence model, due to its inability to correctly solve the shock wave location for this case [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Table 4 – Comparison between computed aerodynamic coefficients and interpolated experimental data for the transonic OAT15A airfoil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Source cL cD BRU3D V00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='69178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='013152 BRU3D AJ0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='69163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='013197 BRU3D A0E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='69183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='013183 Interp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='5949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='0106 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Concluding Remarks Reconstruction of property gradients at cell interfaces is a small portion of the complete discretiza- tion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' However, it can have a profound impact on the quality of finite volume-based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In the present work, essentially no differences are observed in the solutions obtained with the three different schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It is quite likely that such behavior is a result of the use of “well behaved” quadrilat- 12 INFLUENCE OF GRADIENT RECONSTRUCTION OVER THE ACCURACY OF FINITE VOLUME SCHEMES eral meshes for the three test-cases addressed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Disparities between the computed values and experimental data are likely due to limitations of the turbulence model employed, namely, the nega- tive Spalart-Allmaras model, and not due to discretization errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' This argument is further enhanced by comparisons with other codes that implement the same turbulence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' In these comparisons, virtually identical results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Therefore, any of the three gradient reconstruction schemes can, theoretically, be used in the simulation of cases similar to the ones investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' It is important to stress, however, that these conclusions are only valid for the type of mesh con- sidered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Highly stretched hybrid meshes can lead to wildly different results, and the differences between each reconstruction scheme might become more obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' This is precisely what the authors will address in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Acknowledgments The authors wish to express their gratitude to the São Paulo Research Foundation, FAPESP, which has supported the present research under the Research Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2021/00147-8 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2013/07375-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The authors also gratefully acknowledge the support for the present research provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, under the Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 309985/2013-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' The work is further supported by the computational resources of the Center for Mathematical Sciences Applied to Industry (CeMEAI), also funded by FAPESP under the Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2013/07375-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Contact Author Email Address Frederico Bolsoni Oliveira: fredericobolsoni@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='com, Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' : +55 (12) 3947-6488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' João Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' Azevedo: joaoluiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='azevedo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='com, Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' : +55 (12) 3947-6488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' References [1] “Digital Twin: Definition & Value,” an AIAA and AIA Position Paper, available in https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' aia-aerospace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content='org/report/digital-twin-paper/, last accessed in 06 feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNA0T4oBgHgl3EQfGf8S/content/2301.02046v1.pdf'} +page_content=' [2] 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a/btE0T4oBgHgl3EQfnwER/content/tmp_files/2301.02515v1.pdf.txt b/btE0T4oBgHgl3EQfnwER/content/tmp_files/2301.02515v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e46ec37cad60a21e2fc61852c8c01d65fae0acd --- /dev/null +++ b/btE0T4oBgHgl3EQfnwER/content/tmp_files/2301.02515v1.pdf.txt @@ -0,0 +1,1450 @@ +GNN-based Passenger Request Prediction +Aqsa Ashraf Makhdomi1* and Iqra Altaf Gillani1 +1*Department of Information Technology, NIT Srinagar, 190006, +J&K, India. +*Corresponding author(s). E-mail(s): +makhdoomiaqsa@gmail.com; +Contributing authors: iqraaltaf@nitsri.ac.in; +Abstract +Passenger request prediction is essential for operations planning, con- +trol, and management in ride-sharing platforms. While the demand +prediction problem has been studied extensively, the Origin-Destination +(OD) flow prediction of passengers has received less attention from +the research community. This paper develops a Graph Neural Net- +work framework along with the Attention Mechanism to predict the +OD flow of passengers. The proposed framework exploits various lin- +ear and non-linear dependencies that arise among requests originat- +ing from different locations and captures the repetition pattern and +the contextual data of that place. Moreover, the optimal size of the +grid cell that covers the road network and preserves the complexity +and accuracy of the model is determined. Extensive simulations are +conducted to examine the characteristics of our proposed approach +and its various components. The results show the superior perfor- +mance of our proposed model compared to the existing baselines. +Keywords: Ride-sharing, Route recommendation, Demand prediction, OD +prediction, GNN , Context-aware data +1 Introduction +The rapid growth of GPS-enabled services and location based sensors has +resulted in an enormous volume of geo-tagged data which provides informa- +tion about the passenger mobility patterns and vehicular movement. This data +about passenger arrival and departure from different locations can be analyzed +1 +arXiv:2301.02515v1 [cs.LG] 6 Jan 2023 + +2 +GNN-based Passenger Request Prediction +and the patterns can be exploited among them in order to predict the future +areas that will fetch more requests. This will help the ride-hailing platforms +in assigning the requests of nearby passengers to the drivers. It will ultimately +decrease the waiting time of passengers and the cruising distance of drivers +without having a rider in the vehicle. The decrease in cruising distance of vehi- +cles results in more profit for the platform as the passengers are fetched earlier +on the route. It further has a positive impact on the environment as there is +a reduction in the emission of greenhouse gases, owing to the decrease in dis- +tance travelled by the vehicle. Thus the prediction of requests enhances the +service quality of ride-hailing platforms and contributes to a greener environ- +ment. This is an important area of research and the output of these prediction +techniques are used as input by various route recommendation [1] and match- +ing algorithms [2]. It has gathered significant attention from researchers and a +number of works like [3, 4] have been done that have predicted the areas with +more passenger demand. +Most of the works [3, 4] that have been done in this direction have predicted +the areas that will fetch more requests, which is also called demand predic- +tion. However, the mobility patterns of passengers can be better analyzed if +the origin (place from where the request came) as well as destination (place +where the request is headed to) of requests can be predicted simultaneously +and not only their origin. This is particularly useful for ridesharing platforms +where efficient pairing of passengers can be done if the origin and destination +of passengers is known beforehand. It is a new area of interest and various +models have been proposed to predict the Origin-Destination (OD) pair. These +methods include LSTM [5], RNN [5] and GNN based architectures [6]. +The requests that arrive in ride-hailing platforms correlate with the neigh- +boring requests (spatial dependencies) and the requests from previous time +slots (temporal dependencies). In order to predict the future requests, these +dependencies should be captured and patterns should be analyzed among +them. Moreover, there might be some areas that don’t have enough requests, +leading to data sparsity problem. We propose to use GNNs to model the +underlying road network with missing data and spatio-temporal dependen- +cies. The proposed architecture captures the dependencies between the nodes +of the graphs through Graph Attention Networks (GANs) wherein each node +aggregates information from its neighbors based on their assigned weight. By +gathering information from neighbors, the missing data of various locations +is captured and each node has obtained a broader view of the environment. +The neighbors with which information is exchanged include spatial neighbors, +the ones that are geographically or semantically similar to the current node +and the temporal neighbors, ones which follow a regular time-based repetition +pattern. These neighbors capture the spatio-temporal dependencies among +the ride-hailing requests and exchange information with each other until the +information converges and each node has a global view of the network. +All the models so far [4, 5] have analyzed the periodic patterns among ride- +hailing requests by capturing various spatio-temporal dependencies and used + +GNN-based Passenger Request Prediction +3 +that to predict the future areas that will fetch requests. Our proposed model, +in addition to these repeating patterns also aims to capture non-repeating pat- +terns that may result in dependencies between requests. These patterns are +modelled by taking the aggregate information from the data of the preceed- +ing hours. They provide insights into the behavioral patterns of people and +describe the contextual data of that place. For instance, if the requests from +the preceding hours are low in quantity, it reveals that either the area is sparse +in requests or some event has occurred that has reduced the flow of requests +to that area. Moreover, these dependencies also reflect the travelling behavior +of people and thereby indicate the re-appearance of requests after a period of +time. For instance, requests start to recur after the office hours of people are +complete or they have completed the time spent outside their homes. These +dependencies are captured by analyzing the data from previous hours and +determining the time frame after which requests tend to re-appear. Thus the +data from previous hours provides a mechanism to deal with non-linear depen- +dencies that describe the patterns apart from the regular repeating trends that +are found in the existing studies. +Grid size is a crucial factor to consider when modelling these dependencies +since it determines the number of spatial and temporal neighbors at a given +time. When the grid size is large, the neighborhood count decreases which +results in the inability to account for spatio-temporal dependencies. On the +other hand, small grid sizes make it necessary to retain microscopic features, +which ultimately leads to an increase in the complexity of the model. Thus, +the appropriate grid size which captures the passenger mobility accurately in +a time-bound manner is a parameter that needs to be determined. +Our major contributions can be summarized as follows: +• We model the road networks and ride-hailing requests as a graph and use +GNN to capture the complex spatio-temporal dependencies. +• We determine the optimal size of grid cell considering the complexity and +accuracy of the model. +• We analyze the temporal non-linearities in the ride-hailing request sequences +and capture the contextual data of the place. +• Extensive simulations conducted on the real-world dataset demonstrate +superior performance by our proposed model. +The rest of the paper is organized as follows: In Section 2, a review of the +existing work done related to the request prediction is presented. In Section +3, the preliminaries required to understand our proposed model are described. +In Section 4 the details of the proposed model are discussed. In Section 5, the +dataset, evaluation metrics, results, and comparison of algorithms is discussed. +Finally, Section 6 concludes our work and highlights its key contributions. + +4 +GNN-based Passenger Request Prediction +2 Related Work +Ride-hailing services like Ola and Uber generate an enormous volume of data, +which includes information about trajectories, geo-tagged check-ins and ride +source and destination. The patterns among data can be learnt and understood +for the advanced development of these services in recommending optimal routes +[1] or providing matching algorithms [2] to drivers and riders. Various machine +learning and deep learning based models have been used to understand the +patterns among data and utilize those patterns for future prediction of requests +[6, 7]. These prediction algorithms for ride-hailing platforms work in two-fold +directions: predicting the demand at nodes and predicting the OD pair of +requests. Demand prediction methods predict the number of requests that will +arrive at a node and the OD based prediction methods predict the number +of requests that will arrive between a specified origin and destination pair of +vertices. +Various works have been in the direction of demand prediction. These works +have evolved from pure time series based models [7, 8] to the models exploiting +spatial and temporal dependencies [3, 4]. The time series based models exploit +the temporal patterns in data and based on these predict the future areas +that will have more requests. Since ride-hailing requests depend upon both +the spatial and temporal dependencies, some of the recent research works have +started to exploit both these dependencies by using various machine learning +and deep learning based approaches [6, 7]. +Recently, OD prediction has aroused as a potential topic among researchers +as it can enhance the functionality of ride-hailing platforms, in particular ride- +sharing systems. Various works have been done that have used GNN [6] based +models to predict the origin and destination of requests. In this direction, +Hamilton et al.[9] proposed a general induction framework, GraphSAGE, that +used the attribute information to create node embeddings for the vertices of +the graph. However, their proposed model only focused on spatial dependen- +cies and did not capture the temporal patterns in the data. Wang et al. [10] +proposed a grid embedding-based multi-task learning framework, where the +grid embedding models the spatial dependencies that can arise among requests +from different areas, and LSTM-based multi-task learning framework captures +the temporal dependencies in the data. However, their proposed model consid- +ers the requests that originate at node v or are destined to node v as the same. +Whereas, they reflect different patterns and should be captured separately. +In order to overcome the above problems, Wang et al.[6] proposed a repre- +sentation learning-based OD prediction model that leveraged the spatial and +temporal dependencies through the use of three types of neighbors: forward, +backward and geographical. Their proposed model took the directed nature +of requests into account and accordingly considered the neighbors as forward +or backward based upon their precedence of request. However, their proposed +model did not model the non-linear dependencies accurately. Our proposed +model tries to overcome the above limitations by capturing the spatial and +temporal dependencies, in particular the dependencies that can appear due + +GNN-based Passenger Request Prediction +5 +to non-linear patterns like the contextual data or the travelling behavior of +people. It is based on the architecture proposed by [6]. +3 Preliminaries +In this section, we will discuss the preliminaries associated with our proposed +model. +Grid: Our proposed model assumes that the entire city is divided into n +non-overlapping grids, denoted by g = {g1, g2, ..., gn}. Each grid is defined by +its starting coordinate and its ending coordinate. For each cell of the grid, +we predict the number of requests that originate from there and are headed +toward other cells. Figure 1 shows the example of a road network divided in +terms of grids. The grid has 25 cells and each cell i of the grid is defined by +its grid ID gi where i ϵ [1, 25]. Our proposed model predicts the number of +requests that can arrive between any two cells of the grid. Their value is stored +in the OD matrix, where the rows and columns of the matrix denote the grid +cells and the entry of matrix represents the number of requests that can arrive +between those cells. Figure 2 shows the OD matrix corresponding to the road +network described in Figure 1. As can be seen through Figure 2, each entry of +the matrix contains the number of requests (∆ij) that can arrive between the +ith cell of the grid (gi) and the jth cell of the grid (gj), where i, j ϵ [1, 25]. +Fig. 1: Road network represented in the form of grids + +g1 +92 +93 +94 +95 +96 +26 +9: +99 +910 +911 +912 +913 +914 +915 +916 +917 +g18 +919 +920 +921 +922 +923 +924 +9256 +GNN-based Passenger Request Prediction +Fig. 2: OD Matrix of the road network +Time Slots: We split time into 24 distinct slots t = {t1, t2, ..., t24} where +ti represents the time between the ith and (i − 1)th hour. For instance, t1 +corresponds to the time between 12 : 00 A.M. and 1 : 00 A.M.. +Spatio-temporal dependencies: The requests in ride-hailing platforms have +temporal dependencies as the studies have found that there are regular time- +based patterns among requests which could be exploited for further prediction. +For instance, during morning hours, the requests follow the pattern of having +a destination at the office, and during the evening hours, requests originate +from the office. These patterns can be used to predict the future origin and +destination of requests. +Requests also follow spatial dependencies and depend upon the inflow and +outflow of requests from the following areas, i.e, if the neighborhood region +has high requests, it is highly probable that the current region will have more +requests. Similarly, if a region is sparse in requests, its subsequent regions will +certainly have few requests. +4 System model +In this section, we detail out the working of our proposed architecture. As can +be seen through Figure 3, input to our proposed model is a dataset that con- +tains the request sequence R={r1, r2, ..., rm} of ride-hailing platforms, where +ri represents a particular request and m denotes the total number of requests. +Each request is represented as ri=, where rsi and rdi determine +the source and destination of request i respectively, and rt′ +i represents the time +at which the request i is made. This data is pre-processed and divided into +24 slots of 1-hour for each day and a graph G=(V, E, ∆) is created for each +time slot. The vertices V of the graph denote the cells of the grid (see Figure + +Grid +91 +92 +gj +925 +cell +g1 +1 +2 +92 +0 +2 +0 +gi +1 +2 +925 +0 +3 +2GNN-based Passenger Request Prediction +7 +Fig. 3: Framework of our proposed model +1), and the edges E represent the interconnection between different grid cells. +This graph is a complete graph as requests can arrive between any pair of ver- +tices. The adjacency matrix of the graph G is represented by the OD matrix +shown in Figure 2. Each edge of the graph G is associated with a weight ∆ij +that determines the number of requests between the grid cells gi and gj (see +Figure 2). If there are no requests between the two vertices, the correspond- +ing weight is set as 0. The output of the pre-processed stage is a sequence of +graphs G={G1, G2, ..., G24}, where Gi represents the requests that arrive in +the tth +i +time slot i.e, all the requests that arrive between (i − 1)th and ith hour +of the day. This graph sequence is generated for all the days in the dataset. +Each vertex of the graph is represented by an embedding which is its +transformation into a vector space that describes it completely and preserves +the maximal information about the local structure of the graph (connection +between nodes and edges). In our proposed model the initial embedding of +a vertex is a combination of its grid ID (gi for the cell i of grid), row num- +ber, column number, time slot ti, day of the week, in-degree, and out-degree. +The in-degree and out-degree of a node are calculated from the OD matrix. +If we represent each element of the OD matrix in Figure 2 as OD(gi, gj) +where gi represents the row grid cell and gj represents the column grid cell, +then the in-degree of the grid cell gk is �n +i=1 OD(gi, gk) and its out-degree is +�n +j=1 OD(gk, gj) Initially, the embeddings of a node contain the local infor- +mation associated with the node and edge connections. This information is +unstable as it only provides a view of the local neighborhood and does not +provide any indication of the spatial or temporal dependencies that can arise +among the requests. In order to capture the dependencies among the requests +and obtain a global view of the network, the nodes exchange embeddings with + +Neighborhood +information +Feature +Featurevectors +Spatial +Pre-process +Dataset +extraction +Attention layer +Spatialembeddings +Demand +Demandflow +Temporal +Temporal +process +embeddings +Attention layer +Origin +destination +Linear +Non-linear +dependencies +dependencies8 +GNN-based Passenger Request Prediction +Notation +Description +g +Grid +gi +ith grid cell +t +Time Slots +ti +ith time slot +R +Request sequence +ri +ith request +rsi, rdi +Source and destination of request i +rt +i +Time at which request i arrives +G +Request Graph +OD Matrix +Adjacency matrix of G +V +Vertices of graph G +E +Edges of graph G +Gi +Graph G at ti time slot +∆ij, +ˆ +∆ij +Actual and predicted number of requests between grid cells gi and gj +δi, ˆδi +Actual and predicted number of requests at grid cell gi +ft +i , bt +i, qi +Set of forward, backward and geographical neighbors of grid cell gi at time t +et +i +Embedding of node vi at time t +Wc, Ws +Learnable weight matrices +w′ +Pre-weighted aggregator +Y +Vector that concatenates the embedding of neighboring nodes +a +Attention coefficient which maps a vector to a single number +µ +LeakyReLu activation function +ft +ij, bt +ij,qt +ij +Weight of forward, backward and geographical neighbors when embeddings are exchanged +αt +j, βt +j, γt +j +Pre-weighted aggregator for forward, backward and geographical neighbors +D +Graph that stores distance between different grid cells +dij +Distance between grid cells gi and gj +h +Number of hours for which non-linearity is determined +pij +Probability that requests are transferred from grid cell gi to grid cell gj +m +Total number of requests +n +Total number of grids +Table 1: Notations +their spatial and temporal neighbors. These neighbors are selected by the fol- +lowing two layers of our proposed model: spatial attention layer - which selects +the spatial neighbors and the temporal attention layer - which selects the tem- +poral neighbors. After the final embeddings are calculated through these two +layers, we need to predict the demand (ˆδi) that can arrive at the ith grid cell +and the number of requests ( ˆ +∆ij) that may arrive between the ith and jth +grid cells. Here ˆδi and +ˆ +∆ij correspond to the predicted demand at grid cell +i and the predicted number of requests between grid cells gi and gj respec- +tively. Whereas δi and ∆ij represent the actual demand at grid cell i and the +actual number of requests between grid cells gi and gj respectively. We get the +demand at a node and the number of requests between two nodes by feeding +the result of these two layers to the transferring attention layer. The working +of these layers is described in detail in the next subsections. +4.1 Spatial Attention Layer +After the initial embedding of nodes is created, it is fed as input to the spa- +tial attention layer which produces a new embedding that carries information +about all the spatial neighbors. These embeddings are created by exchanging +data with three different types of spatial neighbors: +Forward neighbors - If there are two neighbors gi and gj, and there is at +least one request that originates from gi and is destined to gj (∆ij > 0) then +gj is the forward neighbor of gi. The set of forward neighbors of gi at time slot + +GNN-based Passenger Request Prediction +9 +Fig. 4: Forward and backward neighbors calculated from an instance of graph +G +t is defined mathematically as: +f t +i = {gj|∆t +ij > 0, ∆t +ij ϵ Gt} +(1) +Backward neighbors - If there are two neighbors gi and gj, and there is at +least one request that originates from gj and is destined to gi then gj is the +backward neighbor of gi. The set of backward neighbors of gi at time slot t is +defined mathematically as: +bt +i = {gj|∆t +ji > 0, ∆t +ji ϵ Gt} +(2) +The sequential flow of requests in the network is captured by the forward +and backward neighbors, which are also referred to as semantic neighbors. +These neighbors identify passenger mobility patterns and determine the flow +of requests into and out of the specific region. These neighbors are time- +dependent, as the request flow in a region is not constant across different time +slots. They are calculated from an instance of graph G. +Figure 4 shows the set of forward and backward neighbors of the grid cell +g14 at a particular time slot. The forward neighbors determine the out-flow +of requests from a particular grid cell and the backward neighbors determine +the in-flow of requests to a particular grid cell. As can be seen through Figure +4, {g19,g21, g28} is the set of forward neighbors of grid cell g14 as there are +some requests that originate from g14 and are destined towards g19, g21 and +g28 respectively. Similarly {g7, g8, g9} is the set of backward neighbors of grid +cell g14 as the requests from grid cells g7,g8 and g9 have their destination at +grid cell g14. +Geographical neighbors: Two neighbors gi and gj are said to be geo- +graphically connected if the Haversine distance between their corresponding + +FORWARD +NEIGHBORS +6 +(919,921,928] +919 +9: +914 +921 +66 +928 +BACKWARD +NEIGHBORS +(97.98.99]10 +GNN-based Passenger Request Prediction +latitude/longitude pairs is within a specified threshold distance. +qi = {gj|dij ≤ L, dij ϵ D} +(3) +where L is the threshold distance that determines the size of geographi- +cal neighbors and D is the distance graph that represents the distance (dij) +between nodes gi and gj. For instance, if we consider the threshold distance +equal to the length of one grid, then the set of geographical neighbors of grid +cell g14 for the example shown in Figure 2 is {g8, g9, g10, g13, g15, g18, g19, g20}. +The set of geographical neighbors of gi is constant across all time slots. +The geographical neighbors can be used to aggregate uncertainty in infor- +mation from the areas with few requests (sparse areas). As we know that +requests arrive in negligible quantity in sparse areas, and the forward and back- +ward neighbors capture the dependencies between the neighbors based on the +request flow of a particular region. But, if a region is sparse in requests it will +not have in-flow and out-flow of requests. In that case, there is no information +from forward and backward neighbors. However, geographical neighbors are +always there and can be used to exchange embeddings in that area. +The initial embeddings of nodes are fed as input to the Graph Attention +Network (GAN) which combines the information from the three neighbors +described above and represents it in the form of a unified vector et +i for each node +vi at time t. Some of the earlier models have used Graph Convolution Network +(GCN) [9] for combining information from different neighbors. However, with +GCN all neighbors are assigned the same weightage when embeddings are +merged, which neglects the importance of nodes that have a similar flow of +requests or that are close to each other. We propose to use GAN which samples +different neighbors based on their weight. It assigns a higher value to the +geographical nodes that are in the close vicinity of the current node. Similarly, +it provides more weightage to the semantic neighbors with a higher flow of +requests to/from the current node. In this way, embeddings of nodes that +have more information are prioritized and the noise from redundant nodes is +removed. +In order to calculate the importance of a neighborhood node vj at time t, we +pass the embedding of the current node et +i ϵ Rz∗1 and the neighborhood node +et +j ϵ Rz∗1 through a weight matrix Wc ϵ Rz +′∗z, where z′ > z. This weight matrix +acts as a single-layer neural network and projects the embedding of a node to +a higher dimension. The output of this layer are the updated embedding of +the nodes viand vj at time t and they are equal to +et +i = Wcet +i +(4) +et +j = Wcet +j +(5) +where et +i ϵ Rz +′∗1 and et +j ϵ Rz +′∗1. Figure 5 displays the working of the attention- +based aggregator of GAN. The input to GAN is the embedding et +i and et +j of +nodes vi and vj at time t, after they have been passed through weight matrix + +GNN-based Passenger Request Prediction +11 +eti1 +eti2 +etiz' +etj2 +etjz' +z'*1 +z'*1 +z'*1 +z'*1 +eti1 +eti2 +etiz' +w'etj1 +eti1 +eti2 +w'etj1 +w'etjz' +etiz' +2z'*1 +aT +L +xtij +AN(eti,w'etj) +. +. +. +. +. +. +. +. +. +. +LeakyReLu +eti=Wceti +etj=Wcetj +w'etj +eti +Y=eti w'etj +Pre-weighted +aggregator +etj1 +w'etjz' +w'etj2 +aTY +soft-max +Fig. 5: Graph Attention Network for calculating affinity between neighboring +nodes +Wc. The embedding et +j is passed through a pre-weighted aggregator w +′ which +provides a prior weight to it before the GAN calculates its importance. This +prior weight is based upon the current state of neighbors and is described in +detail in the next subsection. The embedding et +i of node i and the embedding of +the neighborhood node after being passed through a pre-weighted aggregator +(w +′et +j) are concatenated and represented as a single vector Y : +Y = et +i ⊕ w +′et +j +(6) +This vector is then passed through a learnable attention coefficient a ϵ R2z +′∗1 +which maps it to a single number. Thereafter, non-linearity is applied through +the LeakyReLu activation function denoted as µ in Eq.(7). +AN(et +i, w′et +j) = µ(aT Y ) +(7) +This attention-based aggregator function of GAN measures the affinity +between the embedding of node vi and its neighbor vj at time t by learning +the weight matrix Wc and the attention coefficient a, and produces a single +number that determines the weight of neighborhood node vj when it needs to +exchange embedding with the node vi. +However, the output of the neural network is not normalized, which is a +problem since the weights should be on the same scale for exchanging embed- +dings. In order to normalize the weights, we apply the soft-max function to +the output of the attention-based aggregator defined in Eq. (7) which brings +all the weights to the same scale. These normalized weights are denoted as xt +ij +in Figure 5 and they denote the weights of forward neighbors (f t +ij), backward + +12 +GNN-based Passenger Request Prediction +neighbors (bt +ij), and geographical neighbors (gij). They are mathematically +represented as: +f t +ij = +exp(AN(et +i, αt +jet +j)) +� +k ϵf t +i exp(AN(et +i, αt +ket +k)) +bt +ij = +exp(AN(et +i, βt +jet +j)) +� +k ϵbt +i exp(AN(et +i, βt +ket +k)) +qt +ij = +exp(AN(et +i, γjet +j)) +� +k ϵqt +i exp(AN(et +i, γket +k)) +(8) +These weights allow our proposed model to prioritize embeddings that are +geographically and semantically similar to the node vi at time t. Here, αt +j, βt +j, +and γj refer to the pre-weighted functions which are detailed out in the next +subsection. +Since we have obtained the weights that determine the importance of +each node, we exchange embeddings with the set of forward, backward and +geographical neighbors with the weights f t +ij, bt +ij and qt +ij calculated above as: +et +i = Wset +i ⊕ +� +jϵf t +i +f t +ijWset +j ⊕ +� +jϵbt +i +bt +ijWset +j ⊕ +� +jϵqi +qt +ijWset +j +(9) +Here et +i is the final embedding of each node vi at time t and it is obtained by +merging information from different set of neighbors based upon their weights +and Ws is the shared weight matrix that projects all embeddings onto the +same z +′ dimensional space before merging them. +4.1.1 Pre-weighted aggregator +The pre-weighted aggregator provide a prior weight to the embedding of node +vj at time t before its affinity is measured by the attention-based aggregator +defined in Eq. (7). The aggregator determines the current flow of requests, +which is calculated from the edges (∆ij) of the graph Gt, and the geographi- +cal relationship between requests, which is determined through the edges (dij) +of graph D. It allows our proposed model to sense the sparsity or density of +requests in the current time slot and accordingly aggregate the information +from the forward and backward neighbors that have a higher flow of requests. +It also provides an indication of the geographical neighbors which are closer to +the requests that arrive at time t. These pre-weighted aggregators are repre- +sented as αt +j, βt +j and γj for the forward, backward and geographical neighbors +respectively and are defined as: + +GNN-based Passenger Request Prediction +13 +αt +j = +∆ij +� +jϵf t +i ∆ij + s +βt +j = +∆ij +� +jϵbt +i ∆ij + s +γj = +1 +dij +� +jϵqt +i +1 +dij +(10) +s is a small additive term to prevent the case when the denominator is equal +to 0. The weights αt +j and βt +j represent the intensity of passenger demand at +time t and the weight γj corresponds to the distance of request from node i. +These weights help our proposed model to prioritize the embeddings that are +geographically or semantically closer to node vi at time t. +4.2 Temporal Attention Layer +After the data is filtered through spatial attention layer, we get a low-level +vector that carries information about all the neighborhood nodes and has an +overview of the spatial dependencies of the graph. In order to capture the tem- +poral dependencies among the learned representations, these node embeddings +are exchanged with time-based neighbors using a temporal attention layer. The +embeddings are exchanged through the attention-based mechanism described +for spatial attention layer. This layer has 4 channels: the first and second chan- +nels analyze the dependencies among requests that arrive in the preceding and +consecutive hour of the previous 7 days, and the third channel finds out depen- +dencies among requests from the same hour of the previous 7 days. These three +channels capture the linear dependencies that arise due to the regular pat- +terns in data like the morning and evening rush hours. However, there could +be non-linear dependencies which show up due to the recent events that might +have taken place at the preceding time intervals or that reveal the travelling +patterns of people. These dependencies are monitored by taking the data from +previous h hours, where the value of h is determined through experimental +evaluations. Their value provides an indication of customer flow in the region +around the time request arrives and thereby helps our proposed model to cap- +ture the patterns apart from the morning and evening hour data repetition +trends that are found in the existing studies. These non-linear dependencies +are described in detail next. +4.2.1 Context-aware data +Context-aware data refers to the data about the surrounding environment of +a location and it provides an indication of the events that might have taken +place around that location. This data is collected through various software and +hardware tools and it has been used to modify the behavior of various systems. + +14 +GNN-based Passenger Request Prediction +In the case of ride-hailing platforms, context-aware data can be used to direct +the routes to users, considering the local events that might have taken place +around that place. For instance, if the context-aware data displays that there +is road construction in the area, then the vehicles are recommended to divert +their route and travel through a different path in order to ensure the passengers +reach their destination on time. Similarly, if the context-aware data displays +that the area is free from traffic jams, then the vehicles are directed towards +that route to provide the users with a hustle-free transportation service. Var- +ious works have been done that have used context-aware data to direct safe +[11], pleasant [12], or personalized routes [13] to the users. These works have +used this data to modify the behavior of the system and direct users towards +the routes that take into account the local condition of the place. +Our proposed model also uses context-aware data to capture the events +that might have taken place around the location at a particular period of +time. However, in our proposed model the data is not gathered from expensive +software and hardware tools. Rather it is collected by the non-linear channel of +the temporal attention layer. This layer captures all the customer requests that +have arrived in that area for the preceding h hours and uses that to analyze +the local environment of that place. The data from preceding h hours provides +an indication of one of the following things: sparsity or density of requests in +the area or the occurrence of an event at that place. +If the data for the past h hours shows that there are few requests in the area, +there may have been some impulsive event that reduced the flow of requests +in and out of the area. This event could correspond to heavy rainfall, snowfall, +or a security breach in that area. If there are no events scheduled at the place +and the customer demand is still low then the area may have few requests, to +begin with, meaning that people in that area do not use ride-hailing platforms +for their daily commute. In both cases, there is a high probability that the area +will have few requests in the next hour. This context-aware data thus helps our +proposed model to anticipate future requests according to the local conditions +of the place. Similarly, if the previous hours are abundant in requests, there +might be an upcoming event in that area - a football game, festival sale, or +any other event that influences passengers. However, if there are no events +scheduled and the customer demand is high, it reveals that the area is dense +in requests and there is always a higher flow of passengers in and out of that +area. These patterns reveal important contextual information and can be used +to monitor the flow of passengers in the region. They help our proposed model +to capture local events that might have taken place around the location where +the request needs to be predicted. These dependencies are captured by the +non-linear channel of the temporal attention layer. +4.2.2 Travelling Behavior +Non-linear dependencies can also arise due to the travelling patterns of people. +According to a behavioral study, people spend 5 to 6 hours outside of their +homes during holidays [14], which can include time spent in their garden, + +GNN-based Passenger Request Prediction +15 +walking, or travelling in cars. Thus, after this time-period people tend to return +to their original location by using any mode of transportation. This suggests +that requests start to recur after this time and these patterns can be exploited +to predict the future requests. Further, during weekdays people go to their +homes after their office hours are complete. As data from various countries +reveals that people spend on average 2 − 6 hours in their workplace [15] this +pattern can be exploited to predict the travelling behavior of people during +weekdays. Moreover, there are other recurring patterns such as shopping [15], +etc, which can be used to predict the future occurrence of requests. Thus, +the data from previous hours provides an indication of travelling behavior +of people and it reveals their office hours, recreation time, shopping hours, +and other hobbies that can be used to predict future requests. This behavior +is captured by the non-linear channel of the temporal attention layer and it +provides insights into the recurring pattern of requests in ride-hailing platforms +which can be used to predict the future occurrence of requests. +4.3 Transferring Attention Layer +After the data is modelled for spatial and temporal patterns, we pass it through +feed-forward neural network layer to get the number of requests at each node +of a graph which represents the demand in the region. The demand is rep- +resented as a matrix ˆδ = { ˆδ1, ˆδ2, ..., ˆδn}. In order to represent the origin and +destination of requests, transferring attention layer is used which models the +transmission from one node to another based upon the transferring probabil- +ity pij. pij represents the probability that requests are transferred from node i +to node j and it is mathematically represented as +pij = +exp(AN(eT +i , eT +j )) +�n +i=j exp(AN(eT +i , eT +j )) +(11) +OD pair ( ˆ +∆ij) is calculated from demand by using transferring probability pij +as follows: +ˆ +∆ij = ˆδ · pij +(12) +5 Prototype Implementation +In this section, we evaluate our proposed model based upon extensive simula- +tions. +Dataset: We conduct experiments on the real-world taxi dataset. The dataset +is generated for New York City and contains data of the month of February. +Experimental Settings: We evaluate the prediction accuracy of our proposed +model based upon the two widely applied metrics: Mean Absolute Percentage + +16 +GNN-based Passenger Request Prediction +Table 2: Simulation to check the length of a grid cell +Task +Grid +length +(km) +MAPE-0 +MAPE-3 +MAPE-5 +MAE-0 +MAE-3 +MAE-5 +OD +2.3 +2.4 +2.5 +2.6 +2.7 +0.4191 +0.3913 +0.3866 +0.3876 +0.3985 +0.4036 +0.3646 +0.3488 +0.3513 +0.3850 +0.3838 +0.3466 +0.3269 +0.3299 +0.3715 +5.8907 +5.8756 +5.4249 +5.4508 +6.4091 +15.3091 +15.3788 +14.0479 +14.1229 +16.9501 +19.3616 +19.5304 +17.7843 +17.8809 +21.5885 +Demand +2.3 +2.4 +2.5 +2.6 +2.7 +0.4350 +0.4173 +0.4222 +0.4237 +0.4405 +0.3880 +0.3688 +0.3612 +0.3627 +0.3918 +0.3671 +0.3466 +0.3354 +0.3362 +0.3717 +53.7656 +60.7930 +46.7006 +46.7409 +60.4972 +94.1438 +106.6110 +81.6304 +81.6977 +106.0421 +106.8488 +121.0631 +92.5871 +92.6591 +120.4107 +Error (MAPE) and Mean Absolute Error (MAE) which are defined as: +MAPE = 1 +m +m +� +i=1 +���� +ˆyi − yi +yi + 1 +���� +(13) +MAE = 1 +m +m +� +i=1 +�� ˆyi − yi +�� +(14) +where m denotes the number of examples, ˆyi denotes the predicted result and +yi represents the actual result. We estimate the performance of our proposed +model on areas with varying levels of customer base and accordingly calculate +MAPE-0, MAPE-3, and MAPE-5 where 0, 3, and 5 denote the minimum +number of requests in different areas. We have used this threshold to see the +patterns in different areas according to their customer base. +In the experimental work, we used 75% of data for training purposes and +kept the remaining 25% for testing purposes. In the training settings, 10% of +data is used as a validation dataset and is used for hyperparameter testing. We +implement our model with PyTorch 1.11.0 on Python 3.9. All the simulations +were run on Windows i7 with 16 GB RAM for 200 epochs. The model was +trained with a batch size of 1 and a learning rate of 0.001. Based on these +parameters, we evaluate the working of our proposed model. +5.1 Grid length +The length of a grid cell is an important parameter that decides the complexity +and accuracy of the proposed model. If the length is large, then the origin and +destination of the majority of requests will be concentrated within a single grid +cell and the OD prediction will reduce to demand prediction. If it is narrow, +the time complexity of the model increases. Thus the length of a particular +grid cell is an important parameter and needs to be determined. +Table 2 displays the performance of the proposed model based upon the +parameters of MAPE and MAE, with the increase in length of a grid cell. +There is a rapid decrease in the error of the model when the length increases +from 2.3 km to 2.5 km. After that, the error becomes stagnant for the grid cells + +GNN-based Passenger Request Prediction +17 +of length 2.5 km and 2.6 km. Thereafter, the error is found to increase again. +The main perspective behind this behavior can be the change in data size and +the number of neighbors on the varying lengths of grid cells. When the length +of grid cell is small, there are multiple cells with no requests, and the spatial +dependencies are not represented accurately by this size. Moreover, with small +length time complexity of the model is high. However, when the length of grid +cell increases, dataset size decreases, and the number of different neighbors +changes which brings about an increase in error. Further, with an increase in +grid cell length, the OD task is found to converge to demand prediction. Thus +the optimal value of the length of grid cells needs to be determined which +reflects passenger mobility and performs effectively in a time-bound manner. +Based on the experiments we have set the length of the grid cells as 2.5 km +for our proposed model as it performs well under all the evaluation metrics. +5.2 Non-linearities +As already stated, the non-linearities in our proposed model are captured by +the fourth channel of the temporal attention layer. This layer monitors the +data from previous h hours and gives an indication of the events that might +have taken place around the location during these hours. However, to properly +monitor the events around that place, the value of h needs to be determined +through simulations. Figures 6, 7, 8 and 9 show the MAPE and MAE of the +proposed model for OD and demand prediction with varying time hours. As +can be seen through these figures, when the data from the previous 3−6 hours +is used, the model is found to perform well under all the evaluation metrics. +Particularly, the data from the previous 6 hours is found to perform best under +all conditions. +As already stated, the non-linear data provides context-aware information +about the location and determines the average flow of requests in the area over +a period of time. For instance, if the data from previous 3 − 6 hours contains +few requests then that area may be sparse or some event may have occurred +like rainfall, security breach, etc which reduced the flow of requests to that +area. Similarly, if the previous hours have more requests then that area will +either be high demand area or some event might be scheduled at that place +which has increased its demand. +However, among the past 3 − 6 hour data, the data from the previous 6 +hours is found to perform best under all the evaluation metrics. The intuitive +explanation might be that it reveals the travelling behavior of people and +represents the average time spent outside by users. +As the office timings in +different countries vary from 2−6 hours [15] and the demand re-arises after this +time period, it provides one of the explanations for the data from the previous +6 hours to perform well. Moreover, a behavioral study [14] has found that +on average people spend 5 − 6 hours outside their homes during holidays (it +can include time spent in their garden, time doing walk or travelling in cars). +It provides further explanation for using previous 6 hour data to model non- +linearity. Thus, from experimental studies and behavioral patterns of people, + +18 +GNN-based Passenger Request Prediction +5 +10 +15 +20 +0.36 +0.37 +0.38 +0.39 +0.4 +0.41 +0.42 +0.43 +Number of hours +MAPE-0 +(a) MAPE-0 for OD pre- +diction +5 +10 +15 +20 +0.3 +0.32 +0.34 +0.36 +0.38 +0.4 +Number of hours +MAPE-3 +(b) MAPE-3 for OD pre- +diction +0 +5 +10 +15 +20 +25 +0.28 +0.3 +0.32 +0.34 +0.36 +0.38 +Number of hours +MAPE-5 +(c) MAPE-5 for OD predic- +tion +Fig. 6: MAPE for OD prediction +5 +10 +15 +20 +4 +4.5 +5 +5.5 +6 +6.5 +Number of hours +MAE-0 +(a) MAE-0 for OD predic- +tion +5 +10 +15 +20 +11 +12 +13 +14 +15 +16 +Number of hours +MAE-3 +(b) MAE-3 for OD predic- +tion +5 +10 +15 +20 +14 +15 +16 +17 +18 +19 +20 +21 +Number of hours +MAE-5 +(c) MAE-5 for OD predic- +tion +Fig. 7: MAE for OD prediction +5 +10 +15 +20 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +Number of hours +MAPE-0 +(a) MAPE-0 for demand +prediction +5 +10 +15 +20 +0.3 +0.35 +0.4 +0.45 +0.5 +Number of hours +MAPE-3 +(b) MAPE-3 for demand +prediction +0 +5 +10 +15 +20 +25 +0.26 +0.28 +0.3 +0.32 +0.34 +0.36 +0.38 +0.4 +MAPE-5 +Number of hours +(c) MAPE-5 for demand +prediction +Fig. 8: MAPE for demand prediction +5 +10 +15 +20 +35 +40 +45 +50 +55 +60 +65 +MAE-0 +Number of hours +(a) MAE-0 for demand pre- +diction +5 +10 +15 +20 +60 +70 +80 +90 +100 +110 +Number of hours +MAE-3 +(b) +MAE-3 +for +demand +prediction +5 +10 +15 +20 +40 +60 +80 +100 +120 +140 +Number of hours +MAE-0 +(c) MAE-5 for demand pre- +diction +Fig. 9: MAE for demand prediction +we conclude that data from previous 6 hours reflects passenger mobility well +and gives an indication of customer re-appearance over different areas. + +GNN-based Passenger Request Prediction +19 +Table 3: Repetition pattern of requests +Task +Ho- +urs +MAPE- +0 +MAPE- +3 +MAPE- +5 +MAE-0 +MAE-3 +MAE-5 +OD +6 +23 +0.3705 +0.3762 +0.3186 +0.3200 +0.2962 +0.2961 +4.5922 +4.5285 +11.6408 +11.4151 +14.6635 +14.3549 +Demand +6 +23 +0.3781 +0.4493 +0.3201 +0.3797 +0.2928 +0.3479 +39.5250 +40.9023 +69.0315 +71.3015 +78.2562 +80.7730 +Table 4: Comparison on OD and demand prediction for different methods +Task +Method +MAPE- +0 +MAPE- +3 +MAPE- +5 +MAE-0 +MAE-3 +MAE-5 +OD +LSTNet +GallatExt +Proposed +model +0.3969 +0.3866 +0.3705 +0.3650 +0.3488 +0.3186 +0.3445 +0.3269 +0.2962 +5.3667 +5.4249 +4.5922 +13.8395 +14.0479 +11.6408 +17.4884 +17.7843 +14.6635 +Demand +LSTNet +GallatExt +Proposed +model +0.4024 +0.4222 +0.3798 +0.3634 +0.3612 +0.3372 +0.3405 +0.3354 +0.3183 +50.1277 +46.7006 +39.5250 +87.7881 +81.6304 +69.0315 +99.6196 +92.5871 +78.2562 +Data from previous 20−23 hours is also found to perform well, in particular +for OD prediction tasks as can be seen through Table 3 and Figures 6 and +7. The main insight that we get from this pattern is the repeating sequence +of requests the following day. As the 23 hour data captures the average flow +of requests for a day and this pattern repeats the following day, therefore the +origin and destination of requests are predicted accurately. +Thus we conclude that data from previous 6 hours reflects the passenger +travelling behavior and provides the contextual data of that place which can +be used to analyze dependencies and predict the future demand and OD pair +of requests in that area. Whereas the data from previous 23 hours captures +the repeating trend in requests and uses that to predict the future OD pair. +5.3 Comparison with previous models +We evaluate the performance of our proposed model by comparing it with the +following baselines: +LSTNet[5]: It captures the spatio-temporal dependencies in ride-hailing +requests through CNN and LSTM. +GallatExt [6]: It predicts the origin and destination of requests through +GNN. +Table 4 displays the performance of LSTNet, GallatExt, and our proposed +model on all the evaluation parameters. Our proposed model performs better +under all the metrics. This is because we have used context-aware data and +analyzed the travelling pattern of users and used that to predict the future +occurrence of requests. These patterns reveal the reason behind the non-linear +temporal dependencies that arise in data and when these patterns are paired +with the linear patterns like the morning and evening rush hours the model + +20 +GNN-based Passenger Request Prediction +is found to capture different types of dependencies and thereby surpass the +existing models in performance. +6 Conclusion +In this paper, we define passenger mobility through origin-destination pre- +diction that is helpful in directing optimal routes to drivers and providing +matching algorithms for drivers and passengers. OD prediction is complex in +comparison to demand prediction as it predicts the demand in a certain region +and the destination of these demands. We have used GNN to capture the +spatio-temporal dependencies among requests and predicted the OD pair of +requests. We have particularly focused on temporal non-linearities that arise +due to contextual events or the travelling patterns of people. While modelling +these dependencies, the length of the grid cell is an important parameter and +it can regulate the neighborhood count. We have determined its value through +extensive simulations. The parameters that decide the grid cell length and +the non-linearities are estimated for demand prediction models also. Exten- +sive simulations determine superior performance by our proposed model as +compared to the existing baselines. +References +[1] Garg, N., Ranu, S.: Route recommendations for idle taxi drivers: Find +me the shortest route to a customer! In: Proceedings of the 24th ACM +SIGKDD International Conference on Knowledge Discovery & Data Min- +ing. KDD ’18, pp. 1425–1434. Association for Computing Machinery, New +York, NY, USA (2018) +[2] Gao, J., Li, X., Wang, C., Huang, X.: Bm-ddpg: An integrated dispatch- +ing framework for ride-hailing systems. 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Accessed: 2022-10-13 (2020) + diff --git a/btE0T4oBgHgl3EQfnwER/content/tmp_files/load_file.txt b/btE0T4oBgHgl3EQfnwER/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6f6bebdba4996ab13ea80a00c9dc8ce58d6484b --- /dev/null +++ b/btE0T4oBgHgl3EQfnwER/content/tmp_files/load_file.txt @@ -0,0 +1,626 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf,len=625 +page_content='GNN-based Passenger Request Prediction Aqsa Ashraf Makhdomi1* and Iqra Altaf Gillani1 1*Department of Information Technology, NIT Srinagar, 190006, J&K, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' E-mail(s): makhdoomiaqsa@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Contributing authors: iqraaltaf@nitsri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Abstract Passenger request prediction is essential for operations planning, con- trol, and management in ride-sharing platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' While the demand prediction problem has been studied extensively, the Origin-Destination (OD) flow prediction of passengers has received less attention from the research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This paper develops a Graph Neural Net- work framework along with the Attention Mechanism to predict the OD flow of passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The proposed framework exploits various lin- ear and non-linear dependencies that arise among requests originat- ing from different locations and captures the repetition pattern and the contextual data of that place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Moreover, the optimal size of the grid cell that covers the road network and preserves the complexity and accuracy of the model is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Extensive simulations are conducted to examine the characteristics of our proposed approach and its various components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The results show the superior perfor- mance of our proposed model compared to the existing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Keywords: Ride-sharing, Route recommendation, Demand prediction, OD prediction, GNN , Context-aware data 1 Introduction The rapid growth of GPS-enabled services and location based sensors has resulted in an enormous volume of geo-tagged data which provides informa- tion about the passenger mobility patterns and vehicular movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This data about passenger arrival and departure from different locations can be analyzed 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='02515v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='LG] 6 Jan 2023 2 GNN-based Passenger Request Prediction and the patterns can be exploited among them in order to predict the future areas that will fetch more requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This will help the ride-hailing platforms in assigning the requests of nearby passengers to the drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It will ultimately decrease the waiting time of passengers and the cruising distance of drivers without having a rider in the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The decrease in cruising distance of vehi- cles results in more profit for the platform as the passengers are fetched earlier on the route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It further has a positive impact on the environment as there is a reduction in the emission of greenhouse gases, owing to the decrease in dis- tance travelled by the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus the prediction of requests enhances the service quality of ride-hailing platforms and contributes to a greener environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This is an important area of research and the output of these prediction techniques are used as input by various route recommendation [1] and match- ing algorithms [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It has gathered significant attention from researchers and a number of works like [3, 4] have been done that have predicted the areas with more passenger demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Most of the works [3, 4] that have been done in this direction have predicted the areas that will fetch more requests, which is also called demand predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, the mobility patterns of passengers can be better analyzed if the origin (place from where the request came) as well as destination (place where the request is headed to) of requests can be predicted simultaneously and not only their origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This is particularly useful for ridesharing platforms where efficient pairing of passengers can be done if the origin and destination of passengers is known beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It is a new area of interest and various models have been proposed to predict the Origin-Destination (OD) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These methods include LSTM [5], RNN [5] and GNN based architectures [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The requests that arrive in ride-hailing platforms correlate with the neigh- boring requests (spatial dependencies) and the requests from previous time slots (temporal dependencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In order to predict the future requests, these dependencies should be captured and patterns should be analyzed among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Moreover, there might be some areas that don’t have enough requests, leading to data sparsity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We propose to use GNNs to model the underlying road network with missing data and spatio-temporal dependen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The proposed architecture captures the dependencies between the nodes of the graphs through Graph Attention Networks (GANs) wherein each node aggregates information from its neighbors based on their assigned weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' By gathering information from neighbors, the missing data of various locations is captured and each node has obtained a broader view of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The neighbors with which information is exchanged include spatial neighbors, the ones that are geographically or semantically similar to the current node and the temporal neighbors, ones which follow a regular time-based repetition pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These neighbors capture the spatio-temporal dependencies among the ride-hailing requests and exchange information with each other until the information converges and each node has a global view of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' All the models so far [4, 5] have analyzed the periodic patterns among ride- hailing requests by capturing various spatio-temporal dependencies and used GNN-based Passenger Request Prediction 3 that to predict the future areas that will fetch requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Our proposed model, in addition to these repeating patterns also aims to capture non-repeating pat- terns that may result in dependencies between requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These patterns are modelled by taking the aggregate information from the data of the preceed- ing hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' They provide insights into the behavioral patterns of people and describe the contextual data of that place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' For instance, if the requests from the preceding hours are low in quantity, it reveals that either the area is sparse in requests or some event has occurred that has reduced the flow of requests to that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Moreover, these dependencies also reflect the travelling behavior of people and thereby indicate the re-appearance of requests after a period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' For instance, requests start to recur after the office hours of people are complete or they have completed the time spent outside their homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These dependencies are captured by analyzing the data from previous hours and determining the time frame after which requests tend to re-appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus the data from previous hours provides a mechanism to deal with non-linear depen- dencies that describe the patterns apart from the regular repeating trends that are found in the existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Grid size is a crucial factor to consider when modelling these dependencies since it determines the number of spatial and temporal neighbors at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' When the grid size is large, the neighborhood count decreases which results in the inability to account for spatio-temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' On the other hand, small grid sizes make it necessary to retain microscopic features, which ultimately leads to an increase in the complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus, the appropriate grid size which captures the passenger mobility accurately in a time-bound manner is a parameter that needs to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Our major contributions can be summarized as follows: We model the road networks and ride-hailing requests as a graph and use GNN to capture the complex spatio-temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We determine the optimal size of grid cell considering the complexity and accuracy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We analyze the temporal non-linearities in the ride-hailing request sequences and capture the contextual data of the place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Extensive simulations conducted on the real-world dataset demonstrate superior performance by our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The rest of the paper is organized as follows: In Section 2, a review of the existing work done related to the request prediction is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In Section 3, the preliminaries required to understand our proposed model are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In Section 4 the details of the proposed model are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In Section 5, the dataset, evaluation metrics, results, and comparison of algorithms is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Finally, Section 6 concludes our work and highlights its key contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4 GNN-based Passenger Request Prediction 2 Related Work Ride-hailing services like Ola and Uber generate an enormous volume of data, which includes information about trajectories, geo-tagged check-ins and ride source and destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The patterns among data can be learnt and understood for the advanced development of these services in recommending optimal routes [1] or providing matching algorithms [2] to drivers and riders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Various machine learning and deep learning based models have been used to understand the patterns among data and utilize those patterns for future prediction of requests [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These prediction algorithms for ride-hailing platforms work in two-fold directions: predicting the demand at nodes and predicting the OD pair of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Demand prediction methods predict the number of requests that will arrive at a node and the OD based prediction methods predict the number of requests that will arrive between a specified origin and destination pair of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Various works have been in the direction of demand prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These works have evolved from pure time series based models [7, 8] to the models exploiting spatial and temporal dependencies [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The time series based models exploit the temporal patterns in data and based on these predict the future areas that will have more requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Since ride-hailing requests depend upon both the spatial and temporal dependencies, some of the recent research works have started to exploit both these dependencies by using various machine learning and deep learning based approaches [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Recently, OD prediction has aroused as a potential topic among researchers as it can enhance the functionality of ride-hailing platforms, in particular ride- sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Various works have been done that have used GNN [6] based models to predict the origin and destination of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In this direction, Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' [9] proposed a general induction framework, GraphSAGE, that used the attribute information to create node embeddings for the vertices of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, their proposed model only focused on spatial dependen- cies and did not capture the temporal patterns in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' [10] proposed a grid embedding-based multi-task learning framework, where the grid embedding models the spatial dependencies that can arise among requests from different areas, and LSTM-based multi-task learning framework captures the temporal dependencies in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, their proposed model consid- ers the requests that originate at node v or are destined to node v as the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Whereas, they reflect different patterns and should be captured separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In order to overcome the above problems, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' [6] proposed a repre- sentation learning-based OD prediction model that leveraged the spatial and temporal dependencies through the use of three types of neighbors: forward, backward and geographical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Their proposed model took the directed nature of requests into account and accordingly considered the neighbors as forward or backward based upon their precedence of request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, their proposed model did not model the non-linear dependencies accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Our proposed model tries to overcome the above limitations by capturing the spatial and temporal dependencies, in particular the dependencies that can appear due GNN-based Passenger Request Prediction 5 to non-linear patterns like the contextual data or the travelling behavior of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It is based on the architecture proposed by [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 3 Preliminaries In this section, we will discuss the preliminaries associated with our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Grid: Our proposed model assumes that the entire city is divided into n non-overlapping grids, denoted by g = {g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=', gn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Each grid is defined by its starting coordinate and its ending coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' For each cell of the grid, we predict the number of requests that originate from there and are headed toward other cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Figure 1 shows the example of a road network divided in terms of grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The grid has 25 cells and each cell i of the grid is defined by its grid ID gi where i ϵ [1, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Our proposed model predicts the number of requests that can arrive between any two cells of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Their value is stored in the OD matrix, where the rows and columns of the matrix denote the grid cells and the entry of matrix represents the number of requests that can arrive between those cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Figure 2 shows the OD matrix corresponding to the road network described in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As can be seen through Figure 2, each entry of the matrix contains the number of requests (∆ij) that can arrive between the ith cell of the grid (gi) and the jth cell of the grid (gj), where i, j ϵ [1, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 1: Road network represented in the form of grids g1 92 93 94 95 96 26 9: 99 910 911 912 913 914 915 916 917 g18 919 920 921 922 923 924 9256 GNN-based Passenger Request Prediction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 2: OD Matrix of the road network Time Slots: We split time into 24 distinct slots t = {t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=', t24} where ti represents the time between the ith and (i − 1)th hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' For instance, t1 corresponds to the time between 12 : 00 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' and 1 : 00 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='. Spatio-temporal dependencies: The requests in ride-hailing platforms have temporal dependencies as the studies have found that there are regular time- based patterns among requests which could be exploited for further prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' For instance, during morning hours, the requests follow the pattern of having a destination at the office, and during the evening hours, requests originate from the office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These patterns can be used to predict the future origin and destination of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Requests also follow spatial dependencies and depend upon the inflow and outflow of requests from the following areas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='e, if the neighborhood region has high requests, it is highly probable that the current region will have more requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Similarly, if a region is sparse in requests, its subsequent regions will certainly have few requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4 System model In this section, we detail out the working of our proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As can be seen through Figure 3, input to our proposed model is a dataset that con- tains the request sequence R={r1, r2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=', rm} of ride-hailing platforms, where ri represents a particular request and m denotes the total number of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Each request is represented as ri=, where rsi and rdi determine the source and destination of request i respectively, and rt′ i represents the time at which the request i is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This data is pre-processed and divided into 24 slots of 1-hour for each day and a graph G=(V, E, ∆) is created for each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The vertices V of the graph denote the cells of the grid (see Figure Grid 91 92 gj 925 cell g1 1 2 92 0 2 0 gi 1 2 925 0 3 2GNN-based Passenger Request Prediction 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 3: Framework of our proposed model 1), and the edges E represent the interconnection between different grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This graph is a complete graph as requests can arrive between any pair of ver- tices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The adjacency matrix of the graph G is represented by the OD matrix shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Each edge of the graph G is associated with a weight ∆ij that determines the number of requests between the grid cells gi and gj (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' If there are no requests between the two vertices, the correspond- ing weight is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The output of the pre-processed stage is a sequence of graphs G={G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=', G24}, where Gi represents the requests that arrive in the tth i time slot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='e, all the requests that arrive between (i − 1)th and ith hour of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This graph sequence is generated for all the days in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Each vertex of the graph is represented by an embedding which is its transformation into a vector space that describes it completely and preserves the maximal information about the local structure of the graph (connection between nodes and edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In our proposed model the initial embedding of a vertex is a combination of its grid ID (gi for the cell i of grid), row num- ber, column number, time slot ti, day of the week, in-degree, and out-degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The in-degree and out-degree of a node are calculated from the OD matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' If we represent each element of the OD matrix in Figure 2 as OD(gi, gj) where gi represents the row grid cell and gj represents the column grid cell, then the in-degree of the grid cell gk is �n i=1 OD(gi, gk) and its out-degree is �n j=1 OD(gk, gj) Initially, the embeddings of a node contain the local infor- mation associated with the node and edge connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This information is unstable as it only provides a view of the local neighborhood and does not provide any indication of the spatial or temporal dependencies that can arise among the requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In order to capture the dependencies among the requests and obtain a global view of the network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' the nodes exchange embeddings with Neighborhood information Feature Featurevectors Spatial Pre-process Dataset extraction Attention layer Spatialembeddings Demand Demandflow Temporal Temporal process embeddings Attention layer Origin destination Linear Non-linear dependencies dependencies8 GNN-based Passenger Request Prediction Notation Description g Grid gi ith grid cell t Time Slots ti ith time slot R Request sequence ri ith request rsi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' rdi Source and destination of request i rt i Time at which request i arrives G Request Graph OD Matrix Adjacency matrix of G V Vertices of graph G E Edges of graph G Gi Graph G at ti time slot ∆ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' ˆ ∆ij Actual and predicted number of requests between grid cells gi and gj δi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' ˆδi Actual and predicted number of requests at grid cell gi ft i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' bt i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' qi Set of forward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' backward and geographical neighbors of grid cell gi at time t et i Embedding of node vi at time t Wc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Ws Learnable weight matrices w′ Pre-weighted aggregator Y Vector that concatenates the embedding of neighboring nodes a Attention coefficient which maps a vector to a single number µ LeakyReLu activation function ft ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' bt ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='qt ij Weight of forward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' backward and geographical neighbors when embeddings are exchanged αt j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' βt j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' γt j Pre-weighted aggregator for forward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' backward and geographical neighbors D Graph that stores distance between different grid cells dij Distance between grid cells gi and gj h Number of hours for which non-linearity is determined pij Probability that requests are transferred from grid cell gi to grid cell gj m Total number of requests n Total number of grids Table 1: Notations their spatial and temporal neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These neighbors are selected by the fol- lowing two layers of our proposed model: spatial attention layer - which selects the spatial neighbors and the temporal attention layer - which selects the tem- poral neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' After the final embeddings are calculated through these two layers, we need to predict the demand (ˆδi) that can arrive at the ith grid cell and the number of requests ( ˆ ∆ij) that may arrive between the ith and jth grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Here ˆδi and ˆ ∆ij correspond to the predicted demand at grid cell i and the predicted number of requests between grid cells gi and gj respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Whereas δi and ∆ij represent the actual demand at grid cell i and the actual number of requests between grid cells gi and gj respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We get the demand at a node and the number of requests between two nodes by feeding the result of these two layers to the transferring attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The working of these layers is described in detail in the next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='1 Spatial Attention Layer After the initial embedding of nodes is created, it is fed as input to the spa- tial attention layer which produces a new embedding that carries information about all the spatial neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These embeddings are created by exchanging data with three different types of spatial neighbors: Forward neighbors - If there are two neighbors gi and gj, and there is at least one request that originates from gi and is destined to gj (∆ij > 0) then gj is the forward neighbor of gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The set of forward neighbors of gi at time slot GNN-based Passenger Request Prediction 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4: Forward and backward neighbors calculated from an instance of graph G t is defined mathematically as: f t i = {gj|∆t ij > 0, ∆t ij ϵ Gt} (1) Backward neighbors - If there are two neighbors gi and gj, and there is at least one request that originates from gj and is destined to gi then gj is the backward neighbor of gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The set of backward neighbors of gi at time slot t is defined mathematically as: bt i = {gj|∆t ji > 0, ∆t ji ϵ Gt} (2) The sequential flow of requests in the network is captured by the forward and backward neighbors, which are also referred to as semantic neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These neighbors identify passenger mobility patterns and determine the flow of requests into and out of the specific region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These neighbors are time- dependent, as the request flow in a region is not constant across different time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' They are calculated from an instance of graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Figure 4 shows the set of forward and backward neighbors of the grid cell g14 at a particular time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The forward neighbors determine the out-flow of requests from a particular grid cell and the backward neighbors determine the in-flow of requests to a particular grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As can be seen through Figure 4, {g19,g21, g28} is the set of forward neighbors of grid cell g14 as there are some requests that originate from g14 and are destined towards g19, g21 and g28 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Similarly {g7, g8, g9} is the set of backward neighbors of grid cell g14 as the requests from grid cells g7,g8 and g9 have their destination at grid cell g14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Geographical neighbors: Two neighbors gi and gj are said to be geo- graphically connected if the Haversine distance between their corresponding FORWARD NEIGHBORS 6 (919,921,928] 919 9: 914 921 66 928 BACKWARD NEIGHBORS (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='99]10 GNN-based Passenger Request Prediction latitude/longitude pairs is within a specified threshold distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' qi = {gj|dij ≤ L, dij ϵ D} (3) where L is the threshold distance that determines the size of geographi- cal neighbors and D is the distance graph that represents the distance (dij) between nodes gi and gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' For instance, if we consider the threshold distance equal to the length of one grid, then the set of geographical neighbors of grid cell g14 for the example shown in Figure 2 is {g8, g9, g10, g13, g15, g18, g19, g20}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The set of geographical neighbors of gi is constant across all time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The geographical neighbors can be used to aggregate uncertainty in infor- mation from the areas with few requests (sparse areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As we know that requests arrive in negligible quantity in sparse areas, and the forward and back- ward neighbors capture the dependencies between the neighbors based on the request flow of a particular region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' But, if a region is sparse in requests it will not have in-flow and out-flow of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In that case, there is no information from forward and backward neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, geographical neighbors are always there and can be used to exchange embeddings in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The initial embeddings of nodes are fed as input to the Graph Attention Network (GAN) which combines the information from the three neighbors described above and represents it in the form of a unified vector et i for each node vi at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Some of the earlier models have used Graph Convolution Network (GCN) [9] for combining information from different neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, with GCN all neighbors are assigned the same weightage when embeddings are merged, which neglects the importance of nodes that have a similar flow of requests or that are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We propose to use GAN which samples different neighbors based on their weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It assigns a higher value to the geographical nodes that are in the close vicinity of the current node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Similarly, it provides more weightage to the semantic neighbors with a higher flow of requests to/from the current node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In this way, embeddings of nodes that have more information are prioritized and the noise from redundant nodes is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In order to calculate the importance of a neighborhood node vj at time t, we pass the embedding of the current node et i ϵ Rz∗1 and the neighborhood node et j ϵ Rz∗1 through a weight matrix Wc ϵ Rz ′∗z, where z′ > z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This weight matrix acts as a single-layer neural network and projects the embedding of a node to a higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The output of this layer are the updated embedding of the nodes viand vj at time t and they are equal to et i = Wcet i (4) et j = Wcet j (5) where et i ϵ Rz ′∗1 and et j ϵ Rz ′∗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Figure 5 displays the working of the attention- based aggregator of GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=" The input to GAN is the embedding et i and et j of nodes vi and vj at time t, after they have been passed through weight matrix GNN-based Passenger Request Prediction 11 eti1 eti2 etiz' etj2 etjz' z'*1 z'*1 z'*1 z'*1 eti1 eti2 etiz' w'etj1 eti1 eti2 w'etj1 w'etjz' etiz' 2z'*1 aT L xtij AN(eti,w'etj) ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=" LeakyReLu eti=Wceti etj=Wcetj w'etj eti Y=eti w'etj Pre-weighted aggregator etj1 w'etjz' w'etj2 aTY soft-max Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 5: Graph Attention Network for calculating affinity between neighboring nodes Wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The embedding et j is passed through a pre-weighted aggregator w ′ which provides a prior weight to it before the GAN calculates its importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This prior weight is based upon the current state of neighbors and is described in detail in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The embedding et i of node i and the embedding of the neighborhood node after being passed through a pre-weighted aggregator (w ′et j) are concatenated and represented as a single vector Y : Y = et i ⊕ w ′et j (6) This vector is then passed through a learnable attention coefficient a ϵ R2z ′∗1 which maps it to a single number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thereafter, non-linearity is applied through the LeakyReLu activation function denoted as µ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' AN(et i, w′et j) = µ(aT Y ) (7) This attention-based aggregator function of GAN measures the affinity between the embedding of node vi and its neighbor vj at time t by learning the weight matrix Wc and the attention coefficient a, and produces a single number that determines the weight of neighborhood node vj when it needs to exchange embedding with the node vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, the output of the neural network is not normalized, which is a problem since the weights should be on the same scale for exchanging embed- dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In order to normalize the weights, we apply the soft-max function to the output of the attention-based aggregator defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' (7) which brings all the weights to the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These normalized weights are denoted as xt ij in Figure 5 and they denote the weights of forward neighbors (f t ij), backward 12 GNN-based Passenger Request Prediction neighbors (bt ij), and geographical neighbors (gij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' They are mathematically represented as: f t ij = exp(AN(et i, αt jet j)) � k ϵf t i exp(AN(et i, αt ket k)) bt ij = exp(AN(et i, βt jet j)) � k ϵbt i exp(AN(et i, βt ket k)) qt ij = exp(AN(et i, γjet j)) � k ϵqt i exp(AN(et i, γket k)) (8) These weights allow our proposed model to prioritize embeddings that are geographically and semantically similar to the node vi at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Here, αt j, βt j, and γj refer to the pre-weighted functions which are detailed out in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Since we have obtained the weights that determine the importance of each node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' we exchange embeddings with the set of forward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' backward and geographical neighbors with the weights f t ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' bt ij and qt ij calculated above as: et i = Wset i ⊕ � jϵf t i f t ijWset j ⊕ � jϵbt i bt ijWset j ⊕ � jϵqi qt ijWset j (9) Here et i is the final embedding of each node vi at time t and it is obtained by merging information from different set of neighbors based upon their weights and Ws is the shared weight matrix that projects all embeddings onto the same z ′ dimensional space before merging them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='1 Pre-weighted aggregator The pre-weighted aggregator provide a prior weight to the embedding of node vj at time t before its affinity is measured by the attention-based aggregator defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The aggregator determines the current flow of requests, which is calculated from the edges (∆ij) of the graph Gt, and the geographi- cal relationship between requests, which is determined through the edges (dij) of graph D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It allows our proposed model to sense the sparsity or density of requests in the current time slot and accordingly aggregate the information from the forward and backward neighbors that have a higher flow of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It also provides an indication of the geographical neighbors which are closer to the requests that arrive at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These pre-weighted aggregators are repre- sented as αt j, βt j and γj for the forward, backward and geographical neighbors respectively and are defined as: GNN-based Passenger Request Prediction 13 αt j = ∆ij � jϵf t i ∆ij + s βt j = ∆ij � jϵbt i ∆ij + s γj = 1 dij � jϵqt i 1 dij (10) s is a small additive term to prevent the case when the denominator is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The weights αt j and βt j represent the intensity of passenger demand at time t and the weight γj corresponds to the distance of request from node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These weights help our proposed model to prioritize the embeddings that are geographically or semantically closer to node vi at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='2 Temporal Attention Layer After the data is filtered through spatial attention layer, we get a low-level vector that carries information about all the neighborhood nodes and has an overview of the spatial dependencies of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In order to capture the tem- poral dependencies among the learned representations, these node embeddings are exchanged with time-based neighbors using a temporal attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The embeddings are exchanged through the attention-based mechanism described for spatial attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This layer has 4 channels: the first and second chan- nels analyze the dependencies among requests that arrive in the preceding and consecutive hour of the previous 7 days, and the third channel finds out depen- dencies among requests from the same hour of the previous 7 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These three channels capture the linear dependencies that arise due to the regular pat- terns in data like the morning and evening rush hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, there could be non-linear dependencies which show up due to the recent events that might have taken place at the preceding time intervals or that reveal the travelling patterns of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These dependencies are monitored by taking the data from previous h hours, where the value of h is determined through experimental evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Their value provides an indication of customer flow in the region around the time request arrives and thereby helps our proposed model to cap- ture the patterns apart from the morning and evening hour data repetition trends that are found in the existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These non-linear dependencies are described in detail next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='1 Context-aware data Context-aware data refers to the data about the surrounding environment of a location and it provides an indication of the events that might have taken place around that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This data is collected through various software and hardware tools and it has been used to modify the behavior of various systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 14 GNN-based Passenger Request Prediction In the case of ride-hailing platforms, context-aware data can be used to direct the routes to users, considering the local events that might have taken place around that place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' For instance, if the context-aware data displays that there is road construction in the area, then the vehicles are recommended to divert their route and travel through a different path in order to ensure the passengers reach their destination on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Similarly, if the context-aware data displays that the area is free from traffic jams, then the vehicles are directed towards that route to provide the users with a hustle-free transportation service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Var- ious works have been done that have used context-aware data to direct safe [11], pleasant [12], or personalized routes [13] to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These works have used this data to modify the behavior of the system and direct users towards the routes that take into account the local condition of the place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Our proposed model also uses context-aware data to capture the events that might have taken place around the location at a particular period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, in our proposed model the data is not gathered from expensive software and hardware tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Rather it is collected by the non-linear channel of the temporal attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This layer captures all the customer requests that have arrived in that area for the preceding h hours and uses that to analyze the local environment of that place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The data from preceding h hours provides an indication of one of the following things: sparsity or density of requests in the area or the occurrence of an event at that place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' If the data for the past h hours shows that there are few requests in the area, there may have been some impulsive event that reduced the flow of requests in and out of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This event could correspond to heavy rainfall, snowfall, or a security breach in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' If there are no events scheduled at the place and the customer demand is still low then the area may have few requests, to begin with, meaning that people in that area do not use ride-hailing platforms for their daily commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In both cases, there is a high probability that the area will have few requests in the next hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This context-aware data thus helps our proposed model to anticipate future requests according to the local conditions of the place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Similarly, if the previous hours are abundant in requests, there might be an upcoming event in that area - a football game, festival sale, or any other event that influences passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, if there are no events scheduled and the customer demand is high, it reveals that the area is dense in requests and there is always a higher flow of passengers in and out of that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These patterns reveal important contextual information and can be used to monitor the flow of passengers in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' They help our proposed model to capture local events that might have taken place around the location where the request needs to be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These dependencies are captured by the non-linear channel of the temporal attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='2 Travelling Behavior Non-linear dependencies can also arise due to the travelling patterns of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' According to a behavioral study, people spend 5 to 6 hours outside of their homes during holidays [14], which can include time spent in their garden, GNN-based Passenger Request Prediction 15 walking, or travelling in cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus, after this time-period people tend to return to their original location by using any mode of transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This suggests that requests start to recur after this time and these patterns can be exploited to predict the future requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Further, during weekdays people go to their homes after their office hours are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As data from various countries reveals that people spend on average 2 − 6 hours in their workplace [15] this pattern can be exploited to predict the travelling behavior of people during weekdays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Moreover, there are other recurring patterns such as shopping [15], etc, which can be used to predict the future occurrence of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus, the data from previous hours provides an indication of travelling behavior of people and it reveals their office hours, recreation time, shopping hours, and other hobbies that can be used to predict future requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This behavior is captured by the non-linear channel of the temporal attention layer and it provides insights into the recurring pattern of requests in ride-hailing platforms which can be used to predict the future occurrence of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3 Transferring Attention Layer After the data is modelled for spatial and temporal patterns, we pass it through feed-forward neural network layer to get the number of requests at each node of a graph which represents the demand in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The demand is rep- resented as a matrix ˆδ = { ˆδ1, ˆδ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=', ˆδn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In order to represent the origin and destination of requests, transferring attention layer is used which models the transmission from one node to another based upon the transferring probabil- ity pij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' pij represents the probability that requests are transferred from node i to node j and it is mathematically represented as pij = exp(AN(eT i , eT j )) �n i=j exp(AN(eT i , eT j )) (11) OD pair ( ˆ ∆ij) is calculated from demand by using transferring probability pij as follows: ˆ ∆ij = ˆδ · pij (12) 5 Prototype Implementation In this section, we evaluate our proposed model based upon extensive simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Dataset: We conduct experiments on the real-world taxi dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The dataset is generated for New York City and contains data of the month of February.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Experimental Settings: We evaluate the prediction accuracy of our proposed model based upon the two widely applied metrics: Mean Absolute Percentage 16 GNN-based Passenger Request Prediction Table 2: Simulation to check the length of a grid cell Task Grid length (km) MAPE-0 MAPE-3 MAPE-5 MAE-0 MAE-3 MAE-5 OD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6 2.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5871 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6591 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4107 Error (MAPE) and Mean Absolute Error (MAE) which are defined as: MAPE = 1 m m � i=1 ���� ˆyi − yi yi + 1 ���� (13) MAE = 1 m m � i=1 �� ˆyi − yi �� (14) where m denotes the number of examples, ˆyi denotes the predicted result and yi represents the actual result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We estimate the performance of our proposed model on areas with varying levels of customer base and accordingly calculate MAPE-0, MAPE-3, and MAPE-5 where 0, 3, and 5 denote the minimum number of requests in different areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We have used this threshold to see the patterns in different areas according to their customer base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In the experimental work, we used 75% of data for training purposes and kept the remaining 25% for testing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' In the training settings, 10% of data is used as a validation dataset and is used for hyperparameter testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We implement our model with PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='0 on Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' All the simulations were run on Windows i7 with 16 GB RAM for 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The model was trained with a batch size of 1 and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Based on these parameters, we evaluate the working of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='1 Grid length The length of a grid cell is an important parameter that decides the complexity and accuracy of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' If the length is large, then the origin and destination of the majority of requests will be concentrated within a single grid cell and the OD prediction will reduce to demand prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' If it is narrow, the time complexity of the model increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus the length of a particular grid cell is an important parameter and needs to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Table 2 displays the performance of the proposed model based upon the parameters of MAPE and MAE, with the increase in length of a grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' There is a rapid decrease in the error of the model when the length increases from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3 km to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' After that, the error becomes stagnant for the grid cells GNN-based Passenger Request Prediction 17 of length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 km and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thereafter, the error is found to increase again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The main perspective behind this behavior can be the change in data size and the number of neighbors on the varying lengths of grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' When the length of grid cell is small, there are multiple cells with no requests, and the spatial dependencies are not represented accurately by this size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Moreover, with small length time complexity of the model is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, when the length of grid cell increases, dataset size decreases, and the number of different neighbors changes which brings about an increase in error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Further, with an increase in grid cell length, the OD task is found to converge to demand prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus the optimal value of the length of grid cells needs to be determined which reflects passenger mobility and performs effectively in a time-bound manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Based on the experiments we have set the length of the grid cells as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 km for our proposed model as it performs well under all the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='2 Non-linearities As already stated, the non-linearities in our proposed model are captured by the fourth channel of the temporal attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This layer monitors the data from previous h hours and gives an indication of the events that might have taken place around the location during these hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, to properly monitor the events around that place, the value of h needs to be determined through simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Figures 6, 7, 8 and 9 show the MAPE and MAE of the proposed model for OD and demand prediction with varying time hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As can be seen through these figures, when the data from the previous 3−6 hours is used, the model is found to perform well under all the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Particularly, the data from the previous 6 hours is found to perform best under all conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As already stated, the non-linear data provides context-aware information about the location and determines the average flow of requests in the area over a period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' For instance, if the data from previous 3 − 6 hours contains few requests then that area may be sparse or some event may have occurred like rainfall, security breach, etc which reduced the flow of requests to that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Similarly, if the previous hours have more requests then that area will either be high demand area or some event might be scheduled at that place which has increased its demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' However, among the past 3 − 6 hour data, the data from the previous 6 hours is found to perform best under all the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The intuitive explanation might be that it reveals the travelling behavior of people and represents the average time spent outside by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As the office timings in different countries vary from 2−6 hours [15] and the demand re-arises after this time period, it provides one of the explanations for the data from the previous 6 hours to perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Moreover, a behavioral study [14] has found that on average people spend 5 − 6 hours outside their homes during holidays (it can include time spent in their garden, time doing walk or travelling in cars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' It provides further explanation for using previous 6 hour data to model non- linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus, from experimental studies and behavioral patterns of people, 18 GNN-based Passenger Request Prediction 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='43 Number of hours MAPE-0 (a) MAPE-0 for OD pre- diction 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4 Number of hours MAPE-3 (b) MAPE-3 for OD pre- diction 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='38 Number of hours MAPE-5 (c) MAPE-5 for OD predic- tion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 6: MAPE for OD prediction 5 10 15 20 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 Number of hours MAE-0 (a) MAE-0 for OD predic- tion 5 10 15 20 11 12 13 14 15 16 Number of hours MAE-3 (b) MAE-3 for OD predic- tion 5 10 15 20 14 15 16 17 18 19 20 21 Number of hours MAE-5 (c) MAE-5 for OD predic- tion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 7: MAE for OD prediction 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6 Number of hours MAPE-0 (a) MAPE-0 for demand prediction 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5 Number of hours MAPE-3 (b) MAPE-3 for demand prediction 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4 MAPE-5 Number of hours (c) MAPE-5 for demand prediction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 8: MAPE for demand prediction 5 10 15 20 35 40 45 50 55 60 65 MAE-0 Number of hours (a) MAE-0 for demand pre- diction 5 10 15 20 60 70 80 90 100 110 Number of hours MAE-3 (b) MAE-3 for demand prediction 5 10 15 20 40 60 80 100 120 140 Number of hours MAE-0 (c) MAE-5 for demand pre- diction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 9: MAE for demand prediction we conclude that data from previous 6 hours reflects passenger mobility well and gives an indication of customer re-appearance over different areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' GNN-based Passenger Request Prediction 19 Table 3: Repetition pattern of requests Task Ho- urs MAPE- 0 MAPE- 3 MAPE- 5 MAE-0 MAE-3 MAE-5 OD 6 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3705 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3186 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6635 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3549 Demand 6 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='2928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3479 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5250 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='9023 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='0315 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3015 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='2562 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='7730 Table 4: Comparison on OD and demand prediction for different methods Task Method MAPE- 0 MAPE- 3 MAPE- 5 MAE-0 MAE-3 MAE-5 OD LSTNet GallatExt Proposed model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3705 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3186 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='0479 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6408 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4884 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='7843 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6635 Demand LSTNet GallatExt Proposed model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='4024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} 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+page_content='3183 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='1277 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='7006 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5250 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='7881 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6304 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='0315 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='6196 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='5871 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='2562 Data from previous 20−23 hours is also found to perform well, in particular for OD prediction tasks as can be seen through Table 3 and Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The main insight that we get from this pattern is the repeating sequence of requests the following day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' As the 23 hour data captures the average flow of requests for a day and this pattern repeats the following day, therefore the origin and destination of requests are predicted accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Thus we conclude that data from previous 6 hours reflects the passenger travelling behavior and provides the contextual data of that place which can be used to analyze dependencies and predict the future demand and OD pair of requests in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Whereas the data from previous 23 hours captures the repeating trend in requests and uses that to predict the future OD pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content='3 Comparison with previous models We evaluate the performance of our proposed model by comparing it with the following baselines: LSTNet[5]: It captures the spatio-temporal dependencies in ride-hailing requests through CNN and LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' GallatExt [6]: It predicts the origin and destination of requests through GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Table 4 displays the performance of LSTNet, GallatExt, and our proposed model on all the evaluation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Our proposed model performs better under all the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' This is because we have used context-aware data and analyzed the travelling pattern of users and used that to predict the future occurrence of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' These patterns reveal the reason behind the non-linear temporal dependencies that arise in data and when these patterns are paired with the linear patterns like the morning and evening rush hours the model 20 GNN-based Passenger Request Prediction is found to capture different types of dependencies and thereby surpass the existing models in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' 6 Conclusion In this paper, we define passenger mobility through origin-destination pre- diction that is helpful in directing optimal routes to drivers and providing matching algorithms for drivers and passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' OD prediction is complex in comparison to demand prediction as it predicts the demand in a certain region and the destination of these demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We have used GNN to capture the spatio-temporal dependencies among requests and predicted the OD pair of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We have particularly focused on temporal non-linearities that arise due to contextual events or the travelling patterns of people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' While modelling these dependencies, the length of the grid cell is an important parameter and it can regulate the neighborhood count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' We have determined its value through extensive simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' The parameters that decide the grid cell length and the non-linearities are estimated for demand prediction models also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Exten- sive simulations determine superior performance by our proposed model as compared to the existing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' References [1] Garg, N.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} +page_content=' Accessed: 2022-10-13 (2020)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE0T4oBgHgl3EQfnwER/content/2301.02515v1.pdf'} diff --git a/e9E_T4oBgHgl3EQf1xyk/content/2301.08337v1.pdf b/e9E_T4oBgHgl3EQf1xyk/content/2301.08337v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7cdeffe98a887819fcc939b8d119de80e3c78a74 --- /dev/null +++ b/e9E_T4oBgHgl3EQf1xyk/content/2301.08337v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5dd6ec298432d69970ec5423e17a0000bb1802cb770a9c88a8fe8b2b1dfd50aa +size 709709 diff --git a/e9E_T4oBgHgl3EQf1xyk/vector_store/index.pkl b/e9E_T4oBgHgl3EQf1xyk/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..15fa3a4133f2eea7ba092dacb2e0aa7b6284c86e --- /dev/null +++ b/e9E_T4oBgHgl3EQf1xyk/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a4ba1e184b18f2ee092fa8190867aae8e27f5698a644934b59ccc0d5bb3ced11 +size 161407 diff --git a/edAyT4oBgHgl3EQfjfiM/content/tmp_files/2301.00416v1.pdf.txt b/edAyT4oBgHgl3EQfjfiM/content/tmp_files/2301.00416v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..262e4b330ea794638de0c0f57a73f5d1a169653e --- /dev/null +++ b/edAyT4oBgHgl3EQfjfiM/content/tmp_files/2301.00416v1.pdf.txt @@ -0,0 +1,962 @@ +Positron & Gamma influence on RREA multiplication +Eduard Kim,∗ Alexander Sedelnikov,† Daria Zemlianskaya,‡ Oraz Anuaruly,§ and Egor Stadnichuk¶ +(Dated: January 1, 2023) +Recent results in high-energy atmospheric physics link relativistic runaway electron avalanches +(RREA) accelerated by the electric field in thunderclouds to atmospheric electricity, such as the +unique natural phenomenon of Terrestrial Gamma-Ray Flashes (TGF). Research shows that the +mere existence of runaway electron avalanches is not sufficient to generate a TGF. In an attempt +to settle this issue, a model of relativistic feedback mechanism was suggested. In this paper, a way +of theoretically investigating the mutual influence of positron and gamma feedback mechanisms is +proposed. The electron avalanche multiplication factor due to this combined feedback is obtained, +followed by a study of minimal conditions of infinite feedback in a thundercloud. +I. +KEYPOINTS +• Positron +& +Gamma +relativistic +feedback +mechanisms interference and mutual enhancement +are analytically predicted; +• Given the mutual influence of relativistic feedback +mechanisms, their combination causes a minor +deviation in RREA dynamics when compared to +their independent conjunction. +• Relativistic feedback realisation requires the area +of a cloud with a homogeneous break-even electric +field to be larger than what is observable in +experiments. +II. +INTRODUCTION +One of the major unresolved difficulties in high-energy +atmospheric physics is obtaining the necessary conditions +for the generation of lightning in thunderclouds. +Many physical models were developed to describe +the accumulation of electric charges and corresponding +enhancement of electric fields within a thundercloud, as +the latest works can be mentioned [1–3]. Experimental +measurements indicate that the absolute value of the +electric field is an order of magnitude lower than the value +∗ Moscow Institute of Physics and Technology, Moscow, 117303, +Russian Federation; Institute for Nuclear Research of RAS, +Moscow 117312; kim.e@phystech.edu +† Moscow +Institute +of +Physics +and +Technology, +Moscow, +117303, Russian Federation; Lebedev Physical Institute RAS; +sedelnikov.as@phystech.edu +‡ Moscow Institute of Physics and Technology, Moscow, 117303, +Russian Federation; Institute for Nuclear Research of RAS, +Moscow 117312 ; zemlianskay.d@phystech.edu +§ Moscow Institute of Physics and Technology, Moscow, 117303, +Russian Federation; Kurchatov Institute RAS; Lebedev Physical +Institute RAS; orazanuaruly@gmail.com +¶ Moscow Institute of Physics and Technology, Moscow, 117303, +Russian Federation; HSE University, Moscow 101000 Russia; +yegor.stadnichuk@phystech.edu +required for conventional electric breakdown in air [4– +6]. As Gurevich showed [7], these fields are sufficient for +relativistic charged particles to cause runaway relativistic +electron avalanches. +Thunderstorm phenomena are not limited only by +lightning. According to recent research, thunderstorms +are a natural source of gamma-radiation: experiments on +detecting cosmic showers show that gamma-ray particle +flux increases during thunderstorms. Such phenomena as +terrestrial gamma-ray flashes (TGF) and thunderstorm +ground enhancement (TGE) observed with satellites from +space [8, 9] and with detectors on the ground [10] +are shown to be the clear, high-energy manifestations +coming from thunderstorms. According to the most +recent research [11] TGF bursts always occur before +or at the same time as the onset of the optical pulse. +These observations suggest the importance of high- +energy processes for lightning initiation. +Finally, thunderstorms generate the most powerful +natural terrestrial radio bursts in the VHF range [12] +known as Narrow Bipolar Events (NBEs), which are +thought to be a precursor of lightning, as in the case +of TGF. Observations made by Rison [13] suggested +that NBE are produced by volumetrically distributed +positive streamers with apparent speeds close to the +speed of light. Based on this work, the mechanism +of lightning initiation was proposed [14]. According to +this mechanism, runaway electron avalanches trigger +giant streamer bursts, the first stage of streamer-leader +transition, indicating the role of RREA in lightning +initiation. As a result, clarifications on runaway electron +generation must be made in order to compare with +observations. +The first significant changes in understanding of +runaway electron generation were made by Dwyer +[15]. It turned out that Gurevich’s runaway electron +avalanches, born only from the cosmic shower particles, +cannot produce enough bremsstrahlung radiation to +form the observed fluxes of TGF. No other powerful +enough +sources +of +ionization +have +been +observed; +therefore, some electron avalanche flux amplification +mechanism was required. Dwyer proposed the relativistic +feedback mechanism [16] which exponentially increases +the avalanche generation and changes the evolution of +arXiv:2301.00416v1 [physics.ao-ph] 1 Jan 2023 + +2 +RREA. Recent research, however, suggests that infinite +self-generation of RREA by relativistic feedback (RREA +burst) is expected to occur only in the presence of +a localized, excessively strong, large-scale atmospheric +electric field (electric cell) [17]. "Infinite generations" on +a TGF lifespan scale ∼ +ms [18] implies that the +vanishing of the electric field due to relativistic discharge +is ignored. +This paper investigates the possibility of using positron +and gamma combined relativistic feedback mechanisms +to achieve self-sustaining RREA multiplication under +realistic thunderstorm conditions. +III. +STATIONARY REGIME OF RREA +MULTIPLICATION +In +agreement +with +Fig.1, +physical +processes +are +considered in the assumptions presented in Chapter 2 +of [17]. +As a result, a set of equations is developed that +describes the combined feedback between the (k +1) and +k generations of RREA and gamma. +� +N k+1 +RREA = N k +RREAνe− + N k +γ νγe−, +N k+1 +γ += N k+1 +RREAνe−γ +(1) +With initial conditions in a form of: +• N 1 +RREA = N0, +• N 0 +γ = 0. +According to [19], short-term disordered development +of RREA transfers into a stationary mode of RREA +generation. It is critical to discuss the nature of ν... +coefficients before attempting to find the stationary mode +of electron avalanche generation. A close examination +of the equation (1) reveals that no positron-related +processes are taken into account. However, these physics +must be factored into νe−, which represents a positron +feedback mechanism. All gamma generation processes are +hidden in νe−γ. Finally, the gamma feedback mechanism +is taken into account by νγe−, which represents the +reproduction of RREA by gamma feedback in each +generation. Positron and gamma feedback mechanisms +in this section are considered independent mechanisms. +The first three generations of RREA were derived using +equation (1): +• First generation +� +N 1 +RREA = N0, +N 1 +γ = N0νe−γ +(2) +• Second generation +� +N 2 +RREA = N0(νe− + νe−γνγe−), +N 2 +γ = N0(νe−νe−γ + νγe−ν2 +e−γ) +(3) +• Third generation +� +N 3 +RREA = N0(ν2 +e− + 2νe−νe−γνγe− + ν2 +γe−ν2 +e−γ), +N 3 +γ = N0(νe−γν2 +e− + 2νe−ν2 +e−γνγe− + ν2 +γe−ν3 +e−γ) +(4) +In a rewritten form, equation (4) takes form +� +N 3 +RREA = N0(νe− + νγe−νe−γ)2, +N 3 +γ = N0((νe−γν2 +e− + 2νγe−ν2 +e−γνe−) + ν2 +γe−ν3 +e−γ) +(5) +Close look at equations (2)-(5) leads to relations +N 4 +RREA +N 3 +RREA = N 3 +RREA +N 2 +RREA = N 2 +RREA +N 1 +RREA = (νe− + νe−γνγe−). Thus, +the combined feedback mechanism coefficient for RREA +generation is +Γc = (νe− + νe−γνγe−) +(6) +To validate the derived formula, the fourth generation +of feedback must be compared to the third generation. +N 4 +RREA = N0(νe− + νγe−νe−γ)3 +(7) +ΓcN 3 +RREA = N0(νe− + νγe−νe−γ)3 +(8) +As expected, we found a stationary mode of RREA +generation via combined feedback. +A. +Multiplication coefficients +In addition to III, defining ν coefficients will lead to +the final formation of a combined feedback mechanism +[20, 21]. +νe− = Ke− +� +e +L(λanih−λRREA) +λanihλRREA +− 1 − L(λanih − λRREA) +λanihλRREA +� +, +(9) +where Ke−= +Pe−;e+Pe+λRREA +λ2λγλγ→e−e+ +� +λRREAλanih +λanih−λRREA +�2 +. +νe−γ = λRREA +λγ +� +e +z−z0 +λRREA − 1 +� +(10) +Because of the exponential law of growth of RREA and +the fact that formula (10) describes a source-function +of gamma-birth, which is also given in the form of +an exponent, the process γ → e− cannot be simply +expressed through a numerical coefficient. The only +plausible explanation is that νγe− is an integral operator +acting on the νe−γ function. + +3 +e− +RREA +γ +e−e+ +e+ → e− +γ → e− +RREA +γ +e+e− +RREA +γ +e+e− +Fig. 1: Schematic image of positron and gamma relativistic feedbacks producing RREA’s second generation in +supercritical ( E +Ebe ≥ 1) electric fields. Consideration includes such processes as: impact ionization, bremsstrahlung +and electron-positron pair generation. +The result of operator νγe− acting on νe−γ is shown in +[22] +νe−γνγe = Ke−γ,γe +� +e +L(λx−λRREA) +λxλRREA +− 1 − L(λx − λRREA) +λxλRREA +� +, +(11) +where Ke−γ,γe = +PγPe−;γ +λγ→eλγ +� +λRREAλx +λx−λRREA +�2 +. +IV. +AN EXACT SOLUTION OF COMBINED +FEEDBACK +The co-dependence of positron and gamma feedbacks +is +essential +for +obtaining +an +exact +solution +from +combined feedback. For the purpose of simplification +of calculations, one should reconsider the study of +distribution functions [20]. +The primary electron avalanche starts at z0. Parameter +λRREA- is the length of the exponential rise of RREA, λγ- +is the length of a runaway electron before the gamma is +born, λ+- is the length of the gamma before the electron- +positron pair is born. +Number of produced gamma quanta inside the interval +[z0, z]: +fγ(z, z0) = λRREA +λγ +· +� +e +z−z0 +λRREA − 1 +� +(12) +Number of produced positrons inside the interval +[z0, z]: +f+(z, z0) = +� z +z0 +fγ(ζ, z0) dζ +λ+ += λRREA +λγλ+ +� +λRREAe +z−z0 +λRREA − +− λRREA − (z − z0) +� +(13) +Number of gamma quanta deployed within a segment +[z0, z]: +fγ′(z, z0) = Pγfγ(z, z0) +(14) +Where λγ→e is the path length of the gamma before +the runaway electron is born. Let Px be the probability of +the particle x turning around. Consider that the number +of electrons produced by unfolded gamma quanta varies +according to the law dN +dz = +1 +λγ→e e− z +λx , where λx is some +characteristic length. Then the number of secondary +electron avalanches born at coordinate z in thickness dz: +df gamma +2 +(z, z0) = dz · Pγ · Pe− +λγ→e +· +� L +z +dζ ∂fγ(ζ, z0) +∂ζ +e +z−ζ +λx +(15) +Simultaneously, +the +following +is +the +number +of +secondary electron avalanches born in the z coordinate +in the thickness dz: +df pos +2 +(z, z0) = dz · Pe−Pe+ +λ2 +· +� L +z +dζ ∂f+(ζ, z0) +∂ζ +e− +ζ−z +λanih +(16) +Since the dynamics of secondary electron avalanches +are no different from those of the primary avalanche, the +following iterative equation can be written for subsequent +generations: +fi(z, 0) = +� L +0 +(f pos +2 +(z, ζ) + f gamma +2 +(z, ζ)) · +�∂f pos +i−1(ζ, 0) +∂ζ ++ ++∂f gamma +i−1 +(ζ, 0) +∂ζ +� +dζ +(17) +Consistently +solving +this +equation +leads +to +the +following: there is a stationary mode of RREA generation +with the following multiplication factor: + +4 +Γ = νe− + νe−γνγe + +� +Ke−Ke−γ,γe +� +� +� +�λanihλRREAe +L(λanih−λRREA) +λanihλRREA +λanih−λRREA +− +λanihλRREA +λanih − λRREA +− L +� +� + ++ +� +�λxλRREAe +L(λx−λRREA) +λxλRREA +λx − λRREA +− +λxλRREA +λx − λRREA +− L +� +� +� +� +(18) +Figure 2 shows an analysis of the minimal conditions +for an infinite RREA burst (Γ = 1) to estimate the +difference between equations (6) and (18). The electric +field value is given as a quantity ratio to the break-even +field [23]. +V. +DISCUSSION +The derivation of equations (6) and (18) allows the +runaway electrons dynamics and RREA burst parameters +to be studied in a positron and gamma combined +feedback model. It can be seen from the figures 2a- +2b difference between (6) and (18) is not significant, +and consideration of formula (18) can slightly soften +conditions for RREA burst. Thereby, it was shown that +instead of deriving the multiplication feedback coefficient +from direct solutions of the equation (17), considering +independent positron and gamma relativistic feedbacks +is suitable for assessing the dynamics of RREA within a +homogeneous supercritical electric field. +According to [5, 24], experimental observations show +a correlation between lightning strikes and electric field +values close to Ebe. The size of a thundercloud region with +a homogeneous electric field with E ≥ Ebe, on the other +hand, was not measured. Cell length could be estimated +using the equation (18) and results from papers such as +[25, 26]. Table I was obtained under the assumption that +lightning initiation is associated with a RREA burst. +Parameters at the moments of lightning discharges with Γ = 1 +In situ parameters +E/Ebe = 1 E/Ebe = 2 +H, km +E, +kV/m +E/Ebe +L, km +Lmax, +km +Lmin, +km +5.14 +109 +0.99 +27.7 +28 +1 +10 +59 +0.939 +46.8 +49 +1.7 +11.29 +48 +0.89 +60.5 +58 +2 +7.25 +75.3 +0.86 +46.6 +36 +1.3 +7.75 +72.8 +0.886 +42 +38 +1.3 +Table. I: In situ parameters at the moments of lightning +discharges [25] with cell length L derived from equation +(18) under assumption Γ = 1. The smallest Lmin and +largest Lmax values estimations for L corresponding to +experimentally observed range of E/Ebe [24] are +provided. Altitude H is given in kilometers, electric field +E in kV +m and L in kilometers. +In Table I first four rows are filled with parameters +(H- altitude, E- electric field) measured at the moment +of lightning strike [25]. In addition, estimations of the +smallest and largest values for L were calculated, given +by the experimentally observed range of values of the +parameter +E +Ebe += 1 ÷ 2 [24]. These estimations differ +significantly due to the rapid behavior of the minimal +conditions for RREA burst, shown in the figure 2. +To satisfy minimal conditions for RREA burst cell +length has to be of order of 10 km ( E +Ebe = 1) within +a framework of combined relativistic feedback model, or +even by strong overestimation of values of electric fields +( E +Ebe = 2) L is of order of ≳ 1 km. Thus, obtained +results suggest the need of modification of the concept of +relativistic feedback for the theory of lightning initiation. +As such modifications, the study of inhomogeneous +structures of electric fields in thunderstorms [20], as +well as the possible influence of hydrometeors and their +geometry on the generation of runaway electrons [27], are +considered. The main requirement for such modification +should be the mitigation of RREA burst condition for +the observed parameters of thundercloud. +VI. +CONCLUSION +The aim of this study was to propose a method +of strengthening the impact of relativistic feedback on +RREA multiplication in the atmosphere. Interference +between positron- and gamma-based relativistic feedback +was proposed to amplify runaway electron generation, +potentially leading to a decrease in the minimal electric +field required to induce RREA burst. +The influence of RREA’s secondary particles (positron +and gamma quanta) on its dynamics is expressed not +only in their independent combination of relativistic +feedbacks but also in mutual amplification. Thus, the +resulting multiplication factor of combined relativistic +feedback allowed an analysis of the minimal conditions +for RREA burst. Despite all of the above, combined +relativistic feedback does not provide RREA burst under +the conditions observed in numerous experiments. +Further +research +into +the +influence +of +feedback +mechanisms on RREA dynamics in atmospheric electric +field structures will be required to develop a modified +feedback mechanism capable of producing RREA bursts +at electric field values comparable to those observed in +experiments. As well, it is important to study its role in +high-energy atmospheric processes, such as TGF, TGE, +NBE, etc. + +5 +(a) +(b) +Fig. 2: For different altitudes, the minimal conditions for RREA burst (Γ = 1) are calculated in cases of independent +combination (positron and gamma feedbacks are taken as independent mechanisms) and the exact solution of the +equation (17), which includes full consideration of feedbacks within their mutual amplification. + +5 +Independentcombination,5km +E +4.5 +Exact solution,5km +4 +Independentcombination,8km +3.5 +Exactsolution,8km +3 +2.5 +2 +1.5 +0.5 +0 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Cell length, m5 +E +Independentcombination,1okm +4.5 +Exactsolution,10km +4 +Independent combination, 15 km +3.5 +Exactsolution,15km +3 +2.5 +2 +1.5 +0.5 +0 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Cell length, m6 +[1] M. Di Renzo and J. 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Dwyer, The relativistic feedback discharge model +of terrestrial gamma ray flashes, Journal of Geophysical +Research: Space Physics 117 (2012). +[22] A. +S. +Sedelnikov, +D. +I. +Zemlianskaya, +and +E. +M. +Stadnichuk, The criterion for infinite gamma feedback +in the dwyer model, Memoirs of the Faculty of Physics +(2022). +[23] M. Mccarthy and G. Parks, On the modulation of x ray +fluxes in thunderstorms, Journal of Geophysical Research +97 (1992). +[24] T. C. Marshall, M. Stolzenburg, C. R. Maggio, L. M. +Coleman, P. R. Krehbiel, T. Hamlin, R. J. Thomas, +and W. Rison, Observed electric fields associated with +lightning initiation, Geophysical Research Letters 32 +(2005). +[25] T. +C. +Marshall, +M. +P. +McCarthy, +and +W. +D. +Rust, +Electric +field +magnitudes +and +lightning +initiation in thunderstorms, Journal of Geophysical +Research: +Atmospheres +100, +7097 +(1995), +https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/95JD00020. +[26] T. C. Marshall, W. D. Rust, and M. Stolzenburg, +Electrical structure and updraft speeds in thunderstorms +over the southern great plains, Journal of Geophysical +Research: Atmospheres 100, 1001 (1995). +[27] D. Zemlianskaya, E. Stadnichuk, and E. Svechnikova, +Influence +of +hydrometeors +on +relativistic +runaway +electron avalanches (2022). + diff --git a/edAyT4oBgHgl3EQfjfiM/content/tmp_files/load_file.txt b/edAyT4oBgHgl3EQfjfiM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..13a842ffbed5028b5eb8a5cb5e21b4d213838253 --- /dev/null +++ b/edAyT4oBgHgl3EQfjfiM/content/tmp_files/load_file.txt @@ -0,0 +1,423 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf,len=422 +page_content='Positron & Gamma influence on RREA multiplication Eduard Kim,∗ Alexander Sedelnikov,† Daria Zemlianskaya,‡ Oraz Anuaruly,§ and Egor Stadnichuk¶ (Dated: January 1, 2023) Recent results in high-energy atmospheric physics link relativistic runaway electron avalanches (RREA) accelerated by the electric field in thunderclouds to atmospheric electricity, such as the unique natural phenomenon of Terrestrial Gamma-Ray Flashes (TGF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Research shows that the mere existence of runaway electron avalanches is not sufficient to generate a TGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' In an attempt to settle this issue, a model of relativistic feedback mechanism was suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' In this paper, a way of theoretically investigating the mutual influence of positron and gamma feedback mechanisms is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The electron avalanche multiplication factor due to this combined feedback is obtained, followed by a study of minimal conditions of infinite feedback in a thundercloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' KEYPOINTS Positron & Gamma relativistic feedback mechanisms interference and mutual enhancement are analytically predicted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Given the mutual influence of relativistic feedback mechanisms, their combination causes a minor deviation in RREA dynamics when compared to their independent conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Relativistic feedback realisation requires the area of a cloud with a homogeneous break-even electric field to be larger than what is observable in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' INTRODUCTION One of the major unresolved difficulties in high-energy atmospheric physics is obtaining the necessary conditions for the generation of lightning in thunderclouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Many physical models were developed to describe the accumulation of electric charges and corresponding enhancement of electric fields within a thundercloud, as the latest works can be mentioned [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Experimental measurements indicate that the absolute value of the electric field is an order of magnitude lower than the value ∗ Moscow Institute of Physics and Technology, Moscow, 117303, Russian Federation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Institute for Nuclear Research of RAS, Moscow 117312;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='e@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='edu † Moscow Institute of Physics and Technology, Moscow, 117303, Russian Federation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Lebedev Physical Institute RAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' sedelnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='as@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='edu ‡ Moscow Institute of Physics and Technology, Moscow, 117303, Russian Federation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Institute for Nuclear Research of RAS, Moscow 117312 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' zemlianskay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='d@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='edu § Moscow Institute of Physics and Technology, Moscow, 117303, Russian Federation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Kurchatov Institute RAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Lebedev Physical Institute RAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' orazanuaruly@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='com ¶ Moscow Institute of Physics and Technology, Moscow, 117303, Russian Federation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' HSE University, Moscow 101000 Russia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' yegor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='stadnichuk@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='edu required for conventional electric breakdown in air [4– 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' As Gurevich showed [7], these fields are sufficient for relativistic charged particles to cause runaway relativistic electron avalanches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Thunderstorm phenomena are not limited only by lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' According to recent research, thunderstorms are a natural source of gamma-radiation: experiments on detecting cosmic showers show that gamma-ray particle flux increases during thunderstorms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Such phenomena as terrestrial gamma-ray flashes (TGF) and thunderstorm ground enhancement (TGE) observed with satellites from space [8, 9] and with detectors on the ground [10] are shown to be the clear, high-energy manifestations coming from thunderstorms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' According to the most recent research [11] TGF bursts always occur before or at the same time as the onset of the optical pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' These observations suggest the importance of high- energy processes for lightning initiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Finally, thunderstorms generate the most powerful natural terrestrial radio bursts in the VHF range [12] known as Narrow Bipolar Events (NBEs), which are thought to be a precursor of lightning, as in the case of TGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Observations made by Rison [13] suggested that NBE are produced by volumetrically distributed positive streamers with apparent speeds close to the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Based on this work, the mechanism of lightning initiation was proposed [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' According to this mechanism, runaway electron avalanches trigger giant streamer bursts, the first stage of streamer-leader transition, indicating the role of RREA in lightning initiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' As a result, clarifications on runaway electron generation must be made in order to compare with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The first significant changes in understanding of runaway electron generation were made by Dwyer [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' It turned out that Gurevich’s runaway electron avalanches, born only from the cosmic shower particles, cannot produce enough bremsstrahlung radiation to form the observed fluxes of TGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' No other powerful enough sources of ionization have been observed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' therefore, some electron avalanche flux amplification mechanism was required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Dwyer proposed the relativistic feedback mechanism [16] which exponentially increases the avalanche generation and changes the evolution of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='00416v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='ao-ph] 1 Jan 2023 2 RREA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Recent research, however, suggests that infinite self-generation of RREA by relativistic feedback (RREA burst) is expected to occur only in the presence of a localized, excessively strong, large-scale atmospheric electric field (electric cell) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' "Infinite generations" on a TGF lifespan scale ∼ ms [18] implies that the vanishing of the electric field due to relativistic discharge is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' This paper investigates the possibility of using positron and gamma combined relativistic feedback mechanisms to achieve self-sustaining RREA multiplication under realistic thunderstorm conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' STATIONARY REGIME OF RREA MULTIPLICATION In agreement with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='1, physical processes are considered in the assumptions presented in Chapter 2 of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' As a result, a set of equations is developed that describes the combined feedback between the (k +1) and k generations of RREA and gamma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' � N k+1 RREA = N k RREAνe− + N k γ νγe−, N k+1 γ = N k+1 RREAνe−γ (1) With initial conditions in a form of: N 1 RREA = N0, N 0 γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' According to [19], short-term disordered development of RREA transfers into a stationary mode of RREA generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' It is critical to discuss the nature of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' coefficients before attempting to find the stationary mode of electron avalanche generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' A close examination of the equation (1) reveals that no positron-related processes are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' However, these physics must be factored into νe−, which represents a positron feedback mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' All gamma generation processes are hidden in νe−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Finally, the gamma feedback mechanism is taken into account by νγe−, which represents the reproduction of RREA by gamma feedback in each generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Positron and gamma feedback mechanisms in this section are considered independent mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The first three generations of RREA were derived using equation (1): First generation � N 1 RREA = N0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' N 1 γ = N0νe−γ (2) Second generation � N 2 RREA = N0(νe− + νe−γνγe−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' N 2 γ = N0(νe−νe−γ + νγe−ν2 e−γ) (3) Third generation � N 3 RREA = N0(ν2 e− + 2νe−νe−γνγe− + ν2 γe−ν2 e−γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' N 3 γ = N0(νe−γν2 e− + 2νe−ν2 e−γνγe− + ν2 γe−ν3 e−γ) (4) In a rewritten form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' equation (4) takes form � N 3 RREA = N0(νe− + νγe−νe−γ)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' N 3 γ = N0((νe−γν2 e− + 2νγe−ν2 e−γνe−) + ν2 γe−ν3 e−γ) (5) Close look at equations (2)-(5) leads to relations N 4 RREA N 3 RREA = N 3 RREA N 2 RREA = N 2 RREA N 1 RREA = (νe− + νe−γνγe−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Thus, the combined feedback mechanism coefficient for RREA generation is Γc = (νe− + νe−γνγe−) (6) To validate the derived formula, the fourth generation of feedback must be compared to the third generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' N 4 RREA = N0(νe− + νγe−νe−γ)3 (7) ΓcN 3 RREA = N0(νe− + νγe−νe−γ)3 (8) As expected, we found a stationary mode of RREA generation via combined feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Multiplication coefficients In addition to III, defining ν coefficients will lead to the final formation of a combined feedback mechanism [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' νe− = Ke− � e L(λanih−λRREA) λanihλRREA − 1 − L(λanih − λRREA) λanihλRREA � , (9) where Ke−= Pe−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='e+Pe+λRREA λ2λγλγ→e−e+ � λRREAλanih λanih−λRREA �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' νe−γ = λRREA λγ � e z−z0 λRREA − 1 � (10) Because of the exponential law of growth of RREA and the fact that formula (10) describes a source-function of gamma-birth, which is also given in the form of an exponent, the process γ → e− cannot be simply expressed through a numerical coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The only plausible explanation is that νγe− is an integral operator acting on the νe−γ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' 3 e− RREA γ e−e+ e+ → e− γ → e− RREA γ e+e− RREA γ e+e− Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' 1: Schematic image of positron and gamma relativistic feedbacks producing RREA’s second generation in supercritical ( E Ebe ≥ 1) electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Consideration includes such processes as: impact ionization, bremsstrahlung and electron-positron pair generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The result of operator νγe− acting on νe−γ is shown in [22] νe−γνγe = Ke−γ,γe � e L(λx−λRREA) λxλRREA − 1 − L(λx − λRREA) λxλRREA � , (11) where Ke−γ,γe = PγPe−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='γ λγ→eλγ � λRREAλx λx−λRREA �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' AN EXACT SOLUTION OF COMBINED FEEDBACK The co-dependence of positron and gamma feedbacks is essential for obtaining an exact solution from combined feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' For the purpose of simplification of calculations, one should reconsider the study of distribution functions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The primary electron avalanche starts at z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Parameter λRREA- is the length of the exponential rise of RREA, λγ- is the length of a runaway electron before the gamma is born, λ+- is the length of the gamma before the electron- positron pair is born.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Number of produced gamma quanta inside the interval [z0, z]: fγ(z, z0) = λRREA λγ � e z−z0 λRREA − 1 � (12) Number of produced positrons inside the interval [z0, z]: f+(z, z0) = � z z0 fγ(ζ, z0) dζ λ+ = λRREA λγλ+ � λRREAe z−z0 λRREA − − λRREA − (z − z0) � (13) Number of gamma quanta deployed within a segment [z0, z]: fγ′(z, z0) = Pγfγ(z, z0) (14) Where λγ→e is the path length of the gamma before the runaway electron is born.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Let Px be the probability of the particle x turning around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Consider that the number of electrons produced by unfolded gamma quanta varies according to the law dN dz = 1 λγ→e e− z λx , where λx is some characteristic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Then the number of secondary electron avalanches born at coordinate z in thickness dz: df gamma 2 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' z0) = dz · Pγ · Pe− λγ→e � L z dζ ∂fγ(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' z0) ∂ζ e z−ζ λx (15) Simultaneously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' the following is the number of secondary electron avalanches born in the z coordinate in the thickness dz: df pos 2 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' z0) = dz · Pe−Pe+ λ2 � L z dζ ∂f+(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' z0) ∂ζ e− ζ−z λanih (16) Since the dynamics of secondary electron avalanches are no different from those of the primary avalanche,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' the following iterative equation can be written for subsequent generations: fi(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' 0) = � L 0 (f pos 2 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' ζ) + f gamma 2 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' ζ)) · �∂f pos i−1(ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' 0) ∂ζ + +∂f gamma i−1 (ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' 0) ∂ζ � dζ (17) Consistently solving this equation leads to the following: there is a stationary mode of RREA generation with the following multiplication factor: 4 Γ = νe− + νe−γνγe + � Ke−Ke−γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='γe � � � �λanihλRREAe L(λanih−λRREA) λanihλRREA λanih−λRREA − λanihλRREA λanih − λRREA − L � � + + � �λxλRREAe L(λx−λRREA) λxλRREA λx − λRREA − λxλRREA λx − λRREA − L � � � � (18) Figure 2 shows an analysis of the minimal conditions for an infinite RREA burst (Γ = 1) to estimate the difference between equations (6) and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The electric field value is given as a quantity ratio to the break-even field [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' DISCUSSION The derivation of equations (6) and (18) allows the runaway electrons dynamics and RREA burst parameters to be studied in a positron and gamma combined feedback model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' It can be seen from the figures 2a- 2b difference between (6) and (18) is not significant, and consideration of formula (18) can slightly soften conditions for RREA burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Thereby, it was shown that instead of deriving the multiplication feedback coefficient from direct solutions of the equation (17), considering independent positron and gamma relativistic feedbacks is suitable for assessing the dynamics of RREA within a homogeneous supercritical electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' According to [5, 24], experimental observations show a correlation between lightning strikes and electric field values close to Ebe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The size of a thundercloud region with a homogeneous electric field with E ≥ Ebe, on the other hand, was not measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Cell length could be estimated using the equation (18) and results from papers such as [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Table I was obtained under the assumption that lightning initiation is associated with a RREA burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Parameters at the moments of lightning discharges with Γ = 1 In situ parameters E/Ebe = 1 E/Ebe = 2 H, km E, kV/m E/Ebe L, km Lmax, km Lmin, km 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='14 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='99 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='7 28 1 10 59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='939 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='8 49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='29 48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='89 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 58 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='25 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='86 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='6 36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='75 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='886 42 38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='3 Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' I: In situ parameters at the moments of lightning discharges [25] with cell length L derived from equation (18) under assumption Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The smallest Lmin and largest Lmax values estimations for L corresponding to experimentally observed range of E/Ebe [24] are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Altitude H is given in kilometers, electric field E in kV m and L in kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' In Table I first four rows are filled with parameters (H- altitude, E- electric field) measured at the moment of lightning strike [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' In addition, estimations of the smallest and largest values for L were calculated, given by the experimentally observed range of values of the parameter E Ebe = 1 ÷ 2 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' These estimations differ significantly due to the rapid behavior of the minimal conditions for RREA burst, shown in the figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' To satisfy minimal conditions for RREA burst cell length has to be of order of 10 km ( E Ebe = 1) within a framework of combined relativistic feedback model, or even by strong overestimation of values of electric fields ( E Ebe = 2) L is of order of ≳ 1 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Thus, obtained results suggest the need of modification of the concept of relativistic feedback for the theory of lightning initiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' As such modifications, the study of inhomogeneous structures of electric fields in thunderstorms [20], as well as the possible influence of hydrometeors and their geometry on the generation of runaway electrons [27], are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The main requirement for such modification should be the mitigation of RREA burst condition for the observed parameters of thundercloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' CONCLUSION The aim of this study was to propose a method of strengthening the impact of relativistic feedback on RREA multiplication in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Interference between positron- and gamma-based relativistic feedback was proposed to amplify runaway electron generation, potentially leading to a decrease in the minimal electric field required to induce RREA burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' The influence of RREA’s secondary particles (positron and gamma quanta) on its dynamics is expressed not only in their independent combination of relativistic feedbacks but also in mutual amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Thus, the resulting multiplication factor of combined relativistic feedback allowed an analysis of the minimal conditions for RREA burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Despite all of the above, combined relativistic feedback does not provide RREA burst under the conditions observed in numerous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' Further research into the influence of feedback mechanisms on RREA dynamics in atmospheric electric field structures will be required to develop a modified feedback mechanism capable of producing RREA bursts at electric field values comparable to those observed in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' As well, it is important to study its role in high-energy atmospheric processes, such as TGF, TGE, NBE, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' 5 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' 2: For different altitudes, the minimal conditions for RREA burst (Γ = 1) are calculated in cases of independent combination (positron and gamma feedbacks are taken as independent mechanisms) and the exact solution of the equation (17), which includes full consideration of feedbacks within their mutual amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content=' 5 Independentcombination,5km E 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 Exact solution,5km 4 Independentcombination,8km 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 Exactsolution,8km 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Cell length, m5 E Independentcombination,1okm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 Exactsolution,10km 4 Independent combination, 15 km 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 Exactsolution,15km 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} +page_content='5 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Cell length, m6 [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edAyT4oBgHgl3EQfjfiM/content/2301.00416v1.pdf'} 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a/edAzT4oBgHgl3EQf3v41/content/tmp_files/2301.01833v1.pdf.txt b/edAzT4oBgHgl3EQf3v41/content/tmp_files/2301.01833v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..443d253f6d4f47991cf9e68dc6a5275f28d6f6e3 --- /dev/null +++ b/edAzT4oBgHgl3EQf3v41/content/tmp_files/2301.01833v1.pdf.txt @@ -0,0 +1,2160 @@ +(2023) 1–20 +Journal +Logo +Classical multivariate Hermite coordinate interpolation in +n-dimensional grid +Aristides I. Kechriniotisa, Konstantinos K. Delibasisb,∗, Iro P. Oikonomouc, Georgios N. +Tsigaridasd +aDepartment of Physics, University of Thessaly, 3rd Km Old National Road Lamia–Athens 35100, Lamia Greece +bDepartment of Computer Science and Biomedical Informatics, University of Thessaly, 2-4 Papasiopoulou str., P.O. 35131 Lamia, Greece +cDepartment of Informatics and Telecommunications, National and Kapodistrian University of Athens, Panepistimioupolis, Ilisia 157 84, Athens +Greece +dDepartment of Physics School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus +GR-15780 Zografou, Athens Greece +Abstract +In this work, we study the Hermite interpolation on n-dimensional non-equal spaced, rectilinear grids over a field k of characteristic +zero, given the values of the function at each point of the grid and the partial derivatives up to a maximum degree. First, we prove the +uniqueness of the interpolating polynomial, and we further obtain a compact closed form that uses a single summation, irrespective +of the dimensionality. The arithmetic complexity of the derived closed formula compares favourably with the only alternative +closed form for the n-dimensional classical Hermite interpolation [1]. In addition, we provide the remainder of the interpolation. +Finally, we perform illustrative numerical examples to showcase the applicability and high accuracy of the proposed interpolant, +compared to other interpolation methods. +Keywords: polynomial interpolation, multivariate Hermite, classical Hermite interpolation, n-Dimensional grid +PACS: 03.65.Pm, 03.50.De, 41.20.-q +1. Introduction and Notations +This work focuses on multivariate classical Hermite interpolation with support points arranged on an n-dimensional +non-equally spaced rectilinear grid (nD grid), given the value of a function, as well as its derivatives up to an arbitrary +maximum order, defined independently for each point and each dimension. An old survey of Hermite interpolation +methods [2] discusses the issues of uniform, regular and singular interpolation in n dimensions, however no closed +form, or interpolation error is provided for the interpolant polynomial. In addition, a number of works deal with bi- +variate Hermite-like interpolation, where the order of partial derivatives along the two dimensions are not independent +[3, 4], which is a different than the classical Hermite interpolation. +Let N0 the set of non-negative integers, and N the set of positive integers. We first state the generalized univariate +(1D) Hermite interpolation formula proposed by Spitzbart [5]: +∗Corresponding author. Tel.: (+30) 22310 66908. +URL: arisk7@gmail.com (Aristides I. Kechriniotis), kdelibasis@gmail.com (Konstantinos K. Delibasis), +iro.oikonomou99@gmail.com (Iro P. Oikonomou), gtsig@mail.ntua.gr (Georgios N. Tsigaridas) +1 +arXiv:2301.01833v1 [math.NA] 4 Jan 2023 + +bleonlineatwww.sciencedirect.con +ScienceDirect/ (2023) 1–20 +2 +Theorem 1.1. Let A be a finite subset of R, and ν : A → N the multiplicity function. Further, let V(A, ν) be the +R-vector space �p ∈ R [x] : deg p < � +a∈A ν (a)� . Given the real numbers tk +a, a ∈ A, k ∈ {0, 1, ..., ν (a) − 1}, then there +is a unique p ∈ V (A,ν) such that p(k) (a) = tk +a, a ∈ A , k ∈ {0, 1, ..., ν (a) − 1}, given by +p = +� +a∈A +ν(a)−1 +� +k=0 +tk +aHk +a, +(1.1) +where Hk +a (x) = Ha (x) (x − a)k +k! +ν(a)−k−1 +� +t=0 +(x − a)t +t! +dt +dxt +� +1 +Ha (x) +� +(a), with Ha (x) := +� +c∈A +c�a +� x − c +a − c +�ν(a) +. +In the above notation, a denotes any of the support points A (where the values of the unknown function, as well as +the values of its derivatives are given) and the multiplicity function v holds the maximum order of derivative minus 1. +The above Theorem can be extended when A is a subset of any field k of characteristic zero. +In our previous work [6], we derived a new closed form expression of the univariate Hermite interpolating poly- +nomial for the general case of arbitrarily spaced data, that was algebraically significantly simpler than Theorem 1.1, +since it only requires simple matrix multiplications, rather than n-order derivatives of rational polyonomial functions +Hk +z(x) of Eq.(1.1). +The 1D Hermite interpolant of Theorem 1.1 can be easily extended to multivariate interpolation. Namely, in [7] the +two dimensional (2D) interpolation is considered. In [8] we extended our previous work [6] into two dimensions (2D), +deriving closed-form expression that was again much more compact than [9, 7] and applicable in the case of support +points arranged on a non-equidistant grid. For both the 1D and 2D cases, we provided means for computationally +efficient implementations of the proposed Hermite interpolating polynomials that achieved computational complexity +comparable to other popular and much simpler interpolation techniques, such as cubic splines, whereas the measured +error when applied to clinical medical images was superior. +For the generalization of Hermite interpolation into n dimensions, we require the following notations. +Let A be a set, |A| the cardinality of A, and An := A × · · · × A +n−times +. Given the sets A1, ..., An, then A :=A1 × ... × An. Further, +the element (a1, ...an) ∈ A will be denoted by a. Let 0 = (0, 0, ..., 0), 1 = (1, 1, ..., 1) be the zero vector and ones vector, +respectively. Thus, points a = (a1, ...an) are arranged on a non-regular N-dimensional grid A. Let k be a field of +characteristic zero. For a ∈kn, and m ∈Nn +0 we denote am := �n +i=1 ami +i . Let k = (k1, ...kn) be an n-dimensional vector +of non-negative integers, holding the order of partial derivatives of the interpolating polynomial with respect to each +variable. +In Nn +0 we define the relation ”≤ ” as follows: k ≤ m if and only if ki ≤ mi, for every i = 1, ..., n. Clearly +� +Nn +0, ≤ +� +is +a poset (Nn +0 is partially ordered). If k ≤ m and k � m, then [k, m] := +� +l ∈Nn +0 : k ≤ l ≤ m +� += [k1, m1] × · · · × [kn, mn], +and is valid |[k, m]| = �n +i=1 (mi − ki + 1). +Given the finite subsets Ai, i = 1, ..., n of the field k, and the multiplicity functions νi : Ai → N, i = 1, ..., n. Then +for i ∈ {1, ..., n}, and a ∈ Ai we define +H(i,a) (xi) := +� +c∈Ai +c�a +� xi − c +a − c +�νi(a) +∈ K [xi] . +(1.2) +Let ν : A → Nn be the generalized multiplicity function given by ν (a) := (ν1 (a1) , ..., νn (an)). For a ∈ A and +k ∈ [0,ν (a) − 1] we define +H(a,k) (x1, ...xn) := +n +� +i=1 +(xi − ai)ki H(i,ai) (xi) +ki! +∈ k [x1, ..., xn] . +(1.3) +Let V (A,ν) denote the k-vector space +� +f ∈ k [x1, ..., xn] : degi f < � +a∈Ai νi (a) , i = 1, ..., n +� +, a basis of which is the +following +B (A,ν) := B (A,ν) := +�������xk : k ∈ +�������0, − 1+ +� +a∈A +ν (a) +������� +������� . +2 + +/ (2023) 1–20 +3 +Further, let us define Haj +� +x j +� +:= +� +c∈Aj +c�aj +� x j − c +a j − c +�ν(a) +, for a j ∈ Aj, as well as +H +k j +aj +� +x j +� +:= Haj +� +x j +� +� +xj − a j +�k j +kj! +ν(aj)−k j−1 +� +t=0 +� +xj − a j +�t +t! +dt +dxt +j +�������� +1 +Haj +� +x j +� +�������� +� +aj +� +∈ k[x], +and subsequently, Hk +a (x1, ...xn) := +n +� +j=1 +H +kj +a j(x j) ∈ k [x1, ..., xn] . +Let us also define P(A, ν) as follows +P (A, ν) := �H(a,k) : a ∈ A, k ∈ [0,ν (a) − 1]� . +Finally, we define below the partial derivative operator acting on f : +∂k := +n +� +i=1 +∂ki +i , ∂k +a f (x) := ∂k f (x) |x=a , where ∂k +i := ∂k +∂xk +i +. +To our knowledge, the only n-dimensional multivariate generalization of the 1D classical Hermite interpolation in +Spitzbart [5] was proposed very recently in [1], and it is presented in the following Theorem: +Theorem 1.2. Let tk +a ∈ R , a ∈ A := A1 × ... × An, k ∈ [0,ν (a) − 1] . Then there exists a unique p ∈ V (A,ν) such that +∂k +a p = tk +a, a ∈ A, k ∈ [0,ν (a) − 1], given by +p = +� +a∈A +� +m∈[0,ν(a)−1] +tk +aHk +a = +� +a1∈A1 +· · · +� +an∈An +ν1(a1)−1 +� +k1=0 +· · · +νn(an)−1 +� +kn=0 +n +� +j=1 +H +k j +aj +� +xj +� +tk +a. +(1.4) +The authors of that work however did not provide the remainder of the interpolating polynomial. Furthermore, +the algebraic complexity of the Hermite polynomial expressed in Theorem1.2 increases rapidly with the number of +dimensions n, which can be confirmed by comparing equations (1.1) and (1.4). +The present work deals with the Hermite interpolating polynomial into a regular, non-equidistant grid of support +points in arbitrarily high dimensions (nD grids) and provides an elegant, compact closed-form expression with only +one summation over support points and without requiring derivatives of rational polynomial functions. We also +provide the interpolation remainder, which, to our knowledge, has not been given so far. +The article is organized as follows. Having defined some necessary notations, we provide in the next section the +lemmas and remarks required for Theorem 2.2 about the uniqueness of the interpolating polynomial and Theorem +2.5 for the proposed closed formula of the interpolant. The remainder of the interpolating polynomial is also proven +in section 3. In section 4 some properties of the ideal of the interpolation are provided. The notations introduced +in this paper, also facilitate a very short proof of Theorem 1.2, which is provided in section 5 for completeness. Fi- +nally, simple algebraic examples are given, showing that the bilinear and trilinear interpolation are special cases of the +proposed n-dimensional Hermite interpolation. In addition, few arithmetic implementations of the proposed interpo- +lation of known functions are provided, demonstrating its superiority in accuracy against other popular interpolation +techniques. +2. The proposed Hermite interpolation on n-D rectilinear grids +Remark 1. It is easy to verify that +���� +� +ak : a ∈ A, k ∈ [0,ν (a) − 1] +����� = +� +a∈A +� +k∈[0,ν(a)−1] +1 = +� +a∈A +n +� +i=1 +νi (ai) = +n +� +i=1 +� +a∈Ai +νi (a) . +3 + +/ (2023) 1–20 +4 +Remark 2. The cardinality of the set P(A, ν) is given by |P (A,ν)| = +���� +��n +i=1 (x − a)k : a ∈ A, k ∈ [0,ν (a) − 1] +�����, which +can be simplified: +|P (A,ν)| = +� +a∈A +n +� +i=1 +νi (ai) . +The above yields |P (A,ν)| = �n +i=1 +� +ai∈Ai νi (a) = dimk V (A,ν). The set of partial derivative operators has equal +cardinality: +����∂m +a , a ∈ A, m ∈ [0,ν (a) − 1]���� = +n +� +i=1 +� +ai∈Ai +νi (a) . +Remark 3. The set P is a subset of V, P (A,ν) ⊂ V (A,ν), since degi H(a,k) = ki + � +a∈Ai−{ai} νi (a) < � +a∈Ai νi (a). +Remark 4. For a, b ∈ A, and k, m ∈ [0,ν (a) − 1] by using the Leibniz derivative rule we easily get: +∂k +aH(b,m) = +��������������� +0, +if +a � b +0, +if +a = b +and +k < m +0, +if +a = b +and +k, m +are incomparable +1, +if +a = b +and +m = k +. +Lemma 2.1. Let (G, ≤) be a poset, and A a finite subset of G. Then the system of linear equations +� +β≤a +ca,βxβ = da, a ∈ A, +(2.1) +has unique solution, where ca,β ∈ k with ca,a � 0 for all a ∈ A. +Proof. Let |A| = m. Then we number the elements of A as follows: We choose radomly a minimal element of +A which will be denoted by a1. Next we choose a minimal element of A − {a1} denoted by a2, and so on. That is +A = {a1, . . . , am} . Therefore (2.1) can be rewritten as +j +� +i=1 +εi, jcaixai = daj, j = 1, . . . , m, +where +εi,j := +������������� +0, +if aj < ai +0, +if ai, a j are not comparable +1, +if ai ≤ aj, +Clearly, the matrix of the coefficients of the variables xai is lower triagular, and because all its diagonal elements +εj, jcaj are non zero, we conclude that (2.1) has exactly one solution. +We now present the Theorem for the uniqueness of the multivariate Hermit interpolating polynomial. +Theorem 2.2. Given the elements tk +a ∈ k , a ∈ A, k ∈ [0,ν (a) − 1], there exists a unique f ∈ V (A,ν) such that +∂m +a f = tm +a , a ∈ A, m ∈ [0,ν (a) − 1]. +Proof. It is sufficient to show that there are unique elements xm +a ∈ k, a ∈ A, m ∈ [0,ν (a) − 1] , such that the polyno- +mial +f = +� +b∈A +� +k∈[0,ν(b)−1] +xk +bH(b,k) +(2.2) +satisfies the conditions +∂m +a f = tm +a . +4 + +/ (2023) 1–20 +5 +By applying the derivative operators ∂m +a , a ∈ A, m ∈ [0,ν (a) − 1] to (2.2), we get the following system consisting of +a number of � +a∈A +�n +i=1 νi (ai) linear equations with an equal number of variables xk +a, a ∈ A, m ∈ [0,ν (a) − 1]: +∂m +a f = +� +b∈A +� +k∈[0,ν(b)−1] +xk +b∂m +a H(b,k), +or +� +b∈A +� +k∈[0,ν(b)−1] +xk +b∂m +a H(b,k) = tm +a , +which by Remark 3 can be partitioned to the following linear equation systems +� +k∈[0,m] +xk +a∂m +a H(a,k) = tm +a , m ∈ [0,ν (a) − 1] . +(2.3) +Clearly each one of the linear systems (2.3) has a number of |ν (a)| = �n +i=1 νi (ai) variables, xk +a, and consists of +|ν (a)| equations. Since [0,ν (a) − 1] is a finite subset of the partially ordered Zn, and because, by Remark 4 for each +m ∈ [0,ν (a) − 1] is valid ∂m +a H(a,m) = 1,then by Lemma 2.1 follows that the system (2.2) has exact a solution. +As a result of Theorem 2.2 we get the following Corollaries. +Corollary 2.3. The set P (A,ν) is a basis of the k-vector space V (A,ν) . +Proof. By Remarks 1 and 2 it is sufficient to show that the elements of P (A,ν) are linearly independent. Similarly +to the the proof of Theorem 2.2, we get � +k∈[0,m] xk +a∂m +a H(a,k) = 0, m ∈ [0,ν (a) − 1] by substituting tm +a = 0 in (2.3). +Consequently, for any a ∈ A, we have xm +a = 0 for all m ∈ [0,ν (a) − 1] . +Corollary 2.4. If f is an element of V (A,ν), such that ∂m +a f = 0 for each a ∈ A, m ∈ [0,ν (a) − 1], then f = 0. +We will derive an expression for the interpolating polynomial +f in Theorem 2.2. First we will use the degree +reverse lexicographic order, which will be denoted by ≺ . More specifically,(k1, . . . , kn) ≺ (l1, . . . ln), if either of the +following holds: +(i) k1 + · · · + kn < l1 + · · · + ln, or +(ii) k1 + · · · + kn = l1 + · · · + ln and ki > li for the largest i for which ki � li. +For example, the reverse lexicographic order of the elements in [0,ν (a) − 1] is +1a : += +(0, . . . , 0) ≺ 2a := (0, . . . , 0, 1) ≺ 3a := (0, . . . , 0, 1, 0) ≺ · · · ≺ (n + 1)a := (1, 0, . . . , 0) +≺ +(n + 2)a := (0, . . . , 0, 2) ≺ · · · ≺ (|ν (a)| − 1)a := (ν1 (a1) − 1, ..., νn (an) − 1) . +Note that from m ≤ n follows m ⪯ n, and from m < n follows m ≺ n. That means ⪯ is a linear extension of ≤. +The following Theorem provides the closed form of the interpolating polynomial. +Theorem 2.5. The formula of the interpolating polynomial f in (2.2) is the following +f := +� +a∈A +Λ−1 +a TaHa = +� +a∈A +|ν(a)| +� +i=1 +�I|ν(a)| − Λa +� TaHa, +(2.4) +where +Λa += +������������������������ +1 +0 +· · · +0 +0 +ε2,1∂2a +a H(a,1a) +1 +· · · +0 +0 +... +... +... +... +... +ε|ν(a)|−1,1∂(|ν(a)|−1)a +a +H(a,1a) +ε|ν(a)|−1,2∂(|ν(a)|−1)a +a +H(a,2a) +· · · +1 +0 +ε|ν(a)|,1∂|ν(a)|a +a +H(a,1a) +ε|ν(a)|,2∂|ν(a)|a +a +H(a,2a) +· · · +ε|ν(a)|,|ν(a)|−1∂|ν(a)|a +a +H(a,(|ν(a)|−1)a) +1 +������������������������ +, (2.5) +xa += +������������� +x1a +a +... +x|ν(a)| +a +������������� +, Ta = +������������� +t1a +a +... +t|ν(a)| +a +������������� +, Ha = +������������� +H(a,1a) +... +H(a,|ν(a)|) +������������� +, |ν (a)| = +n +� +i=1 +νi (ai) . +5 + +/ (2023) 1–20 +6 +Proof. The interpolating polynomial f in (2.2) can be equivalently written as +f = +� +a∈A +|ν(a)| +� +i=1 +xiaa H(a,ia) +or +f = +� +a∈A +xaHa, +(2.6) +where xa = +� +xiaa +� +is a column matrix of size |ν (a)| × 1, and Ha = �H(a,ia) +� is a row matrix of size 1 × |ν (a)| . Then the +system of linear equations becomes, +j +� +i=1 +εi,j∂ ja +a H(a,ia)xiaa = t ja +a , j = 1, ..., |ν (a)| , +where +εi,j := +������������� +0, +if ja < ia +0, +if ia, ja are not comparable +1, +if ia ≤ ja, +Now we write this system of linear equations in matrix form: +Λaxa = Ta, +(2.7) +where Λa = +� +εi, j∂ja +a H(a,ia) +� +is a lower unitriagular matrix of size |ν (a)| × |ν (a)|, and Ta = +� +tiaa +� +is a column matrix of +size |ν (a)| × 1. From (2.7) we have xa = Λ−1 +a Ta, or by Lemma 2.1 in ref 2, +xa = +|ν(a)| +� +i=1 +�I|ν(a)| − Λa +�i Ta, +(2.8) +where I|ν(a)| is unit matrix of size |ν (a)|. Finally, substituting (2.8) in (2.6) yileds the required formula. +3. Remainder of the interpolation for k = R or C +In this section, we derive an expression for the error of the interpolation formula given in Theorem 2.5, which is +identical to the error of Theorem 1.2, for the class of real functions f (x1, . . . , xn), which can be continued analytically +as a single valued, regular function f (z1, . . . , zn) of n complex variables in a certain cross-product region Dz1×· · ·×Dzn. +In the case of a single variable x, let C be a closed contour in the region Dz of analytic continuation of a real +function f(x) containing the points a ∈ A in its interior. Let as denote +H (x) = +� +a∈A +(x − a)ν(a) . +By applying the residue theorem to the contour integral +1 +2πi +� +C +f (z) +(z − x) H (z)dz, +we obtain (see [7]) +f(x) = p (x) + H (x) +2πi +� +C +f (z) +(z − x) H (z)dz, +(3.1) +where +p (x) = +� +a∈A +ν(a)−1 +� +k=0 +f (k) (a) Hk +a, +6 + +/ (2023) 1–20 +7 +From (3.1) we have +1 +2πi +� +C +f (z) +z − xdz = p (x) − H (x) +2πi +� +C +f (z) +(z − x) H (z)dz, +or equivalently +1 +2πi +� +C +H (z) − H (x) +H (z) (z − x) f (z) dz = p (x) . +(3.2) +In the following, we assume that C j, j = 1, . . . , n are simple closed contours in the regions Dj containing Aj in +their interior, of analyticity of f +� +x1, . . . , xj−1, z j, xj+1, . . . , xn +� +, where x1, . . . , xj−1, xj+1, . . . , xn are fixed. Further, we +assume that f (z1, . . . , zn) is simultaneously analytic in Dz1 × · · · × Dzn. +Finally, we introduce the following notation: +Hr(wr) = +� +a∈Ar +(wr − a)νr(a), r = 1, . . . , n. +Lemma 3.1. The following identity holds: +1 +(2πi)n +� +C1 +· · · +� +Cn +�n +r=1 (Hr (zr) − Hr (xr)) +�n +r=1 (zr − xr) Hr (zr) f (z1, . . . , zn) dz1 . . . dzn = +� +a∈A +� +k∈[0,ν(a)−1] +Hk +a∂k +a f. +(3.3) +Proof. We will prove by induction with respect to n: For n = 1, (3.3) reduces to (3.1). Let (3.3) hold for n − 1. Then +1 +(2πi)n +� +C1 +· · · +� +Cn +�n +r=1 (Hr (zr) − Hr (xr)) +�n +r=1 (zr − xr) Hr (zr) f (z1, . . . , zn) dz1 . . . dzn += +1 +2πi +� +Cn +Hn (zn) − Hn (xn) +(zn − xn) Hn (zr) +������ +1 +(2πi)n−1 +� +C1 +· · · +� +Cn−1 +�n−1 +r=1 (Hr (zr) − Hr (xr)) +�n−1 +r=1 (zr − xr) Hr (zr) +f (z1, . . . , zn) dz1 . . . dzn−1 +������ dzn += +1 +2πi +� +Cn +Hn (zn) − Hn (xn) +(zn − xn) Hn (zr) +� +an∈A1 +· · · +� +an∈An +ν1(a1)−1 +� +k1=0 +· · · +νn−1(an−1)−1 +� +kn−1=0 +n−1 +� +j=1 +H +kj +a j +� +xj +� +∂k1+···+kn−1 +∂xk1 +1 . . . ∂xkn−1 +n−1 +f �a1, . . . , an−1, zn +� dzn += +� +an∈A1 +· · · +� +an∈An +ν1(a1)−1 +� +k1=0 +· · · +νn−1(an−1)−1 +� +kn−1=0 +n−1 +� +j=1 +H +k j +aj(x j) +������� +1 +2πi +� +Cn +Hn (zn) − Hn (xn) +(zn − xn) Hn (zr) +∂k1+···+kn−1 +∂xk1 +1 . . . ∂xkn−1 +n−1 +f �a1, . . . , an−1, zn +� dzn +������� . +Thus, using (3.2) we obtain: +1 +(2πi)n +� +C1 +· · · +� +Cn +�n +r=1 (Hr (zr) − Hr (xr)) +�n +r=1 (zr − xr) Hr (zr) f (z1, . . . , zn) dz1 . . . dzn += +� +a1∈A1 +· · · +� +an∈An +ν1(a1)−1 +� +kn=0 +· · · +νn(an)−1 +� +kn=0 +n +� +j=1 +H +k j +aj +� +x j +� +∂kn+···+k1 +∂xkn +n . . . ∂xk1 +1 +f (a1, . . . , an) += +� +a∈A +� +k∈[0,ν(a)−1] +Hk +a∂k +a f. +The interpolation error is derived in the following Theorem. +Theorem 3.2. The error of the interpolation formula of Theorem 2.5 is defined as +R := f − +� +a∈A +� +k∈[0,ν(a)−1] +tk +aHk +a, +where tk +a = ∂k +a f , is given by: +R = +1 +(2πi)n +� +C1 +· · · +� +Cn +�n +r=1 Hr (zr) − �n +r=1 (Hr (zr) − Hr (xr)) +�n +r=1 (zr − xr) Hr (zr) +f (z1, . . . , zn) dz1 . . . dzn. +(3.4) +7 + +/ (2023) 1–20 +8 +Proof. From (3.3) we have +1 +(2πi)n +� +C1 +· · · +� +Cn +�n +r=1 Hr (zr) − �n +r=1 (Hr (zr) − Hr (xr)) +�n +r=1 (zr − xr) Hr (zr) +f (z1, . . . , zn) dz1 . . . dzn +(3.5) += +1 +(2πi)n +� +C1 +· · · +� +Cn +1 +�n +r=1 (zr − xr) f (z1, . . . , zn) dz1 . . . dzn − +� +a∈A +� +k∈[0,ν(a)−1] +Hk +a∂k +a f. +By applying the residue Theorem n-times, we obtain +1 +(2πi)n +� +C1 +· · · +� +Cn +1 +�n +r=1 (zr − xr) f (z1, . . . , zn) dz1 . . . dzn = f (x1, . . . , xn) . +(3.6) +From relations (3.5) and (3.6) we get (3.4). +Remark 5. Applying Theorem 3.2 for n = 2 we obtain the error formula for the bivariate Hermite polynomial, as +given in [7, Section 5]. +Remark 6. Lemma 3.1 provides an alternative way of generating the Hermite polynomial that interpolates any func- +tion f, provided that the integral can be calculated. A relevant example will be given in the next section. +Remark 7. Lemma 3.1 and Theorem 3.2 hold also for complex functions of complex variables and for finite subsets +A1, . . . , An of C. +Remark 8. Let us denote by H �Dz1 × · · · × Dzn +� the ring of all holomorphic functions on Dz1 × · · · × Dzn and by +(h1, . . . , hn) the ideal generated by the polynomials h1, . . . , hn. Consider the linear operator T(A,ν) : H �Dz1 × · · · × Dzn +� → +V (A,ν) given by +T(A,ν) ( f) = +1 +(2πi)n +� +C1 +· · · +� +Cn +�n +r=1 (hr (zr) − hr (xr)) +�n +r=1 (zr − xr) hr (zr) f (z1, . . . , zn) dz1 . . . dzn. +Clearly by Lemma 3.1 we have that T(A,ν) is an epimorphism and +ker T(A,ν) = �g ∈ H �Dz1 × · · · × Dzn +� : ∂m +a g = 0, a ∈ A, m ∈ [0,ν (a) − 1]� . +By using Leibniz derivative rule for multivariable functions we have that for any f ∈ H �Dz1 × · · · × Dzn +� and any +g ∈ ker T(A,ν) hold ∂m +a ( fg) = 0. Therefore ker T(A,ν) is an ideal of H �Dz1 × · · · × Dzn +�. In other words, T(A,ν) is an ideal +projector. Further by residue Theorem we get T(A,ν) (hr) = 0, r = 1, . . . , n . Therefore (h1, . . . , hn) ⊆ ker T(A,ν). +In the next section will be in generally clarified the connection between the ideals (h1, . . . , hn) and ker T(A,ν) which +will be denoted by J (A,ν). +4. On the ideal of the interpolation +Let us consider the polynomials fi ∈ k [xi] , i = 1, . . . , n, Ai the sets of their roots in the algebraic closure k of +k, and νi (a) the multiplicity of a ∈ Ai. Therefore fi (xi) = � +a∈Ai (xi − a)νi(a) , i = 1, . . . , n. Furthermore, we denote +by I the ideal of k [x1, . . . , xn] generated by f1, . . . , fn, and by I the ideal of k [x1, . . . , xn] generated by f1, . . . , fn. +For g ∈ k [x1, . . . , xn] let us denote by �g the residue class of g in k [x1, . . . , xn] /I , and by g the residue class of g in +k [x1, . . . , xn] /I and J (A,ν) := +� +f ∈ k [x1, . . . , xn] : ∂m +a f = 0, a ∈ A, m ∈ [0,ν (a) − 1] +� +. Similarly to Remark 8 it can +be shown that J (A,ν) is an ideal of k [x1, . . . , xn] . +Lemma 4.1. The k-algebra k [x1, . . . , xn] /J (A,ν) as a k−vector space is isomorphic to V (A,ν), and +dimk k [x1, . . . , xn] /J (A,ν) = +n +� +i=1 +� +ai∈Ai +νi (a) . +8 + +/ (2023) 1–20 +9 +Proof. Let φ : k [x1, . . . , xn] → k [x1, . . . , xn] /J (A,ν) be the canonical map and B (A,ν) := +� +xk : k ∈ �0, − 1+ � +a∈A ν (a)�� +a basis of the vector space V (A,ν). It is sufficient to show that φ (B (A,ν)) is a basis of K [x1, . . . , xn] /J (A,ν). Starting +from � +k∈[0,−1+ � +a∈A ν(a)] ckφ +� +xk� += 0, we have � +k∈[0,−1+ � +a∈A ν(a)] ckxk ∈ J (A,ν), or equivalently ∂m +a +� +k∈[0,−1+ � +a∈A ν(a)] ckxk = +0, so according to Corollary 2.4 � +k∈[0,−1+ � +a∈A ν(a)] ckxk = 0. That is ck = 0, k ∈ �0, − 1+ � +a∈A ν (a)�, which means that +φ +� +xk� +, k ∈ �0, − 1+ � +a∈A ν (a)� are linearly indepedend. Now we will show that φ +� +xk� +, k ∈ �0, − 1+ � +a∈A ν (a)� gener- +ate k [x1, . . . , xn] /J (A,ν): By Theorem 2.2, for any g ∈ k [x1, . . . , xn] there is exactly one f = � +k∈[0,−1+ � +a∈A ν(a)] ckxk ∈ +V (A,ν) such that ∂m +a f = ∂m +a g, a ∈ A, m ∈ [0,ν (a) − 1]. Consequently, φ (g) = φ ( f) or +φ (g) = +� +k∈[0,−1+ � +a∈A ν(a)] +ckφ +� +xk� +. +Lemma 4.2. If ( f1, . . . , fn) is the ideal of k [x1, . . . , xn] generated by the polynomials f1 ∈ k [x1] , . . . , fn ∈ k [xn], then +the quotient ring k [x1, . . . , xn] /I as a k-vector space is isomorphic to V (A,ν), and +dimk k [x1, . . . , xn] /I = +n +� +i=1 +deg fi = +n +� +i=1 +� +ai∈Ai +νi (a) . +Proof. Clearly, it is sufficient to prove that for any polynomial f ∈ k [x1, . . . , xn] there exist unique c(k1,...,kn) ∈ K, 0 ≤ +k1 < deg f1, . . . , 0 ≤ kn < deg fn, such that +f = +deg f1−1 +� +k1=0 +· · · +deg fn−1 +� +kn=0 +c(k1,...,kn)xk1 +1 . . . xkn +n + I. +To this end we will apply the method of induction over n. For n = 1 obviously the claim is valid. Let the claim +be valid for n − 1, n > 1. From f ∈ k [x1, . . . , xn−1] [xn], by dividing f +by fn (xn), we have f = pfn + q, where +p, q ∈ k [x1, . . . , xn] are uniquely determined by degn q < deg fn. Therefore +f = +− deg fn−1 +� +i=0 +qixi +n + ( fn) , +(4.1) +where qi are uniquely determined polynomials of k [x1, . . . , xn−1], and ( fn) is the ideal of k [x1, . . . , xn] generated by +fn. Now by induction the polynomials qi can be written as, +qi = +� +0≤k10 +r1+···+rm=n +k(a) +±r1k(a) +±,r2 . . . k(a) +±,rm. +(3.24) +– 10 – + +Similarly, we can write the zero modes as +ψ(a) ++,0 = exp +� +−βh1k(a) +0 +� += H−k(a) +0 +1 +, +ψ(a) +−,0 = exp +� +βh1k(a) +0 +� += Hk(a) +0 +1 +. +(3.25) +We shall refer to the modes k(a) +r +(r ∈ Z) as Heisenberg modes. There could also be different +conventions to define these modes as discussed in Appendix C. +In terms of the Heisenberg modes, we can rewrite the relations involving ψ± as +� +k(a) +0 , k(b) +s +� += 0, +� +k(a) +r̸=0, k(b) +s +� += δr+s,0 +1 +r +� +C−r − Cr� +H−rMab +2 +� +HrAab +1 +− H−rAab +1 +� +, +(3.26) +� +k(a) +0 , e(b) +n +� += −Aabe(b) +n , +� +k(a) +0 , f(b) +n +� += Aabf(b) +n , +(3.27) +� +k(a) +r̸=0, e(b) +n +� += 1 +rC−|r|/2H−rMab +2 +� +HrAab +1 +− H−rAab +1 +� +e(b) +n+r, +(3.28) +� +k(a) +r̸=0, f(b) +n +� += −1 +rC|r|/2H−rMab +2 +� +HrAab +1 +− H−rAab +1 +� +f(b) +n+r. +(3.29) +Moreover, we have +� +e(a) +n , f(b) +−n +� += δab +� +Hk(a) +0 +1 +− H−k(a) +0 +1 +� +. +(3.30) +It would also be helpful to notice that +� +e(a) +±1, f(b) +0 +� += ∓δabC1/2H∓k(a) +0 +1 +k(a) +±1, +� +e(a) +0 , f(b) +±1 +� += ∓δabC−1/2H∓k(a) +0 +1 +k(a) +±1. +(3.31) +Coproduct +We can also write the coproduct above using k(a) +r : +∆ +� +k(a) +r +� += +� +Cr ⊗ k(a) +r ++ k(a) +r +⊗ 1, +r ≥ 0 +k(a) +r +⊗ Cr + 1 ⊗ k(a) +r , +r < 0 +. +(3.32) +In particular, ∆ +� +k(a) +0 +� += 1 ⊗ k(a) +0 ++ k(a) +0 +⊗ 1. +Grading +Likewise, for the aforementioned grading, we have deg +� +k(a) +r +� += (0, r). In [21, 22], +such gradings were useful in the quantum double construction of the universal R-matrix +for certain toroidal algebra associated to gl1. For toroidal BPS algebras associated to any +non-chiral quivers here, a naive generalization would be R = R(0)R(1)R(2) with +R(0) = +� +C−1 ⊗ D−1� � +D−1 ⊗ C−1� � +a +� +ψ(a) ++,0 ⊗ +� +D(a)�−1� �� +D(a)�−1 +⊗ ψ(a) ++,0 +� +, +R(1) = exp +� +�� +r≥1 +r +� +a +k(a) +r +⊗ k(a) +−r +� +� , +R(2) = 1 ⊗ 1 + +� +n∈Z +� +a +e(a) +n +⊗ f(a) +−n + . . . , +(3.33) +where the ellipsis in R(2) indicates terms with hdeg ≥ 1, and pdeg +� +R(2)� +should be 0. How- +ever, whether these naive expressions would work and/or what modifications (such as proper +normalizations etc.) are needed would still require further investigations in future. +– 11 – + +3.2 +Toric Duality +Now let us try to construct the transformations of the generators under toric duality. As +mentioned in Appendix A.3, only fermionic nodes can be dualized. If the node ϝ is dualized, +then we just need to add an adjoint loop to ϝ±1 if |ϝ±1| = 0 or remove the existing adjoint +loop on ϝ ± 1 if |ϝ ± 1| = 1. As a result, +ς′ +a = +� +−ςa, +a = ϝ, ϝ + 1 +ςa, +otherwise +, +(3.34) +where the primed notation stands for the one after performing the duality. Therefore, we +have +A′ +ab = +� +� +� +� +� +� +� +−Aab, +(a, b) = (ϝ ± 1, ϝ), (ϝ, ϝ ± 1) +Aaa + 2Aaϝ, +a = b = ϝ ± 1 +Aab, +otherwise +(3.35) +and +M′ +ab = +� +� +� +� +� +� +� +−Mab, +a = ϝ − 1, ϝ, b = a + 1 +−Mab, +a = ϝ, ϝ + 1, b = a − 1 +Mab, +otherwise +. +(3.36) +Analogous to the rational case, the ke and kf commutation relations can be used to +express higher e, f using lower modes9. The higher modes of k can in turn be obtained using +the ef relations. In fact, the relations involving higher modes can also be derived from those +with lower modes. Therefore, the toroidal BPS algebras for non-chiral quivers are finitely +presented with the relations involving e0, e±1, f0, f±1, k0, k±1 (or equivalently, ψ±,0, ψ±,1). +Hence, it suffices to find the transformations for these modes10. +We would like to mimic the isomorphisms for the rational case in [17], which was in turn +found by virtue of the odd reflections of the underlying affine Lie superalgebras. As all but +three of the nodes are unaffected, we would expect the modes to be invariant for a ̸= ϝ, ϝ±1. +Therefore, from their relations, we have +C′ = C. +(3.37) +Now, let us first consider the zero modes. For a = ϝ, the k′ +0 modes should be determined +only by k0 themselves, possibly with changes of minus signs (such as multiplication by −1), +while the e0 and f0 modes should get swapped. In the rational case, the ψ′ +0 mode is a sum +of ψ(a) +0 +and ψ(ϝ) +0 +for a = ϝ ± 1. Here, our ansatz for ψ0 would still be a combination of ψ(a) +0 +and ψ(ϝ) +0 +, but we expect it to be a multiplication instead of addition as we are dealing with +9Here, by higher (resp. lower) modes, we mean the modes with larger (resp. smaller) absolute values |n|. +10As pointed out in [17], there is a subtlety for the case xy = z2w2. For one of the two toric phases, i.e. +the one with only fermionic nodes, it seems that the Serre relations can not be fully recovered from the Serre +relations for modes with n = 0, ±1. However, we may still verify its transformation when using the currents +as will be discussed later. +– 12 – + +the trigonometric case (and hence addition for k0). On the other hand, for e′(a) +0 +, the ansatz +would be a linear combination of e(a) +0 e(ϝ) +0 +and e(ϝ) +0 +e(a) +0 +(and likewise for f′(a) +0 +). +By computing the supercommutators [x, y} with +x = e(ϝ) +0 +e(a) +0 , e(a) +0 e(ϝ) +0 +and y = f(a) +0 f(ϝ) +0 +, f(ϝ) +0 +f(a) +0 , +(3.38) +we find that for a = ϝ ± 1, +ψ′(a) +±,0 = ψ(a) +±,0ψ(ϝ) +±,0, +k′(a) +0 += k(a) +0 ++ k(ϝ) +0 +, +e′(a) +0 += e(ϝ) +0 +e(a) +0 +− (−1)|a|HAaϝ +1 +e(a) +0 e(ϝ) +0 +, +f′(a) +0 += +1 +HAaϝ +1 +− H−Aaϝ +1 +� +f(a) +0 f(ϝ) +0 +− (−1)|a|H−Aaϝ +1 +f(ϝ) +0 +f(a) +0 +� +(3.39) +would verify the corresponding ef relation. Likewise, checking the ef relation for a = ϝ, we +have +ψ′(ϝ) +±,0 = ψ(ϝ) +∓,0, +k′(ϝ) +0 += −k(ϝ) +0 +, +(3.40) +and e′(ϝ) +0 += f(ϝ) +0 +, f′(ϝ) +0 += −e(ϝ) +0 +. However, to be compatible with the ee and ff relations that +contain modes with n = 0, ±1, we need to multiply them by some extra factors: +e′(ϝ) +0 += ψ(ϝ) ++,0f(ϝ) +0 += H−k(ϝ) +0 +1 +f(ϝ) +0 +, +f′(ϝ) +0 += −ψ(ϝ) +−,0e(ϝ) +0 += −Hk(ϝ) +0 +1 +e(ϝ) +0 +. +(3.41) +Notice that they would still recover the transformations of the Chevalley generators under +odd reflections in the limit β → 0. One may check that these transformations are consistent +with all the other relations involving zero modes. +Next, let us consider the modes with n = ±1. By considering the commutator of k′(b̸=ϝ) +1 +and e′(a) +0 +with b = a ± 1 (which is always possible since there are at least four nodes in the +quiver), we find that for a = ϝ ± 1, +e′(a) +1 += e(ϝ) +0 +e(a) +1 +− (−1)|a|HAaϝ +1 +e(a) +1 e(ϝ) +0 +. +(3.42) +Likewise, +f′(a) +1 += +1 +HAaϝ +1 +− H−Aaϝ +1 +� +f(a) +1 f(ϝ) +0 +− (−1)|a|H−Aaϝ +1 +f(ϝ) +0 +f(a) +1 +� +. +(3.43) +Again, computing [x, y} with +x = e(ϝ) +0 +e(a) +1 , e(a) +1 e(ϝ) +0 +and y = f(a) +1 f(ϝ) +0 +, f(ϝ) +0 +f(a) +1 , +(3.44) +we find that +ψ′(a) ++,1 = ψ(ϝ) ++,0ψ(a) ++,1 − C1/2H−Maϝ +2 +� +HAaϝ +1 +f(ϝ) +1 +e(ϝ) +0 ++ H−Aaϝ +1 +e(ϝ) +0 +f(ϝ) +1 +� +ψ(a) ++,0, +(3.45) +ψ′(a) +−,1 = ψ(ϝ) +−,0ψ(a) +−,1 − C−1/2HMaϝ +2 +� +HAaϝ +1 +e(ϝ) +−1 f(ϝ) +0 ++ H−Aaϝ +1 +f(ϝ) +0 +e(ϝ) +−1 +� +ψ(a) +−,0. +(3.46) +– 13 – + +In terms of the Heisenberg modes, we have +k′(a) +1 += k(a) +1 +− C1/2H−Maϝ +2 +� +HAaϝ +1 +f(ϝ) +1 +e(ϝ) +0 ++ H−Aaϝ +1 +e(ϝ) +0 +f(ϝ) +1 +� +Hk(ϝ) +0 +1 +, +(3.47) +k′(a) +−1 = k(a) +−1 − C−1/2HMaϝ +2 +� +HAaϝ +1 +e(ϝ) +−1 f(ϝ) +0 ++ H−Aaϝ +1 +f(ϝ) +0 +e(ϝ) +−1 +� +H−k(ϝ) +0 +1 +. +(3.48) +By considering the commutation relations of k′(ϝ±1) +1 +and e′(ϝ) +0 +, we find that +e′(ϝ) +1 += CH−2Maϝ +2 +Hk(ϝ) +0 +1 +f(ϝ) +1 +, +(3.49) +where a can either be ϝ + 1 or ϝ − 1 as Maϝ would be the same. Likewise, +f′(ϝ) +1 += H−2Maϝ +2 +� +−C−1e(ϝ) +1 ++ C−1/2k(ϝ) +1 +e(ϝ) +0 +� +Hk(ϝ) +0 +1 +, +(3.50) +e′(ϝ) +−1 = H2Maϝ +2 +� +Cf(ϝ) +−1 − C1/2k(ϝ) +−1 f(ϝ) +0 +� +H−k(ϝ) +0 +1 +, +(3.51) +f′(ϝ) +−1 += −C−1H2Maϝ +2 +H−k(ϝ) +0 +1 +e(ϝ) +−1 . +(3.52) +Using the ef relations, we get +ψ′(ϝ) +±,1 = −H∓2Maϝ +2 +� +ψ(ϝ) +∓,0 +�2 +ψ(ϝ) +±,1, +k′(ϝ) +±1 = −H∓2Maϝ +2 +k(ϝ) +±1 . +(3.53) +One may check that these transformations are consistent with all the other relations. +From the above discussions, we may also derive the transformations in terms of currents. +By applying the k±1 modes successively, it is not hard to see that +e′(a)(U) = e(ϝ) +0 +e(a)(U) − (−1)|a|HAaϝ +1 +e(a)(U)e(ϝ) +0 +, +(3.54) +f′(a)(U) = +1 +HAaϝ +1 +− H−Aaϝ +1 +� +f(a)(U)f(ϝ) +0 +− (−1)|a|H−Aaϝ +1 +f(ϝ) +0 +f(a)(U) +� +(3.55) +for a = ϝ ± 1. Then by considering their supercommutator, we find that each term contains +some formal delta function with other terms being cancelled. This yields +ψ′(a) +± (U) =e(ϝ) +0 +ψ(a) +± (U)f(ϝ) +0 +− (−1)|a|HAaϝ +1 +e(ϝ) +0 +f(ϝ) +0 +ψ(a) +± (U) +− H−Aaϝ +1 +ψ(a) +± (U)e(ϝ) +0 +f(ϝ) +0 +− f(ϝ) +0 +ψ(a) +± (U)e(ϝ) +0 +. +(3.56) +It is less straightforward to write down the currents for ϝ. Nevertheless, we can write +some conjectural expressions by computing a few more higher modes and then verify them +using the current relations. The perturbative calculations show that +e′(ϝ) +0 +(U) = f(ϝ) +>0 +� +C−1U +� +ψ +(ϝ) ++ +� +C−1/2H2Maϝ +2 +U +� ++ f(ϝ) +≤0 (CU) ψ +(ϝ) +− +� +C1/2H2Maϝ +2 +U +� +, +(3.57) +f′(ϝ) +0 +(U) = −e(ϝ) +≥0 (CU) ψ +(ϝ) ++ +� +C1/2H2Maϝ +2 +U +� +− e(ϝ) +<0 +� +C−1U +� +ψ +(ϝ) +− +� +C−1/2H2Maϝ +2 +U +� +, +(3.58) +– 14 – + +where +f(ϝ) +>0 (U) = +� +n>0 +f(ϝ) +n +U −n, +f(ϝ) +≤0 (U) = +� +n≤0 +f(ϝ) +n +U −n, +e(ϝ) +≥0 (U) = +� +n≥0 +e(ϝ) +n U −n, +e(ϝ) +<0 (U) = +� +n<0 +e(ϝ) +n U −n, +(3.59) +and +ψ +(ϝ) ++ (U) = +� +ψ(ϝ) +−,0 +�2 +� +� +�ψ(ϝ) ++,0 − +ψ(ϝ) ++,1 +U +− +ψ(ϝ) ++,2 − +� +ψ(ϝ) ++,1 +�2 +ψ(ϝ) +−,0 +U 2 +− +ψ(ϝ) ++,3 + +� +ψ(ϝ) ++,1 +�3 � +ψ(ϝ) +−,0 +�2 +U 3 +− . . . +� +� +� , +(3.60) +ψ +(ϝ) +− (U) = +� +ψ(ϝ) ++,0 +�2 � +ψ(ϝ) +−,0 − ψ(ϝ) +−,1U − +� +ψ(ϝ) +−,2 − +� +ψ(ϝ) +−,1 +�2 +ψ(ϝ) ++,0 +� +U 2 +− +� +ψ(ϝ) +−,3 + +� +ψ(ϝ) +−,1 +�3 � +ψ(ϝ) ++,0 +�2� +U 3 − . . . +� +. +(3.61) +In fact, we find that the perturbative expressions here coincide with the “inverse currents”,that +is, +ψ +(ϝ) +± (U) = ψ(ϝ) +± (U)−1. +(3.62) +Then we have +ψ′(ϝ) +± +(U) = ψ(ϝ) +± +� +H2Maϝ +2 +U +�−1 +. +(3.63) +Indeed, one may verify these expressions using the current relations. It is also worth noting +that +k′(ϝ) +n += −H2nMaϝ +2 +k(ϝ) +n . +(3.64) +3.3 +Higgsing +As reviewed in Appendix A.2, the toric quiver gauge theories have nice features under the +Higgs-Kibble mechanism. It is then natural to wonder if their BPS algebras are also connected +via blowing up/down the singularities, or more precisely, if there is a subalgebra structure for +a higgsed theory from a parent theory. +As the higgsing process always merges the two neighbouring nodes, say a and a + 1, in +the quiver for any toric CY without compact divisors, we expect the generators associated +with other nodes (and the central element C) to be invariant. Of course, there is a relabelling +for b > a + 1 as the number of nodes is reduced by one after higgsing. +For x′(a) (x = e, f, ψ, k), where the primed letters indicate the generators for the higgsed +theory, it should be a combination of x(a) and x(a+1). As discussed in Appendix A.3, the +parity should satisfy +��x′(a)�� = +��x(a)�� + +��x(a+1)��. +Therefore, for the zero modes, a natural +– 15 – + +candidate would be a combination of e(a) +0 e(a+1) +0 +and e(a+1) +0 +e(a) +0 +(and likewise for f). Similar to +the construction for toric duality, we find that +e′(a) +0 += e(a+1) +0 +e(a) +0 +− (−1)|a||a+1|HAa,a+1 +1 +e(a) +0 e(a+1) +0 +, +(3.65) +f′(a) +0 += +1 +HAa,a+1 +1 +− H−Aa,a+1 +1 +� +f(a) +0 f(a+1) +0 +− (−1)|a||a+1|H−Aa,a+1 +1 +f(a+1) +0 +f(a) +0 +� +, +(3.66) +ψ′(a) +±,0 = ψ(a) +±,0ψ(a+1) +±,0 +, +k′(a) +0 += k(a) +0 ++ k(a+1) +0 +(3.67) +would give the expected subalgebra structure for the zero modes. This is precisely the trans- +formation for a = ϝ ± 1 in the above discussions of toric duality with ϝ replaced by a + 1. +In fact, in the rational limit β → 0, this gives the surjection map of the Chevalley generators +of the corresponding affine Lie superalgebras. +However, when we use k′(a−1) +±1 += k(a) +±1 or k′(a+1) +±1 += k(a+2) +±1 +to get the higher modes from +e′(a) +0 +(resp. f′(a) +0 +), the expressions are not symmetric in e(a) and e(a+1) (resp. f(a) and f(a+1)) +any more. Indeed, for instance, +� +k(a−1) +1 +, e′(a) +0 +� +yields +e′(a) +1 += e(a+1) +0 +e(a) +1 +− (−1)|a||a+1|HAa,a+1 +1 +e(a) +1 e(a+1) +0 +(3.68) +while +� +k(a+2) +1 +, e′(a) +0 +� +leads to +e′(a) +1 += e(a+1) +1 +e(a) +0 +− (−1)|a||a+1|HAa,a+1 +1 +e(a) +0 e(a+1) +1 +. +(3.69) +They are not equal to each other as can be seen from the ee relation. Explicitly, +e(a+1) +1 +e(a) +0 −(−1)|a||a+1|HAa,a+1 +1 +e(a) +0 e(a+1) +1 += HMa,a+1 +2 +� +e(a+1) +0 +e(a) +1 +− (−1)|a||a+1|HAa,a+1 +1 +e(a) +1 e(a+1) +0 +� +. +(3.70) +Due to the non-trivial factor HMa,a+1 +2 +, this transformation does not give the subalgebra struc- +ture. Nevertheless, when H2 = 1, the quiver BPS algebras reduce to a one-parameter algebra, +and the above two expressions for e′(a) +1 +would coincide. +Therefore, at least when h2 = 0, for non-chiral quivers11, the toroidal BPS algebra +contains the ones for the higgsed theories as its subalgebras. The surjection for the generators +associated with a and a + 1 are the same as the transformations for a = ϝ ± 1 under toric +duality with ϝ replaced by a ∓ 1 12. Of course, a + 1 (as well as a) can be either bosonic +or fermionic. This is also the case for the rational quiver Yangians, where the surjection +map is most conveniently expressed in the J presentation. See (4.5) in [17] (with conventions +therein). It is not clear whether higgsing would still lead to the subalgebra structure for +generic h2, and if so, what the surjection map would be. Physically, the two parameters of +the algebra are related to the Ω-background that is used to resolve the singular target space +11For C3/(Z2 × Z2) which can be higgsed to the suspended pinch point, this should also be true. The +discussions here do not cover C3, C × C2/Z2 and the conifold although we still expect this to hold. +12As a result, this gives two transformations, but they should essentially be the same up to a normalization +factor. +– 16 – + +of the supersymmetric quantum mechanics. In particular, the scalars in the vector multiplets +would also have non-zero VEVs. Therefore, the algebra structure under higgsing could be +closely related to the localizations of the Higgs and Coulomb branches [2]. +4 +Elliptic Algebras and Chiral Quivers +Now, let us have a discussion on the remaining cases including the elliptic algebras for non- +chiral quivers and the algebras for chiral quivers. Unlike the rational and toroidal algebras +for non-chiral quivers, it is more difficult to work with modes. This is due to the existence +of q-Pochhammer symbols in the elliptic case while for chiral quivers, different CYs/quivers +would have different “minimalistic” presentations. Therefore, we shall mainly consider the +more unified current relations. +4.1 +Elliptic Algebras for Non-Chiral Quivers +Given a generalized conifold xy = zMwN with M + N ≥ 3, the elliptic quiver algebra E has +the relations +ψ(a) +± (U)ψ(b) +± (V ) = ψ(b) +± (V )ψ(a) +± (U), +(4.1) +ψ(a) +± (U)ψ(b) +∓ (V ) = +� +UCV −1HAab +1 +HMab +2 +; q +� +∞ +� +qU −1C−1V H−Aab +1 +H−Mab +2 +; q +� +∞ +� +U −1C−1V HAab +1 +H−Mab +2 +; q +� +∞ +� +qUCV −1H−Aab +1 +HMab +2 +; q +� +∞ +� +U −1CV HAab +1 +H−Mab +2 +; q +� +∞ +� +qUC−1V −1H−Aab +1 +HMab +2 +; q +� +∞ +� +UC−1V −1HAab +1 +HMab +2 +; q +� +∞ +� +qU −1CV H−Aab +1 +H−Mab +2 +; q +� +∞ +ψ(b) +∓ (V )ψ(a) +± (U) +(4.2) +ψ(a) +± (U)e(b)(V ) = HAab +1 +� +U −1C∓ 1 +2 V H−Aab +1 +H−Mab +2 +; q +� +∞ +� +qUC± 1 +2 V −1HAab +1 +HMab +2 +; q +� +∞ +� +U −1C∓ 1 +2 V HAab +1 +H−Mab +2 +; q +� +∞ +� +qUC± 1 +2 V −1H−Aab +1 +HMab +2 +; q +� +∞ +e(b)(V )ψ(a) +± (U) +(4.3) +ψ(a) +± (U)f(b)(V ) = H−Aab +1 +� +U −1C± 1 +2 V HAab +1 +H−Mab +2 +; q +� +∞ +� +qUC∓ 1 +2 V −1H−Aab +1 +HMab +2 +; q +� +∞ +� +U −1C± 1 +2 V H−Aab +1 +H−Mab +2 +; q +� +∞ +� +qUC∓ 1 +2 V −1HAab +1 +HMab +2 +; q +� +∞ +f(b)(V )ψ(a) +± (U) +(4.4) +e(a)(U)e(b)(V ) = (−1)|a||b|HAab +1 +� +U −1V H−Aab +1 +H−Mab +2 +; q +� +∞ +� +qUV −1HAab +1 +HMab +2 +; q +� +∞ +� +U −1V HAab +1 +H−Mab +2 +; q +� +∞ +� +qUV −1H−Aab +1 +HMab +2 +; q +� +∞ +e(b)(V )e(a)(U) +(4.5) +– 17 – + +f(a)(U)f(b)(V ) = (−1)|a||b|H−Aab +1 +� +U −1V HAab +1 +H−Mab +2 +; q +� +∞ +� +qUV −1H−Aab +1 +HMab +2 +; q +� +∞ +� +U −1V H−Aab +1 +H−Mab +2 +; q +� +∞ +� +qUV −1HAab +1 +HMab +2 +; q +� +∞ +f(b)(V )f(a)(U) +(4.6) +� +e(a)(U), f(b)(V ) +� += −δab +� +δ +� +UV −1C−1� +ψ(a) ++ +� +UC−1/2� +− δ +� +UV −1C +� +ψ(a) +− +� +V C−1/2�� +. +(4.7) +Similar to the toroidal case, for any fermionic node ϝ, we have ψ(ϝ) +± (U)e(ϝ)(V ) = +e(ϝ)(V )ψ(ϝ) +± (U), e(ϝ)(U)e(ϝ)(V ) = −e(ϝ)(V )e(ϝ)(U) etc. Moreover, when the central charge +is trivial, that is, C = 1, ψ(a) ++ (U) commutes with ψ(b) +− (V ). +4.1.1 +More on Mode Expansions +Although we would like to work with the currents directly, it would still be helpful to have +a look at their mode expansions. There are infinitely many groups of relations as α can be +any non-negative integer, but there are finitely many terms in each relation at each order. At +order q0, for instance, the ee relations read +e(a) +m+1,0e(b) +n,0 − HAab +1 +H−Mab +2 +e(a) +m,0e(b) +n+1,0 = (−1)|a||b| � +HAab +1 +e(b) +n,0e(a) +m+1,0 − H−Mab +2 +e(b) +n+1,0e(a) +m,0 +� +, (4.8) +which coincide with the ee relations for the toroidal algebra. In fact, all the relations at q0 +are the same as those in the toroidal case. Therefore, the elliptic subalgebra E0 at order q0 +is isomorphic to the toroidal algebra T. This is expected as the elliptic algebra E reduces to +T in the limit q → 0. +As another example, let us also write the ψe relations at order q1 here: +� +HMab +2 +U − HAab +1 +V +� �� +−H−Aab +1 +HMab +2 +UV −1 − HAab +1 +H−Mab +2 +V U −1� +ψ(a) +±,0 +� +C∓1/2U +� +e(b) +0 (V ) ++ψ(a) +±,1 +� +C∓1/2U +� +e(b) +0 (V ) + ψ(a) +±,0 +� +C∓1/2U +� +e(b) +1 (V ) +� += +� +HAab +1 +HMab +2 +U − V +� �� +−HAab +1 +HMab +2 +UV −1 − H−Aab +1 +H−Mab +2 +V U −1� +ψ(a) +±,0 +� +C∓1/2U +� +e(b) +0 (V ) ++ψ(a) +±,1 +� +C∓1/2U +� +e(b) +0 (V ) + e(b) +1 (V )ψ(a) +±,0 +� +C∓1/2U +�� +, +(4.9) +from which we can write the corresponding mode relations. +The other relations can be +obtained in a similar manner. For relations at higher orders of q, there would be more terms +with larger ranges of modes in the coefficients. In general, at order qα, the ψ± +� +C∓1/2U +� +e(V ) +relations read +� +HMab +2 +U − HAab +1 +V +� +α +� +γ=0 +� +α1,α2 +α1+α2=α−γ +Kγ(Aab)ψ(a) +±,α1 +� +C∓1/2U +� +e(b) +α2(V ) += +� +HAab +1 +HMab +2 +U − V +� +α +� +γ=0 +� +α1,α2 +α1+α2=α−γ +Kγ(−Aab)e(b) +α2(V )ψ(a) +±,α1 +� +C∓1/2U +� +(4.10) +– 18 – + +for some functions Kγ coming from the expansions of (the product of) the q-Pochhammer +symbols. Here, we have suppressed the other indices and arguments in Kγ for brevity. In +particular, K0 = 1. The e(U)e(V ) relations have the same coefficients (with an extra sign +factor (−1)|a||b|) while for the ψ± (C∓U) f(V ) and f(U)f(V ) relations, we simply have Aab ↔ +−Aab on both sides. We can then write the mode relations at each order of q from these current +relations. +Heisenberg modes +Similar to the discussions on the toroidal algebras above, as well as +some elliptic deformed algebras in [23], we may expand the ψ± modes as +ψ(a) ++ (U) = H−k(a) +0 +1 +exp +� +�� +n̸=0 +k(a) +n U −n +� +� , +ψ(a) +− (U) = Hl(a) +0 +1 +exp +� +�� +n̸=0 +l(a) +−nU n +� +� . +(4.11) +For convenience, we shall still refer to the k and l modes as Heisenberg modes. Notice that +the sums are now over Z\{0}. Moreover, +ψ(a) ++,n = H−k(a) +0 +1 +� +� +� +� +∞ +� +m=0 +1 +m! +� +ri̸=0 +r1+···+rm=n +kr1kr2 . . . krm +� +� +� +� , +(4.12) +ψ(a) +−,n = H−l(a) +0 +1 +� +� +� +� +∞ +� +m=0 +1 +m! +� +ri̸=0 +r1+···+rm=n +lr1lr2 . . . lrm +� +� +� +� . +(4.13) +In particular, k(a) +0 +and l(a) +0 +are not equal to ψ(a) +±,0 (or ψ(a) +±,0,0) here. Nevertheless, the Heisenberg +modes may still play the role that raises or lowers the e, f modes. More explicitly, +� +k(a) +r , k(b) +s +� += +� +l(a) +r , l(b) +s +� += +� +k(a) +0 , l(b) +s +� += +� +k(a) +r , l(b) +0 +� += 0, +(4.14) +� +k(a) +r̸=0, l(b) +s +� += δr+s,0 +1 +r +1 +1 − qr +� +C−r − Cr� +H−rMab +2 +� +HrAab +1 +− qrH−rAab +1 +� +, +(4.15) +� +k(a) +0 , e(b) +n +� += −Aabe(b) +n , +� +k(a) +0 , f(b) +n +� += Aabf(b) +n , +(4.16) +� +l(a) +0 , e(b) +n +� += Aabe(b) +n , +� +l(a) +0 , f(b) +n +� += −Aabf(b) +n , +(4.17) +� +k(a) +r̸=0, e(b) +n +� += 1 +r +1 +1 − qr C−r/2H−rMab +2 +� +HrAab +1 +− H−rAab +1 +� +e(b) +n+r, +(4.18) +� +k(a) +r̸=0, f(b) +n +� += −1 +r +1 +1 − qr Cr/2H−rMab +2 +� +HrAab +1 +− H−rAab +1 +� +f(b) +n+r, +(4.19) +� +l(a) +r̸=0, e(b) +n +� += 1 +r +1 +1 − qr C−r/2H−rMab +2 +� +HrAab +1 +− H−rAab +1 +� +e(b) +n−r, +(4.20) +� +l(a) +r̸=0, f(b) +n +� += −1 +r +1 +1 − qr Cr/2H−rMab +2 +� +HrAab +1 +− H−rAab +1 +� +f(b) +n−r. +(4.21) +However, the ef relations in terms of k and l would be quite different from those of the +toroidal cases. This is one of the difficulties when discussing toric duality for elliptic algebras. +– 19 – + +4.1.2 +Toric Duality +Let us have a brief discussion on toric duality for the elliptic cases. In fact, as discussed +in Appendix B, the dressed currents E(a)(u), F (a)(u) and Ψ(a) +± (u) introduced therein have +the same relations as those of the toroidal cases. Therefore, the previous transformations +should also apply to the elliptic cases using the dressed currents (with products replaced by +correlators or normal orderings). Moreover, by comparing these relations with the ones using +the “bare” generators at each order qα, we may write the correlators ⟨XY ⟩α in the expansion +of q. For instance, from (4.10), we have +� +Ψ(a) +± +� +C∓1/2U +� +E(b) (V ) +� +α = +α +� +γ=0 +� +α1,α2 +α1+α2=α−γ +Kγ(Aab)ψ(a) +±,α1 +� +C∓1/2U +� +e(b) +α2(V ), +(4.22) +� +E(b) (V ) Ψ(a) +± +� +C∓1/2U +�� +α = +α +� +γ=0 +� +α1,α2 +α1+α2=α−γ +Kγ(−Aab)e(b) +α2(V )ψ(a) +±,α1 +� +C∓1/2U +� +. +(4.23) +Nevertheless, let us still take a look at the original bare generators e, f, ψ± directly in the +followings for completeness. +Suppose that the node ϝ is dualized. Then the currents associated to a ̸= ϝ, ϝ ± 1 +(and hence C) should remain invariant. For a = ϝ ± 1, we expect the currents to have a +combination of a and ϝ currents/modes similar to the ones in the toroidal cases. Let us recall +that for the toroidal algebras, we have +e′(a)(U) = +� +e(ϝ) +0 +, e(a)(U) +� +H +Aaϝ +1 +, +(4.24) +where the deformed bracket is given by [x, y}q = xy −(−1)|x||y|qyx. Likewise, for the rational +algebras, we have +e′(a)(U) = +� +e(ϝ) +0 +, e(a)(U) +� +. +(4.25) +As a result, each transformation is determined by its corresponding version of the bracket. +Moreover, these are preciously the brackets that appear in their Serre relations. Therefore, +we propose that the elliptic version of the bracket is used here: +e′(a)(U) = +� +e(ϝ)(V ), e(a)(U) +� +χ +���� +V 0 , +(4.26) +where χ represents the elliptic deformed bracket as in Appendix B and V 0 indicates that we +only take the terms of order V 0. More explicitly, using the q-binomial theorem, we have +e′(a)(U) = +∞ +� +n=0 +� +H2Aaϝ +1 +; q +� +n +(q; q)n +� +qH−Aaϝ +1 +HMaϝ +2 +U +�n +� +e(ϝ) +−ne(a)(U) − (−1)|a|HAaϝ +1 +H−2nMaϝ +2 +U −2ne(a)(U)e(ϝ) +n +� +. +(4.27) +– 20 – + +Likewise, +f′(a)(U) = +∞ +� +n=0 +� +H−2Aaϝ +1 +; q +� +n +(q; q)n +� +qHAaϝ +1 +H−Maϝ +2 +U −1�n +HAaϝ +1 +− H−Aaϝ +1 +� +f(a)(U)f(ϝ) +n +− (−1)|a|H−Aaϝ +1 +H2nMaϝ +2 +U 2nf(ϝ) +−n f(a)(U) +� +. +(4.28) +For the node ϝ, we expect that ψ′ +± are still given by the inverse currents, that is, +ψ′(ϝ) +± +(U) = ψ(ϝ) +± +� +H2Maϝ +2 +U +�−1 +. +(4.29) +Analogously, it is natural to conjecture that e′(ϝ) and f′(ϝ) would have the same forms as in +the toroidal algebras. In other words e′ = f>0ψ−1 ++ + f≤0ψ−1 +− , f′ = −e≥0ψ−1 ++ − e<0ψ−1 +− , where +we have omitted the different arguments in different factors for brevity. +Indeed, the inverse currents are consistent with the relations under toric duality. For +instance, the e′(a)e′(ϝ) relation contains +e(a)(U)E(ϝ)F(ϝ) +± ψ(ϝ) +± +� +C∓1/2H2Maϝ +2 +V +�−1 +=(−1)|a|U −1V HMaϝ +2 +� +UV −1H−Aaϝ +1 +H−Maϝ +2 +; q +� +∞ +� +U −1V H−Aaϝ +1 +HMaϝ +2 +; q +� +∞ +� +qU −1V HAaϝ +1 +HMaϝ +2 +; q +� +∞ +� +qUV −1HAaϝ +1 +H−Maϝ +2 +; q +� +∞ +F(ϝ) +± ψ(ϝ) +± +� +C∓1/2H2Maϝ +2 +V +�−1 +e(a)(U)E(ϝ) + . . . , +(4.30) +where E(ϝ) (resp. +F(ϝ) +± ) sketchily indicates the factors containing only e(ϝ) (resp. +f(ϝ)) +modes. The ellipsis stands for the extra terms coming from exchanging these factors which +should be cancelled in the whole expression. Recall that A′ +aϝ = −Aaϝ and M′ +aϝ = −Maϝ. +As we can see, this recovers the correct coefficient for the e′(a)e′(ϝ) relation. +Higgsing +Similar to the rational and toroidal cases, the surjection (if it exists) induced +from higgsing should leave the central element C and all but two (say, a and a + 1) currents +invariant (with a possible relabelling of nodes). However, due to the complication at higher +orders of q, it is more difficult to write the currents associated to a′ in terms of those for a +and a + 1. Nevertheless, we may still conjecture that higgsing would also give subalgebras in +the elliptic case, at least in certain one-parameter degeneracy. +4.2 +Comments on Chiral Quivers +We shall now make some comments on the cases for chiral quivers. As mentioned in Appendix +A.1, only nodes with two arrows in and two arrows out will be dualized. Then all the possible +cases are listed in Figure 4.1. However, as we are now going to discuss, we will only consider +the cases (a), (c) and (d) here. +For the last two cases, (e) and (f), the quivers would remain the same after dualizing the +red node (assuming that all the arrows added get integrated out). Therefore, their quiver +– 21 – + +(a) +(b) +(c) +(d) +(e) +(f) +Figure 4.1: The six possible configurations for the dualized node in the quiver. The node to be dualized is +coloured red. The dashed nodes indicate that they can be connected to the remaining part of the quiver. +BPS algebras are trivially invariant, and we only need to focus on the remaining four cases. +Moreover, for toric CYs with compact divisors, as the quiver nodes do not have adjoints (at +least for all the known examples to our best knowledge), all the e and f modes/currents are +fermionic. In particular, this means that (b) should be excluded as the node with two arrows +(one in and one out) connected to the dualized node will get an adjoint loop that cannot be +integrated out under duality. Indeed, as far as we know, including the examples classified in +[24–26], there is no case (b) appearing. For the remaining three cases, their quivers under +toric duality are illustrated in Figure 4.2. +As a preliminary attempt of constructing the transformations, let us consider certain +expressions similar to the cases for non-chiral quivers. Of course, the central element C and +the currents associated to the nodes that are not connected to the dualized node should be +invariant. +Suppose that the node ϝ is dualized. As before, we expect e′(ϝ)(u) to be a combination +of F(ϝ) +± (u)ψ(ϝ) +± (−u ∓ c/2)−1 for some F(ϝ) +± (u) (and likewise for f′(ϝ)(u)). For simplicity, let +us just take e′(ϝ)(u) = f(ϝ)(u)ψ(ϝ) ++ (−u − c/2)−1 as an illustration. +Indeed, as ϝ has all +its arrow(s) connected to a being reversed, e(a)(u)f(ϝ)(u)ψ(ϝ) ++ (−u − c/2)−1 would give the +required prefactor from the e(a) � +ψ(ϝ)�−1 relation while e(a)f(ϝ) would be responsible for the +minus sign. +For the nodes connected to ϝ, since they would remain fermionic after toric duality, we +cannot multiply them by e(ϝ) or f(ϝ) as in the non-chiral quiver cases. Let us first consider +– 22 – + +(a) +(d) +(c) +Figure 4.2: How the arrows and hence the ζ factors would change under toric duality for (a), (c), (d). The +two types of dashed lines indicate the arrows connecting the orange nodes (before possible cancellations). +the cases (a) and (c). Suppose that we take +e′(a)(u) = +� +e(a)(−u)ψ(ϝ) ++ (−u − c/2), +a ↠ ϝ or ϝ ↠ a +e(a)(−u), +a → ϝ or ϝ → a +, +(4.31) +where a → b and a ↠ b indicate the number of arrows from a to b. Then we have +e′(a)(u)e′(ϝ)(v) = e′(a)(u)f(ϝ)(v)ψ(ϝ) ++ (−v − c/2)−1 += −φa⇐ϝ(−u, −v)−1f(ϝ)(v)ψ(ϝ) ++ (−v − c/2)−1e′(a)(u) += − (UV )− t +2 χaϝ φa⇐ϝ(v − u)−1f(ϝ)(v)ψ(ϝ) ++ (−v − c/2)−1e′(a)(u), +(4.32) +which recovers the correct numbers of ζ in the relations as χ′ +aϝ = −χaϝ. For instance, if +a ↠ ϝ in the original quiver, then we have ϝ ↠ a after toric duality, and +φa⇐ϝ(v − u)−1 = +1 +ζ +� +h1 +aϝ − u + v +� +ζ +� +h2 +aϝ − u + v +�. +(4.33) +One may also check that the other e′e′ relations would also give the correct numbers of the ζ +factors. For the case (d), we may choose +e′(a)(u) = +� +e(a)(−u)ψ(ϝ) ++ (−u − c/2), +a → ϝ +e(a)(−u), +ϝ → a +. +(4.34) +However, only checking the numbers of ζ in the relations is not sufficient, and astute +readers may have already found the following problems: +– 23 – + +• Recall that for the rational quiver Yangians, the equalities in the current relations are up +to some umvn terms. The transformations in terms of the currents may not incorporate +these terms in general. Whether the transformations in terms of currents would work +or how corrections should be made would probably require more detailed delibrations +on the relations of modes, which can be much more intricate. +• From the transformations of e and f, we may obtain ψ′ +± from the ef relations. However, +unlike the toroidal and elliptic algebras for non-chiral quivers discussed above, there +would be terms that do not have formal delta functions. Although we could still in +principle put them at the right orders of U, V (and q) in the mode expansions of ψ′ +±, +there could be ambiguities in this process. This subtlety should also be related to the +ambiguities of multiplying ψ(ϝ) +± +in the above transformations. +• Most importantly, when we check the ζ factors above, we have not taken the correct +charge assignment for the dual algebra into account. For instance, hi +aϝ in (4.33) may +not be the right charges for the arrows in the dualized quiver. In fact, by checking some +examples, it is not hard to find that even if (4.33) gives the correct charges, the arrows +connecting the orange nodes do not have the required charges after the transformations. +In fact, such transformations would only work when the two parameters h1,2 are both +zero. One may consider possible shifts of the spectral parameters, such as e(a)(−u + +ϵ1)ψ(ϝ)(−u − c/2 + ϵ2) etc., in the above transformations. However, it would yield a set +of homogeneous equations only with the trivial solution as there are more independent +constraints than variables. +Therefore, the transformations for the dual algebras require a more careful construction. +Finding such maps may require more sophisticated ways, and we leave it to future work. +Likewise, for higgsing, simple multiplications of the currents for the merged nodes would +only give subalgebra structure in the trivial case with vanishing parameters. Moreover, given +a chiral quiver, it can be higgsed to either chiral or non-chiral quivers. There can also be +more than one pair of nodes to be merged. Although we still expect such surjection maps +under higgsing (at least for one-parameter degeneracies), it could be very different from the +above cases involving only non-chiral quivers. +Heisenberg modes +Similar to the discussions for non-chiral quivers, we may also take the +mode expansions as +ψ(a) ++ (U) = exp +�� +n∈Z +k(a) +n U −n +� +, +ψ(a) +− (U) = exp +�� +n∈Z +l(a) +−nU n +� +. +(4.35) +We shall still refer to k and l as Heisenberg modes. Notice that the conventions when writing +k0 and l0 are slightly different from before, and the sums are over Z. +Consider two nodes a and b in any chiral quiver. Suppose that there are |a → b| = r and +|b → a| = s. Then +� +k(a) +0 , l(b) +0 +� += log +� +C−r−s� += −(r + s)βc, +(4.36) +– 24 – + +� +k(a) +0 , k(b) +0 +� += − +� +l(a) +0 , l(b) +0 +� += log +� +Cr−s� += (r − s)βc, +(4.37) +� +k(a) +m̸=0, k(b) +n +� += +� +l(a) +m̸=0, l(b) +n +� += +� +k(a) +0 , l(b) +n̸=0 +� += +� +k(a) +m̸=0, l(b) +0 +� += 0. +(4.38) +Moreover, we have +� +k(a) +0 , e(b)(V ) +� += +� +l(a) +0 , e(b)(V ) +� += +� +� +� +� +� +� +� +log +� +HabV −(r−s)� +e(b)(V ), +r > s +log +� +−HabV −(r−s)� +e(b)(V ), +r < s +log ((−1)rHab) e(b)(V ), +r = s +, +(4.39) +� +k(a) +0 , f(b)(V ) +� += +� +l(a) +0 , f(b)(V ) +� += +� +� +� +� +� +� +� +− log +� +HabV −(r−s)� +f(b)(V ), +r > s +− log +� +−HabV −(r−s)� +f(b)(V ), +r < s +− log ((−1)rHab) f(b)(V ), +r = s +, +(4.40) +where Hab = +r� +i=1 +H1/2 +ab,i +s� +j=1 +H1/2 +ba,j. It would be more useful to write them as +e± +1 +r−s k(a) +0 e(b) +n e∓ +1 +r−s k(a) +0 += sgn(r, s)H +± +1 +r−s +ab +e(b) +n∓1 +(r ̸= s), +(4.41) +e± +1 +r−s k(a) +0 f(b) +n e∓ +1 +r−s k(a) +0 += sgn(r, s)H +∓ +1 +r−s +ab +f(b) +n±1 +(r ̸= s), +(4.42) +ek(a) +0 e(b) +n e−k(a) +0 += sgn(r, s)Habe(b) +n +(r = s), +(4.43) +ek(a) +0 f(b) +n e−k(a) +0 += sgn(r, s)Habf(b) +n +(r = s), +(4.44) +(4.45) +and likewise for l(a) +0 , where we have defined +sgn(r, s) = +� +� +� +� +� +� +� +1, +r > s +(−1)r, +r = s +−1, +r < s +. +(4.46) +The remaining relations would be different for the toroidal and the elliptic cases. For the +toroidal algebras, we have +� +k(a) +m , e(b) +n +� += 1 +mC−m/2 +� +�� +j +Hm +ba,j − +� +i +H−m +ab,i +� +� e(b) +n+m +(m > 0), +(4.47) +� +k(a) +m , f(b) +n +� += − 1 +mCm/2 +� +�� +j +Hm +ba,j − +� +i +H−m +ab,i +� +� f(b) +n+m +(m > 0), +(4.48) +� +l(a) +−m, e(b) +n +� += 1 +mCm/2 +� +�� +j +Hm +ba,j − +� +i +H−m +ab,i +� +� e(b) +n+m +(m > 0), +(4.49) +– 25 – + +� +l(a) +−m, f(b) +n +� += − 1 +mC−m/2 +� +�� +j +Hm +ba,j − +� +i +H−m +ab,i +� +� f(b) +n+m +(m > 0), +(4.50) +� +k(a) +m , e(b) +n +� += +� +k(a) +m , f(b) +n +� += +� +l(a) +−m, e(b) +n +� += +� +l(a) +−m, f(b) +n +� += 0 +(m < 0), +(4.51) +� +k(a) +m , l(b) +n +� += δm+n,0 +1 +m +� +C−m − Cm� +� +�δm>0 +� +j +Hm +ba,j + δm<0 +� +i +H−m +ab,i +� +� +(m ̸= 0), +(4.52) +where δcond is 1 when the condition cond is satisfied and 0 otherwise. Notice that we would +only raise the e, f modes using the non-zero Heisenberg modes. +If we take t = 1 in the +balancing factor (UV ) +t +2 χab for the toroidal algebras, then only km and l−m with m < 0 would +lower the e, f modes while the other non-zero Heisenberg modes would commute with them. +This would also make certain changs in (4.41)∼(4.44). +For the elliptic algebras, we have +� +k(a) +m , e(b) +n +� += 1 +m +1 +1 − qm C−m/2 +� +�� +j +Hm +ba,j − +� +i +H−m +ab,i +� +� e(b) +n+m, +(4.53) +� +k(a) +m , f(b) +n +� += − 1 +m +1 +1 − qm Cm/2 +� +�� +j +Hm +ba,j − +� +i +H−m +ab,i +� +� f(b) +n+m, +(4.54) +� +l(a) +−m, e(b) +n +� += 1 +m +1 +1 − qm Cm/2 +� +�� +j +Hm +ba,j − +� +i +H−m +ab,i +� +� e(b) +n+m, +(4.55) +� +l(a) +−m, f(b) +n +� += − 1 +m +1 +1 − qm C−m/2 +� +�� +j +Hm +ba,j − +� +i +H−m +ab,i +� +� f(b) +n+m, +(4.56) +� +k(a) +m , l(b) +n +� += δm+n,0 +1 +m +1 +1 − qm +� +C−m − Cm� +� +�� +i +H−m +ab,i − +� +j +Hm +ba,j +� +� , +(4.57) +where m ̸= 0. If we take t = 1 in the balancing factor (UV ) +t +2 χab, then 1/(1 − qm) would be +changed to 1/ (q−m − 1). +5 +Discussions +Let us mention some properties of the toric quivers that are not discussed above. +They +should be closely related to the truncations of the quiver BPS algebras, which could lead to +important physical consequences. +Specular duality +There is another duality for toric quiver gauge theories known as the +specular duality as proposed in [24, 27]. Many concepts and quantities enjoy nice properties +– 26 – + +under such duality [28–30]. In general, specular duality does not preserve the mesonic moduli +space (except self-dual cases) although the Hilbert series would agree up to some fugacity +maps. Instead, it is a duality that preserves the master space [31, 32]. Therefore, we do +not expect the quiver BPS algebras to be isomorphic under specular duality. However, it +exchanges the internal and external perfect matchings, which are associated to the internal +and external points of the toric diagram respectively, of the dual brane tilings. +As each arrow in the quiver can be written in terms of a product of some perfect match- +ings, the arrows also have a one-to-one correspondence for specular dual theories. It is then +natural to wonder if the charge assignments would also follow this correspondence of the ar- +rows. However, we have checked several examples and this is not the case, even for self-dual +ones13. Nevertheless, as argued in [1], the perfect matchings can be used to determine cer- +tain truncations of the quiver Yangian. This is because such truncations come from adding +D4-branes to the divisors of the toric CY threefold, which correspond to the lattice points +of the toric diagram. In [1], such truncations were only identified for external (or more pre- +cisely, corner) perfect matchings. It could be possible that the truncations from D4-branes +associated to internal points can be studied from the specular dual case, where the internal +perfect matchings are mapped to the external ones14. +Deformed VOAs +The truncations of quiver BPS algebras are of particular interest since +they are expected to be related to certain vertex operator algebras (VOAs), and hence im- +plement the AGT (aka BPS/CFT) correspondence [33, 34]. Indeed, the truncations of the +rational algebras give rise to the (universal enveloping algebras of) rectangular W-algebras +[17, 35, 36]. We expect that the truncations of the toroidal and even elliptic quiver BPS +algebras would lead to deformations of the rational VOAs. In particular, the toroidal algebra +for C3 is shown to be a q-deformation of the W1+∞-algebra in [37]. We conjecture that there +exist certain q-deformations of the WM|N×∞-algebras such that for toroidal BPS algebras T +associated to the generalized conifolds, we have the following commutative diagram which +would give the 5d AGT correspondence: +T +U(qWM|N×l) +T�⊗T +U(qWM|N×l1)�⊗U(qWM|N×l2) +Φl +Φl1 ⊗ Φl2 +∆ +� +∆l1,l2 +, +(5.1) +where Φl are some surjections and the hats denote the completions of the algebras. On the +BPS algebra side, this would require a detailed study on the so-called horizontal representa- +tions of the algebras with non-trivial central element C so that we can get vertex operators +from the generators. On the VOA side, we need to find some suitable deformations of the +13For a self-dual quiver, an arrow would often be mapped to a different arrow in the quiver. +14Of course, there can also be external lattice points that are not at the corners. Moreover, for non-reflexive +polygons, specular duality can relate brane tilings on Riemann surfaces with higher genus [28]. +– 27 – + +WM|N×∞-algebras studied in [38, 39]. It would also be helpful to know more about their free +field realizations. +Acknowledgement +I am grateful to Ian Cheung, Yang-Hui He, Vishnu Jejjala, Jian-Rong Li and in particular, +Deshuo Liu and Rak-Kyeong Seong for helpful discussions. The research is supported by a +CSC scholarship. +A +Toric Quiver Gauge Theories +In this appendix, we give a quick recap on some properties of 4d N = 1 quiver gauge theories +from toric geometry. More details can be found in the references mentioned below. See also +[40–42] for reviews. +A.1 +Toric Duality +Given a quiver gauge theory with its associated brane tiling, one can study many of its +salient features. Let us first consider quivers that are related by Seiberg duality [43] in the +toric phase, which is also known as the toric duality [8–12]. +In short, picking a node j in the quiver to dualize, we replace all the arrows connected +to j with their conjugate (flavour) by reversing their orientations. Then we add a meson, +that is, an arrow from i to k, to the new quiver for each 2-path i → j → k in the original +quiver. This process is depicted in Figure A.1. In cluster algebra, this is exactly the mutation +i +j +k +Contents +1 +Seiberg Duality +1 +1 +Seiberg Duality +In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 +gauge theories []. We will phrase our discussion in the language of quivers, since all the +theories considered in this paper are of this type. +Let us consider dualizing a node j in the quiver, which does not have adjoint chiral +fields.1 The transformation of the gauge theory can be summarized in the following +rules: +1. Flavors. +In physics, the arrows connected to the mutated node are usually referred +to as flavors. The flavors transform by simply reversing their orientation, namely: +1.a) Replace every incoming arrow i ! j with the outgoing arrow j ! i. Calling Xij +the incoming arrow, we replace it by the dual flavor ˜Xji +1.b) Replace every outgoing arrow j ! k with the incoming arrow k ! j. Calling +Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj +This is the quiver implementation of the fact that the magnetic flavors are in the +complex conjugate representations, of both the dualized gauge group and the spectator +nodes, of the original flavors.2 +2. Mesons. +Next we add mesons to the quiver, i.e. composite arrows, as follows. For +every 2-path i ! j ! k we add a new arrow i ! k. This meson Mik can be regarded +as the composition of the flavors i ! j and j ! k of the original theory. In other +words, we generate all possible composite arrows consisting of incoming and outgoing +chiral fields. +1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- +tions (see e.g. []). +2In our discussion, including the points that follow, we allow for the possibility of chiral fields +connecting pairs of nodes in both directions. +– 1 – +Contents +1 +Seiberg Duality +1 +1 +Seiberg Duality +In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 +gauge theories []. We will phrase our discussion in the language of quivers, since all the +theories considered in this paper are of this type. +Let us consider dualizing a node j in the quiver, which does not have adjoint chiral +fields.1 The transformation of the gauge theory can be summarized in the following +rules: +1. Flavors. +In physics, the arrows connected to the mutated node are usually referred +to as flavors. The flavors transform by simply reversing their orientation, namely: +1.a) Replace every incoming arrow i ! j with the outgoing arrow j ! i. Calling Xij +the incoming arrow, we replace it by the dual flavor ˜Xji +1.b) Replace every outgoing arrow j ! k with the incoming arrow k ! j. Calling +Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj +This is the quiver implementation of the fact that the magnetic flavors are in the +complex conjugate representations, of both the dualized gauge group and the spectator +nodes, of the original flavors.2 +2. Mesons. +Next we add mesons to the quiver, i.e. composite arrows, as follows. For +every 2-path i ! j ! k we add a new arrow i ! k. This meson Mik can be regarded +as the composition of the flavors i ! j and j ! k of the original theory. In other +words, we generate all possible composite arrows consisting of incoming and outgoing +chiral fields. +1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- +tions (see e.g. []). +2In our discussion, including the points that follow, we allow for the possibility of chiral fields +connecting pairs of nodes in both directions. +– 1 – +Contents +1 +Seiberg Duality +1 +1 +Seiberg Duality +In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 +gauge theories []. We will phrase our discussion in the language of quivers, since all the +theories considered in this paper are of this type. +Let us consider dualizing a node j in the quiver, which does not have adjoint chiral +fields.1 The transformation of the gauge theory can be summarized in the following +rules: +1. Flavors. +In physics, the arrows connected to the mutated node are usually referred +to as flavors. The flavors transform by simply reversing their orientation, namely: +1.a) Replace every incoming arrow i ! j with the outgoing arrow j ! i. Calling Xij +the incoming arrow, we replace it by the dual flavor ˜Xji +1.b) Replace every outgoing arrow j ! k with the incoming arrow k ! j. Calling +Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj +This is the quiver implementation of the fact that the magnetic flavors are in the +complex conjugate representations, of both the dualized gauge group and the spectator +nodes, of the original flavors.2 +2. Mesons. +Next we add mesons to the quiver, i.e. composite arrows, as follows. For +every 2-path i ! j ! k we add a new arrow i ! k. This meson Mik can be regarded +as the composition of the flavors i ! j and j ! k of the original theory. In other +words, we generate all possible composite arrows consisting of incoming and outgoing +chiral fields. +1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- +tions (see e.g. []). +2In our discussion, including the points that follow, we allow for the possibility of chiral fields +connecting pairs of nodes in both directions. +– 1 – +Contents +1 +Seiberg Duality +1 +1 +Seiberg Duality +In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 +gauge theories []. We will phrase our discussion in the language of quivers, since all the +theories considered in this paper are of this type. +Let us consider dualizing a node j in the quiver, which does not have adjoint chiral +fields.1 The transformation of the gauge theory can be summarized in the following +rules: +1. Flavors. +In physics, the arrows connected to the mutated node are usually referred +to as flavors. The flavors transform by simply reversing their orientation, namely: +1.a) Replace every incoming arrow i ! j with the outgoing arrow j ! i. Calling Xij +the incoming arrow, we replace it by the dual flavor ˜Xji +1.b) Replace every outgoing arrow j ! k with the incoming arrow k ! j. Calling +Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj +This is the quiver implementation of the fact that the magnetic flavors are in the +complex conjugate representations, of both the dualized gauge group and the spectator +nodes, of the original flavors.2 +2. Mesons. +Next we add mesons to the quiver, i.e. composite arrows, as follows. For +every 2-path i ! j ! k we add a new arrow i ! k. This meson Mik can be regarded +as the composition of the flavors i ! j and j ! k of the original theory. In other +words, we generate all possible composite arrows consisting of incoming and outgoing +chiral fields. +1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- +tions (see e.g. []). +2In our discussion, including the points that follow, we allow for the possibility of chiral fields +connecting pairs of nodes in both directions. +– 1 – +Contents +1 +Seiberg Duality +1 +1 +Seiberg Duality +In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 +gauge theories []. We will phrase our discussion in the language of quivers, since all the +theories considered in this paper are of this type. +Let us consider dualizing a node j in the quiver, which does not have adjoint chiral +fields.1 The transformation of the gauge theory can be summarized in the following +rules: +1. Flavors. +In physics, the arrows connected to the mutated node are usually referred +to as flavors. The flavors transform by simply reversing their orientation, namely: +1.a) Replace every incoming arrow i ! j with the outgoing arrow j ! i. Calling Xij +the incoming arrow, we replace it by the dual flavor ˜Xji +1.b) Replace every outgoing arrow j ! k with the incoming arrow k ! j. Calling +Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj +This is the quiver implementation of the fact that the magnetic flavors are in the +complex conjugate representations, of both the dualized gauge group and the spectator +nodes, of the original flavors.2 +2. Mesons. +Next we add mesons to the quiver, i.e. composite arrows, as follows. For +every 2-path i ! j ! k we add a new arrow i ! k. This meson Mik can be regarded +as the composition of the flavors i ! j and j ! k of the original theory. In other +words, we generate all possible composite arrows consisting of incoming and outgoing +chiral fields. +1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- +tions (see e.g. []). +2In our discussion, including the points that follow, we allow for the possibility of chiral fields +connecting pairs of nodes in both directions. +– 1 – +i +j +k +Figure A.1: A sketch of how quivers transform under Seiberg duality. Figure taken from [44]. +for quivers (without adjoint loops and 2-cycles) [45]. The superpotential and the ranks of +the gauge groups would be transformed accordingly. In particular, factors XijXjk should be +replaced with Mik in the superpotential, and terms Mik � +Xkj � +Xji need to be added. Moreover, +fields that acquire masses, i.e., quadratic terms in the superpotential, should be integrated +out using their F-term relations [11]. As we shall consider toric quivers, only nodes with two +arrows and two arrows out would be dualized. In terms of brane tilings, this can be performed +by the urban renewal [14, 46]. +A.2 +The Higgs-Kibble Mechanism +As studied in [13], higgsing of a theory corresponds to blowing down a compact 2-cycle to +a point in the toric geometry while unhiggsing blows up a point to a compact 2-cycle. The +– 28 – + +(a) +(b) +1 +2 +3 +1 +2 +1 +Figure A.2: The example of higgsing dP0 (aka KP2, C3/Z3 (1, 1, 1)) to C3. +process of higgsing can also be nicely encoded in the toric diagrams and in the quivers. An +example is depicted in Figure A.2. +By turning on a VEV of a bifundamental in the quiver at each step, some fields would +acquire masses which should be integrated out. +Using their equations of motion, we can +obtain the superpotential after higgsing similar to the process of performing toric duality. In +the above example, the superpotential changes as +W =I1 +12I3 +23I2 +31 + I1 +12I1 +23I3 +31 + I2 +23I1 +31I3 +12 − I2 +12I3 +23I1 +31 − I1 +12I2 +23I3 +31 − I2 +23I2 +31I3 +12 +⟨X1 +23⟩=1 +−−−−−→I1 +22I3 +21I3 +12I2 +22 − I2 +22I3 +21I3 +12I1 +22 +⟨X3 +12⟩=1 +−−−−−→I1 +11 +� +I3 +11, I2 +11 +� +, +(A.1) +where we have omitted the traces. +In terms of brane tilings, turning on a VEV of a bifundamental removes an edge, and +integrating out massive fields corresponds to removing (and combining) certain nodes in the +dimer. An illustration can be found in [24, Figure 50]. +A.3 +Generalized Conifolds +Recall that the toric diagram of any generalized conifold xy = zMwN is a trapezium on the +lattice of height one with two horizontal lines of lengths M and N. The quiver (in any toric +phase) can essentially be viewed as the “tripled” quiver15 of a Dynkin diagram associated to +the underlying untwisted affine Lie superalgebra �slM|N. +The quivers in different toric phases with different numbers of bosonic and fermionic +nodes are encoded by the triangulations of the corresponding toric diagram [47, 48]. Each +simplex in a given triangulation corresponds to a sign ςa = ±1. Together, they form a parity +sequence ς = {ςa|a ∈ Z/(M + N)Z}. Overall, the numbers of plus and minus ones are given +by M and N. If two simplices are aligned side by side, then ςa = ςa+1. If they are aligned in +the alternative way, then ςa and ςa+1 have opposite signs. Some illustrations can be found in +Figure A.3. +Besides a pair of opposite arrows connecting each pair of nodes a and a+1, the quiver has +a self-loop on each bosonic node. If ςa = ςa+1, then the node a is bosonic/even. Otherwise, +15Here, by “tripled”, we mean that we first add an opposite arrow for each existing arrow in the Dynkin +quiver. Then we only add adjoint loops to the bosonic nodes. +– 29 – + +(a) +(b) +(c) +Figure A.3: +Figure taken from [17]. +We have (a) ς = {−1, +1}, (b) ς = {−1, −1} and (c) ς = +{−1, −1, +1, +1, −1, +1, +1, +1}. +it is fermionic/odd. The superpotential is composed of terms +� +ςatr(Ia,aIa,a−1Ia−1,a − Ia,aIa,a+1Ia+1,a), +ςa = ςa+1 +ςatr(Ia,a+1Ia+1,aIa,a−1Ia−1,a), +ςa = −ςa+1 +. +(A.2) +Following the above rule of toric duality, it is straightforward to see that we can only +dualize fermionic nodes in the toric phase. This would just change the parity of the two nodes +connected to the dualized node by adding or removing the adjoint loops. Correspondingly, +the Dynkin diagrams of the underlying affine Lie superalgebra are related by odd reflections. +A generalized conifold with a larger polygon can be higgsed to one with a smaller polygon. +This can be decomposed into a sequence of higgsings. For each single higgsing, the leftmost +or rightmost simplex is removed. In the quiver, we merge two adjacent nodes. The two nodes +can be either bosonic or fermionic. Suppose that the nodes a and a + 1 are merged, then +|a′| = |a| + |a + 1|, where a′ denotes the corresponding node after higgsing. Let us list how +the Cartan matrices would change for the three possible cases: + + + + + + + + + + + +... +· · · 2 −1 · · · +· · · −1 2 −1 · · · +· · · −1 2 −1 · · · +−1 2 · · · +... + + + + + + + + + + + +|a| = |a + 1| = 0 : +a +a + 1 +a′ + + + + + + + + + +... +· · · 2 −1 · · · +· · · −1 2 −1 · · · +−1 2 · · · +... + + + + + + + + + +, +(A.3) + + + + + + + + + + + +... +· · · 2 −1 · · · +· · · −1 2 −1 · · · +· · · −1 0 +1 · · · +1 −2 · · · +... + + + + + + + + + + + +|a| = 0, |a + 1| = 1 : +a +a + 1 +a′ + + + + + + + + + +... +· · · 2 −1 · · · +· · · −1 0 +1 · · · +1 −2 · · · +... + + + + + + + + + +, +(A.4) +– 30 – + + + + + + + + + + + + +... +· · · 2 −1 · · · +· · · −1 0 +1 +· · · +· · · +1 +0 −1 · · · +−1 2 · · · +... + + + + + + + + + + + +|a| = |a + 1| = 1 : +a +a + 1 +a′ + + + + + + + + + +... +· · · 2 −1 · · · +· · · −1 2 −1 · · · +−1 2 · · · +... + + + + + + + + + +. +(A.5) +B +Serre Relations +Besides the relations listed in §2, the quiver BPS algebras also have Serre relations. Here, we +will only discuss the cases for non-chiral quivers with M + N ≥ 3, MN ̸= 2. Although the +Serre relations for general chiral quivers are still not known, examples can be found in [5]. It +is observed that the Serre relations (for either chiral or non-chiral quivers) are closely related +to the superpotential of the theory [1]. +For the rational algebras, we have +Sym +u1,u2 +� +e(a)(u1), +� +e(a)(u2), e(a±1)(v) +�� += 0 +(|a| = 0), +(B.1) +Sym +u1,u2 +� +e(a)(u1), +� +e(a+1)(v1) +� +e(a)(u2), e(a−1)(v2) +��� += 0 +(|a| = 1), +(B.2) +Sym +u1,u2 +� +f(a)(u1), +� +f(a)(u2), f(a±1)(v) +�� += 0 +(|a| = 0), +(B.3) +Sym +u1,u2 +� +f(a)(u1), +� +f(a+1)(v1) +� +f(a)(u2), f(a−1)(v2) +��� += 0 +(|a| = 1). +(B.4) +For the toroidal algebras, the Serre relations are +Sym +u1,u2 +� +e(a)(u1), +� +e(a)(u2), e(a±1)(v) +� +H1 +� +H1 += 0 +(|a| = 0), +(B.5) +Sym +u1,u2 +� +e(a)(u1), +� +e(a+1)(v1) +� +e(a)(u2), e(a−1)(v2) +� +H1 +� +H1 +� +H1 += 0 +(|a| = 1), +(B.6) +Sym +u1,u2 +� +f(a)(u1), +� +f(a)(u2), f(a±1)(v) +� +H−1 +1 +� +H−1 +1 += 0 +(|a| = 0), +(B.7) +Sym +u1,u2 +� +f(a)(u1), +� +f(a+1)(v1) +� +f(a)(u2), f(a−1)(v2) +� +H−1 +1 +� +H−1 +1 +� +H−1 +1 += 0 +(|a| = 1). +(B.8) +Here, the q-graded bracket is given by �x, y�q = xy − (−1)|x||y|q(x,y)yx, where (x, y) is the +root pairing stemmed from the underlying affine Lie superalgebra. For instance, the pairing +of two simple roots gives the corresponding entry in the Cartan matrix. +– 31 – + +As we can see, both of the two types of the algebras have their versions of the brackets. +Therefore, we would also like to use an “elliptic bracket” to write the Serre relations for the +elliptic cases. Let us introduce the operators χa(u) and ξa(u) that commute with all e, f, ψ± +generators in the elliptic algebras. They have the following correlators: +e⟨χa(u)χb(v)⟩ = +� +qHAab +1 +H−Mab +2 +U −1V ; q +� +∞ +� +qH−Aab +1 +H−Mab +2 +U −1V ; q +� +∞ +, +(B.9) +e⟨ξa(u)ξb(v)⟩ = +� +qH−Aab +1 +H−Mab +2 +U −1V ; q +� +∞ +� +qHAab +1 +H−Mab +2 +U −1V ; q +� +∞ +, +(B.10) +e⟨χa(u)ξb(v)⟩ = 1. +(B.11) +Then using the correlators of the “dressed” operators +E(a)(u) = eχa(u)e(a)(u), +F (a)(u) = eξa(u)f(a)(u), +Ψ(a) +± (u) = eχa(u±c/2)eξa(u∓c/2)ψ(a) +± (u), +(B.12) +the relations of the elliptic algebras can be written in the same forms as those of the toroidal +algebras. For instance, the ee relations of the elliptic algebras now become +� +HMab +2 +U − HAab +1 +V +� � +E(a)(u)E(b)(v) +� += (−1)|a||b| � +HAab +1 +HMab +2 +U − V +� � +E(b)(v)E(a)(u) +� +. +(B.13) +Therefore, the Serre relations of the elliptic algebras can simply be obtained by taking the +ones of the toroidal algebras. Then we replace the toroidal generators with the dressed elliptic +generators and take the correlators of the whole expressions. For brevity, we shall write them +using the “elliptic brackets” as +Sym +u1,u2 +� +e(a)(u1), +� +e(a)(u2), e(a±1)(v) +� +χ +� +χ += 0 +(|a| = 0), +(B.14) +Sym +u1,u2 +� +e(a)(u1), +� +e(a+1)(v1) +� +e(a)(u2), e(a−1)(v2) +� +χ +� +χ +� +χ += 0 +(|a| = 1), +(B.15) +Sym +u1,u2 +� +f(a)(u1), +� +f(a)(u2), f(a±1)(v) +� +ξ +� +ξ += 0 +(|a| = 0), +(B.16) +Sym +u1,u2 +� +f(a)(u1), +� +f(a+1)(v1) +� +f(a)(u2), f(a−1)(v2) +� +ξ +� +ξ +� +ξ += 0 +(|a| = 1). +(B.17) +C +Conventions of Heisenberg Modes +In the main context, we introduced the modes kr (and lr) for the ψ± currents. Here, we +mention some alternative convention to define these Heisenberg modes. It could be possible +that this would be more convenient when considering certain aspects of the algebras such as +their representations and the AGT correspondence. +– 32 – + +Let us consider the toroidal algebras for non-chiral quivers as an example. The other +cases can be redefined in a similar manner. First, we rescale the e, f modes as +e(a) +n += +� +q − q−1�1/2 e(a) +n , +f(a) +n += +� +q − q−1�1/2 f(a) +n , +(C.1) +where we have suggestively written q = exp(βh1) = H1. Notice that this does not change the +ee and ff relations. Then the e0f0 relations (as well as the enf−n relations) would become +� +e(a) +0 , f(a) +0 +� += δab +qk(a) +0 +− q−k(a) +0 +q − q−1 += δab +� +k(a) +0 +� +q . +(C.2) +Here, [x]q = qx−q−x +q−q−1 +is the standard q-number. On the other hand, the k0en (resp. k0fn) +relations remain the same as the ones for k0en (resp. k0fn). As we can see, the relations +among the zero modes resemble the ones appeared in quantum groups. +Likewise, we can write +ψ(a) +± (U) = ψ(a) +±,0 exp +� +� +q − q−1� ∞ +� +n=0 +k(a) +±nU ∓n +� +(C.3) +such that k(a) +r += +� +q − q−1� +k(a) +r . 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Phys. 14 no. 4, (2010) 1147–1181, arXiv:0910.5479 +[hep-th]. +– 36 – + diff --git a/f9AyT4oBgHgl3EQfxPl4/content/tmp_files/load_file.txt b/f9AyT4oBgHgl3EQfxPl4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..48510776efb658214c7c7263b4765f75689a2f59 --- /dev/null +++ b/f9AyT4oBgHgl3EQfxPl4/content/tmp_files/load_file.txt @@ -0,0 +1,1721 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf,len=1720 +page_content='A Survey of Toric Quivers and BPS Algebras Jiakang Baoa,b aDepartment of Mathematics, City, University of London, EC1V 0HB, UK bLondon Institute for Mathematical Sciences, Royal Institution, London W1S 4BS, UK E-mail: jiakang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='bao@city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='uk Abstract: In this note, we discuss some properties of the quiver BPS algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We consider how they would transform under different operations on the toric quivers, such as dualities and higgsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='00663v1 [hep-th] 2 Jan 2023 Contents 1 Introduction and Summary 1 2 Quiver BPS Algebras 3 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 28 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3 Generalized Conifolds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 29 B Serre Relations 31 C Conventions of Heisenberg Modes 32 References 33 1 Introduction and Summary Given a toric Calabi-Yau (CY) threefold, D-branes wrapping its holomorphic cycles give rise to BPS bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The 4d N = 1 gauge theory can be beautifully summarized in the language of toric diagrams, quivers and brane tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As a realization of the BPS algebras in such supersymmetric gauge theories, quiver Yangians were introduced in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Physically, – 1 – they can be derived via localization techniques in supersymmetric quantum mechanics [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' See [4] for a recent summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Later in [5] (see also [6, 7]), this was extended to the trigonometric and elliptic counter- parts of the rational quiver Yangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Such algebras, dubbed rational (toroidal, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' elliptic) quiver BPS algebras, can be realized by 3d N = 2 (2d N = (2, 2), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 1d N = 4) quantum field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' These theories are low-energy effective theories on the D-branes that probe the CY threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In particular, the three types of algebras can be uniformly described by some bond factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the elliptic algebras, the bond factor is composed of certain theta function Θq(u), where q is the square of the nome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In other words, it is related to the modulus τ of the torus by q = exp(2πiτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Under dimensional reduction, this gives the trigonometric version of the algebras whose bond factor is determined by Sinβ(u) := 2 sinh(βu/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In the limit where the radius β of the circle goes to 0, one reaches the rational case with the bond factor consisting of the rational function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' All these algebras have two parameters h1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' On the other hand, it is well-known that for supersymmetric gauge theories on toric CYs, many features, such as dualities and higgsing, can be nicely described using quivers and brane tilings [8–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It is then natural to ask what properties the quiver BPS algebras would have under these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As each quiver has its associated quiver BPS algebras, it is conjectured that the corre- sponding quiver BPS algebras are isomorphic under Seiberg/toric duality1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In the case of rational quiver Yangians for toric CY threefolds without compact divisors, this was proven in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Here, we shall have a discussion on the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For toroidal algebras from non-chiral quivers, we can construct the transformations of the generators under toric duality that are similar to the rational cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As such construction is based on the modes of the algebra, this approach becomes more difficult in the elliptic cases as the current relations involve q-Pochhammer symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, we would like to work with the current relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' There are four types of currents, e, f and ψ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As shown in [5], all the other current relations can be derived from the ee, ff and ef relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Hence, it is sufficient to consider these relations to construct the transformations for dual algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Suppose the node ϝ is dualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Although the elliptic cases are rather involved, we propose that the transformations of the currents associated to ϝ ± 1 can be determined by the corresponding “brackets” of the three types of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For chiral quivers that are associated to toric CYs with compact divisors, there do not seem to have underlying Kac-Moody superalgebras for the quiver BPS algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The patterns of the mode relations would also vary from case to case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As a result, it is not easy to study the connections of toric dual algebras even for the rational quiver Yangian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As they are quite different from the non-chiral cases even if we directly consider the current relations, we shall only give a naive construction for the currents here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As a result, this would only give 1In general, Seiberg duality can also take quivers outside the toric phases, leading to the phenomenon called duality cascades [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' One can still define the algebras for these quivers following §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, whether they would give rise to the corresponding BPS algebras needs further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, we shall only focus on toric quivers here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 2 – a valid transformation in the trivial case where the two parameters of the algebra vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' A proof of isomorphisms for general cases might require more sophisticated methods, and it might also be helpful to first start with some specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Nevertheless, we hope that the discussions here would provide some basic ideas of constructing such transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Although the exact maps are still not known in general, we can summarize some common features for both chiral and non-chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Suppose that the node ϝ is dualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Roughly speaking, the roles of e(ϝ) and f(ϝ) should get swapped under toric duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For ψ(ϝ) ± (u), we expect them to become their inverses ψ(ϝ) ± (�u)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is not surprising as the arrows connected to ϝ would get reversed under toric duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, for nodes that are connected to the dualized node, their associated generators should always be combined with certain generators for ϝ under toric duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The transformations of the toric dual algebras would also shed light on the discussions re- lated to the Higgs-Kibble mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Mathematically, this corresponds to blowing up/down the singularities in the toric setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In the rational and toroidal cases for non-chiral quivers, when they degenerate to one-parameter algebras, there is a surjection of the algebras from the parent theory to the higgsed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The construction is similar to the one discussed in toric duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We conjecture that higgsing would still give the subalgebra structure for general cases (at least in the one-parameter degeneracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In §2, after defining the quiver BPS algebras, we give some properties of the algebras, including coproducts and gradings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In §3, we consider how the toroidal algebras would transform under toric duality and higgsing for non-chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We then have some discussions on the elliptic cases, as well as the algebras for chiral quivers, in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In §5, we mention some prospects regarding specular duality, truncations and vertex operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We review some basic aspects of quiver gauge theories in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In Appendix B, we list the Serre relations for algebras associated to (most) non-chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In Appendix C, we make some supplementary comments on the modes of the quiver BPS algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2 Quiver BPS Algebras The quiver BPS algebras are generated by three types of currents, ψ(a) ± (u), e(a)(u) and f(a)(u), where a denotes the nodes of a given quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' When there is an odd number of adjoint loops on a node a, we say that it is bosonic with |a| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Otherwise, it is fermionic with |a| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This naturally endows the algebras with a Z2-grading such that ��e(a)(u) �� = ��f(a)(u) �� = |a| – 3 – while ψ(a) ± (u) are always bosonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The currents have the following mode expansions2: x(a)(u) = � � � � � � � � � � � � � � � � n∈Z+ x(a) n un , rational � n∈Z x(a) n Un , trigonometric � n∈Z x(a) n Un = � n∈Z � α∈Z≥0 x(a) n,α Un qα, elliptic , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1) where x = e, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For non-chiral quivers, ψ(a) ± (u) = � � � � � � � � � � � � � � � � � � n∈Z≥0 ψ(a) n un , rational � n∈Z≥0 ψ(a) ±,n U±n , trigonometric � n∈Z ψ(a) ±,n U±n = � n∈Z � α∈Z≥0 ψ(a) ±,n,α U±n qα, elliptic (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2) with ψ(a) ±,n<0,0 = 0 in the elliptic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For chiral quivers, ψ(a) ± (u) = � � � � � � � � � � � � � � � � n∈Z ψ(a) n un , rational � n∈Z ψ(a) ±,n U±n , trigonometric � n∈Z ψ(a) ±,n U±n = � n∈Z � α∈Z≥0 ψ(a) ±,n,α U±n qα, elliptic .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3) Notice that in the rational case, ψ+ = ψ− = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, the expansions for trigonometric and elliptic cases are in terms of U rather than u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The letters in the upper case are related to those in the lower case by3 X = eβx, (x, X) = (u, U), (v, V ), (c, C), (hI, HI), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='4) Henceforth, we will use the upper and lower cases interchangeably (such as e(a)(U) = e(a)(u)) in the arguments of the currents for trionometric and elliptic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For convenience, we may also write e(a)(U) = � α∈Z≥0 e(a) α (U)qα (and likewise for the other currents) in the elliptic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' To write their relations, we first need to introduce some necessary concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' To distin- guish chiral and non-chiral quivers, we define the chirality parameter as χab = |a → b| − |b → a| (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='5) 2One can also consider shifted quiver BPS algebras that would introduce an extra shift parameter to (some part of) the mode expansion of ψ± [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is closely related to the crystal representations and the framings of the quivers [3, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, we shall not consider this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 3As a result, β can be absorbed under a redefinition of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Nevertheless, we shall keep it here due to its physical origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 4 – for each pair of nodes a, b in the quiver, where |a → b| denotes the number of arrows from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, we shall define the formal delta function as δ(u) = � � � 1/u, rational � n∈Z U n, otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='6) The key factor in the definition of the algebras is the bond factor ϕa⇐b(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In this paper, we shall write it as4 ϕa⇐b(u) = � a→b ζ(hI + u) � b→a ζ(hI − u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='7) where hI is the parameter/charge associated to the arrow I in the quiver, and ζ(u) = � � � � � � � � � u, rational Sinβ(u) := 2 sinh βu 2 = U 1/2 − U −1/2, trigonometric Θq(u) := −U −1/2θq(u) = � U 1/2 − U −1/2� ∞ � k=0 � 1 − U −1qk� (1 − Uqk), elliptic .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='8) Here, θq(u) = (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q)∞ � qU −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ in terms of the q-Pochhammer symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' From the expres- sions for ζ, we have ζ(u) = −ζ(−u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='9) It is straightforward to see that in the rational limit β → 0, the trigonometric case reduces to the rational one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Likewise, when q → 0, the elliptic one reduces to the trigonometric one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This will also be the limits that relate the three types of quiver BPS algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' To get rid of the powers with half-integers, we will take the balanced bond factor φa⇐b(u, v) = (UV ) t 2 χabϕa⇐b(u − v), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='10) where t is 0 in the rational case and −1 otherwise5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, this balancing would only affect chiral quivers in the trigonometric and elliptic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As can be seen from its expression, the bond factor satisfies ϕa⇐b(u)ϕb⇐a(−u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='11) Therefore, φa⇐b(u, v)φb⇐a(v, u) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='12) Moreover, we have φa⇐b(u + s, v) = stχabφa⇐b(u, v − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='13) 4This is slightly different from the notion in [5] when both |a → b| and |b → a| are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Nevertheless, the bond factor here should still be legitimate as it satisfies the reciprocity condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='11) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 5Notice that this is slightly different from the original one in [5] where t was defined to be 1 for the trigonometric and elliptic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is only a choice for our later discussions on mode expansions for chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Since the balancing factor (UV ) t 2 χab is used to get rid of the half-integer powers in the Laurent expansions of the expressions, this should just be a matter of convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 5 – With this (balanced) bond factor, the three types of quiver Yangians can be presented in a unified way as [1, 5] ψ(a) ± (u)ψ(b) ± (v) ≃ C±tχabψ(b) ± (v)ψ(a) ± (u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='14) ψ(a) + (u)ψ(b) − (v) ≃ φa⇐b(u + c/2, v − c/2) φa⇐b(u − c/2, v + c/2)ψ(b) − (v)ψ(a) + (u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='15) ψ(a) ± (u)e(b)(v) ≃ φa⇐b(u ± c/2, v)e(b)(v)ψ(a) ± (u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='16) ψ(a) ± (u)f(b)(v) ≃ φa⇐b(u ∓ c/2, v)−1f(b)(v)ψ(a) ± (u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='17) e(a)(u)e(b)(v) ≃ (−1)|a||b|φa⇐b(u, v)e(b)(v)e(a)(u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='18) f(a)(u)f(b)(v) ≃ (−1)|a||b|φa⇐b(u, v)−1f(b)(v)f(a)(u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='19) � e(a)(u), f(b)(v) � ≃ −δab � δ(u − v − c)ψ(a) + (u − c/2) − δ(u − v + c)ψ(a) − (v − c/2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='20) Here, c is the central element of the algebra which is 0 for the rational case (while it can be non-trivial for the other two cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In the last relation, we have used the supercommutator [x, y} = xy − (−1)|x||y|yx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the rational quiver Yangians, “≃” indicates that the equalities are up to some sporadic umvn terms6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the trigonometric and elliptic cases, it means that the Laurent expansion on the two sides should agree, and we shall henceforth simply write it as “=”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As shown in [5], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='14)∼(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='17) can be derived from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='18)∼(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, when discussing the current relations, it suffices to consider the ee, ff and ef relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Besides the above relations, we also need the Serre relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It is believed that the Serre relations are closely related to the superpotential although the precise relations are still not known in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For generalized conifolds and some chiral quivers, their Serre relations were given in [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' See also Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In general, these quiver BPS algebras are two-parameter algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is because the charges hI should satisfy the loop and vertex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Due to the uniquely determined superpotential in the toric setting, each of its monomial term L, whose arrows form a closed loop in the quiver, yields one loop constraint: � I∈L hI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, the algebra also has shift automorphisms hI → hI + sgna(I)ϵa for some parameters ϵa, where sgna(I) is 1 (−1, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 0) if I ∈ {a → b|b ̸= a} (I ∈ {b → a|b ̸= a}, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We can use the vertex constraint � I sgnahI = 0 to mod out these gauge symmetry redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Overall, we have two free parameters, say h1 and h2, left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Together with the R-symmetry, they give the U(1)3 isometry of the toric CY threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 Coproducts The coproducts of the quiver BPS algebras are of particular interest due to their crucial role in the construction of R-matrices and the study of Bethe/gauge correspondence [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For rational quiver Yangians of certain non-chiral quivers, the coproduct was given in [17] using 6This is to ensure that the mode relations, which can be found for example in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='20) in [1], would be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 6 – the techniques developed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, this is still not known for chiral quivers7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In contrast, the coproducts for trigonometric and elliptic cases are more straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' One may verify that the following gives a coassociative homomorphism (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' [6]): ∆ � e(a)(U) � = e(a)(U) ⊗ 1 + ψ(a) � C1/2 1 U � ⊗ e(a)(C1U), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='21) ∆ � f(a)(U) � = 1 ⊗ f(a)(U) + f(a)(C2U) ⊗ f(a) � C1/2 2 U � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='22) ∆ � ψ(a) + (U) � = ψ(a) + (U) ⊗ ψ(a) + � C−1 1 U � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='23) ∆ � ψ(a) − (U) � = ψ(a) − � C−1 2 U � ⊗ ψ(a) − (U), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='24) ∆(C) = C ⊗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='25) Here, C1 = C ⊗ 1 and C2 = 1 ⊗ C are the conventional notations that indicate where the C factors should be in the mode expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' More explicitly, for the toroidal algebras associated to non-chiral quivers, we have ∆ � e(a) n � = e(a) n ⊗ 1 + ∞ � j=0 C−n−j/2ψ(a) −,j ⊗ e(a) n+j, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='26) ∆ � f(a) n � = 1 ⊗ f(a) n + ∞ � j=0 f(a) n−j ⊗ C−n+j/2ψ(a) +,j, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='27) ∆ � ψ(a) +,n � = n � j=0 Cn−jψ(a) +,j ⊗ ψ(a) +,n−j, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='28) ∆ � ψ(a) −,n � = n � j=0 ψ(a) −,n−j ⊗ C−n+jψ(a) −,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='29) For the elliptic algebras and the algebras for chiral quivers, we just need to replace all ∞ � j=0 and n� j=0 with � j∈Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In the elliptic case, it is also straightforward to write down this in terms of x(a) n,α (x = e, f, ψ±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We simply replace xn ⊗ 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 1 ⊗ xn) with xn,α ⊗ 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 1 ⊗ xn,α) and xm ⊗ yn with α� γ=0 xm,γ ⊗ yn,α−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Hopf algebras Together with the above coproduct in terms of the currents, we can have a counit and an antipode such that the algebra is endowed with the Hopf (super)algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The counit reads ϵ � e(a)(U) � = ϵ � f(a)(U) � = 0, ϵ � ψ(a) ± (U) � = ϵ(C) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='30) 7The reason is that the quiver Yangians for chiral quivers do not seem to have underlying Kac-Moody superalgebras which are quite heavily relied on when finding the coproduct for the non-chiral quiver cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Due to the complication in the current relations, we also need to write the coproduct in terms of modes, which is more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 7 – The antipode is an anti-homomorphism, that is, S(xy) = (−1)|x||y|S(y)S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Assuming that ψ(a) ± (U) are invertible in the algebra, then S � e(a)(U) � = −ψ(a) − � C−3/2U �−1 e(a)(CU), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='31) S � f(a)(U) � = −f(a)(CU)ψ(a) + � C−3/2U �−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='32) S � ψ(a) ± (U) � = ψ(a) ± � C−1U �−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='33) S(C) = C−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='34) It is also straightforward to write them in terms of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' One may check that they satisfy the properties of Hopf algebras, such as M ◦ (S × id) ◦ ∆ = M ◦ (id × S) ◦ ∆ = η ◦ ϵ, where M and η denote the multiplication and the unit map respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 Gradings Similar to the discussions in [21, 22], we can assign different gradings to the quiver BPS algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The degree of an element x can be written as deg(x) = (pdeg(x), hdeg(x)), where pdeg(x) = � pdeg(a)(x) � is a vector known as the principal degree and hdeg(x) is a number called the homogeneous degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We can introduce some invertible elements D(a) and D such that D(a)x � D(a)�−1 = eβpdeg(a)(x)x and DxD−1 = e−βhdeg(x)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We have deg � e(a) n � = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' , 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' , 0, n) and deg � f(a) n � = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' , −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' , 0, n), where ±1 is at the ath entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' On the other hand, deg � ψ(a) ±,n � = (0, ±n) and deg(C) = deg � D(a)� = deg(D) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For elliptic algebras, we may further consider the degree with respect to q, as well as an operator Dq, such that the modes at order α would have q-deg equal to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 3 Toroidal Algebras for Non-Chiral Quivers The first examples we shall discuss are the toroidal algebras for non-chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Here, we will mainly focus on the generalized conifolds xy = zMwN with M + N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In particular, it suffices to consider these cases in the discussions of toric duality as the other cases all have one single toric phase8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We shall use the same convention as in [5] for the two parameters h1,2 of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the arrow pointing from a to b, its charge is hab = Aabh1 + Mabh2 = � � � � � � � � � � � � � 2ςah1, a = b ςb(−h1 − h2), a + 1 = b ςa(−h1 + h2), a = b + 1 0, otherwise , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1) 8Of course, for M + N ≥ 3, all the triangles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=', MN = 0) and the suspended pinch point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=', (M, N) = (2, 1), (1, 2)) have one single toric phase as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 8 – where the definition of the signs ςa, as well as the construction of the quivers from a given toric diagram, can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Equivalently, we can write Hab = HAab 1 HMab 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Here, Aab is the Cartan matrix Aab = (ςa + ςa+1)δab − ςaδa,b+1 − ςbδa+1,b, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2) and Mab is defined as Mab = ςaδa,b+1 − ςbδa+1,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3) Therefore, Aab is symmetric while Mab is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The relations for the toroidal quiver algebra T can then be explicitly written as ψ(a) ± (U)ψ(b) ± (V ) = ψ(b) ± (V )ψ(a) ± (U), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='4) HMab 2 HAab 1 U − CV HMab 2 U − HAab 1 CV ψ(a) ± (U)ψ(b) ∓ (V ) = HMab 2 HAab 1 CU − V HMab 2 CU − HAab 1 V ψ(b) ∓ (V )ψ(a) ± (U), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='5) � HMabC±1/2U − HAab 1 V � ψ(a) ± (U)e(b)(V ) = � HMab 2 HAab 1 C±1/2U − V � e(b)(V )ψ(a) ± (U), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='6) � HMabC∓1/2U − H−Aab 1 V � ψ(a) ± (U)f(b)(V ) = � HMab 2 H−Aab 1 C∓1/2U − V � f(b)(V )ψ(a) ± (U), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='7) � HMabU − HAab 1 V � e(a)(U)e(b)(V ) = (−1)|a||b| � HMab 2 HAab 1 U − V � e(b)(V )e(a)(U), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='8) � HMabU − H−Aab 1 V � f(a)(U)f(b)(V ) = (−1)|a||b| � HMab 2 H−Aab 1 U − V � f(b)(V )f(a)(U), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='9) � e(a)(U), f(b)(V ) � = −δab � δ � UV −1C−1� ψ(a) + � UC−1/2� − δ � UV −1C � ψ(a) − � V C−1/2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='10) The Serre relations are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, when the central charge is trivial, that is, when C = 1, ψ+ would commute with the ψ− as can be seen directly from their current relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 More on Mode Expansions We can also express (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='4)∼(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='10) in terms of modes: ψ(a) ±,mψ(b) ±,n = ψ(b) ±,nψ(a) ±,m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='11) H2Mab 2 HAab 1 Cψ(a) +,m+2ψ(b) −,n − HMab 2 � H2Aab 1 + C2� ψ(a) +,m+1ψ(b) −,n−1 − HAab 1 Cψ(a) +,mψ(b) −,n−2 = H2Mab 2 HAab 1 Cψ(b) −,nψ(a) +,m+2 − HMab 2 � H2Aab 1 C2 + 1 � ψ(b) −,n−1ψ(a) +,m+1 − HAab 1 Cψ(b) −,n−2ψ(a) +,m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='12) – 9 – HMab 2 C1/2ψ(a) +,m+1e(b) n − HAab 1 ψ(a) +,me(b) n+1 = HMab 2 HAab 1 C1/2e(b) n ψ(a) +,m+1 − e(b) n+1ψ(a) +,m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='13) HMab 2 C−1/2ψ(a) −,me(b) n − HAab 1 ψ(a) −,m+1e(b) n+1 = HMab 2 HAab 1 C−1/2e(b) n ψ(a) −,m − e(b) n+1ψ(a) −,m+1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='14) HMab 2 C−1/2ψ(a) +,m+1f(b) n − H−Aab 1 ψ(a) +,mf(b) n+1 = HMab 2 H−Aab 1 C−1/2f(b) n ψ(a) +,m+1 − f(b) n+1ψ(a) +,m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='15) HMab 2 C1/2ψ(a) −,mf(b) n − H−Aab 1 ψ(a) −,m+1f(b) n+1 = HMab 2 H−Aab 1 C1/2f(b) n ψ(a) −,m − f(b) n+1ψ(a) −,m+1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='16) HMab 2 e(a) m+1e(b) n − HAab 1 e(a) m e(b) n+1 = (−1)|a||b| � HMab 2 HAab 1 e(b) n e(a) m+1 − e(b) n+1e(a) m � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='17) HMab 2 f(a) m+1f(b) n − H−Aab 1 f(a) m f(b) n+1 = (−1)|a||b| � HMab 2 H−Aab 1 f(b) n f(a) m+1 − f(b) n+1f(a) m � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='18) � e(a) m , f(b) n � = −δab � C(m−n)/2ψ(a) +,m+n − C−(m−n)/2ψ(a) −,−m−n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='19) Notice that ψ±,l<0 simply vanishes such as in the ef relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In particular, the ψe and ψf relations include ψ(a) ±,0e(b) n = H±Aab 1 e(b) n ψ(a) ±,0, ψ(a) ±,0f(b) n = H∓Aab 1 f(b) n ψ(a) ±,0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='20) by setting m = −1 and HMab 2 C1/2ψ(a) +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1e(b) n − HAab 1 ψ(a) +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0e(b) n+1 = HMab 2 HAab 1 C1/2e(b) n ψ(a) +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 − e(b) n+1ψ(a) +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' HMab 2 C−1/2ψ(a) −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0e(b) n − HAab 1 ψ(a) −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1e(b) n+1 = HMab 2 HAab 1 C−1/2e(b) n ψ(a) −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0 − e(b) n+1ψ(a) −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' HMab 2 C−1/2ψ(a) +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1f(b) n − H−Aab 1 ψ(a) +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0f(b) n+1 = HMab 2 H−Aab 1 C−1/2f(b) n ψ(a) +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 − f(b) n+1ψ(a) +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' HMab 2 C1/2ψ(a) −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0f(b) n − H−Aab 1 ψ(a) −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1f(b) n+1 = HMab 2 H−Aab 1 C1/2f(b) n ψ(a) −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0 − f(b) n+1ψ(a) −,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='21) by setting m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Likewise, the ψ+ψ− relation includes ψ(a) +,mψ(b) −,0 = ψ(b) −,0ψ(a) +,m, ψ(a) +,0ψ(b) −,n = ψ(b) −,nψ(a) +,0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='22) by taking n = 0 and m = −2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, ψ±,0 commute with all the modes of ψ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It is worth noting that given a fixed fermionic node ϝ, the ψ(ϝ) ± modes commute with all ϝ modes, and the e(ϝ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' f(ϝ)) modes anticommute with the e(ϝ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' f(ϝ)) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In fact, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='20), it is not hard to see that ψ(a) +,0ψ(a) −,0 is central for any node a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Write these central elements as C(a) = ψ(a) +,0ψ(a) −,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then we can write ψ(a) ±,0 = C(a) � ψ(a) ∓,0 �−1 with a mild assumption that � ψ(a) ±,0 �−1 are also in the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For convenience, we shall rescale them to be 1, that is, ψ(a) +,0 = � ψ(a) −,0 �−1 , in the following discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Like many toroidal algebras, it is instructive to write the ψ(a) ± (U) currents as ψ(a) ± (U) = ψ(a) ±,0 exp � ∞ � n=1 k(a) ±nU ∓n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='23) Therefore, ψ(a) ±,n = ψ(a) ±,0 n � m=1 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' � r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=',rm>0 r1+···+rm=n k(a) ±r1k(a) ±,r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k(a) ±,rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='24) – 10 – Similarly, we can write the zero modes as ψ(a) +,0 = exp � −βh1k(a) 0 � = H−k(a) 0 1 , ψ(a) −,0 = exp � βh1k(a) 0 � = Hk(a) 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='25) We shall refer to the modes k(a) r (r ∈ Z) as Heisenberg modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' There could also be different conventions to define these modes as discussed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In terms of the Heisenberg modes, we can rewrite the relations involving ψ± as � k(a) 0 , k(b) s � = 0, � k(a) r̸=0, k(b) s � = δr+s,0 1 r � C−r − Cr� H−rMab 2 � HrAab 1 − H−rAab 1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='26) � k(a) 0 , e(b) n � = −Aabe(b) n , � k(a) 0 , f(b) n � = Aabf(b) n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='27) � k(a) r̸=0, e(b) n � = 1 rC−|r|/2H−rMab 2 � HrAab 1 − H−rAab 1 � e(b) n+r, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='28) � k(a) r̸=0, f(b) n � = −1 rC|r|/2H−rMab 2 � HrAab 1 − H−rAab 1 � f(b) n+r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='29) Moreover, we have � e(a) n , f(b) −n � = δab � Hk(a) 0 1 − H−k(a) 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='30) It would also be helpful to notice that � e(a) ±1, f(b) 0 � = ∓δabC1/2H∓k(a) 0 1 k(a) ±1, � e(a) 0 , f(b) ±1 � = ∓δabC−1/2H∓k(a) 0 1 k(a) ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='31) Coproduct We can also write the coproduct above using k(a) r : ∆ � k(a) r � = � Cr ⊗ k(a) r + k(a) r ⊗ 1, r ≥ 0 k(a) r ⊗ Cr + 1 ⊗ k(a) r , r < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='32) In particular, ∆ � k(a) 0 � = 1 ⊗ k(a) 0 + k(a) 0 ⊗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Grading Likewise, for the aforementioned grading, we have deg � k(a) r � = (0, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In [21, 22], such gradings were useful in the quantum double construction of the universal R-matrix for certain toroidal algebra associated to gl1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For toroidal BPS algebras associated to any non-chiral quivers here, a naive generalization would be R = R(0)R(1)R(2) with R(0) = � C−1 ⊗ D−1� � D−1 ⊗ C−1� � a � ψ(a) +,0 ⊗ � D(a)�−1� �� D(a)�−1 ⊗ ψ(a) +,0 � , R(1) = exp � �� r≥1 r � a k(a) r ⊗ k(a) −r � � , R(2) = 1 ⊗ 1 + � n∈Z � a e(a) n ⊗ f(a) −n + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='33) where the ellipsis in R(2) indicates terms with hdeg ≥ 1, and pdeg � R(2)� should be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' How- ever, whether these naive expressions would work and/or what modifications (such as proper normalizations etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=') are needed would still require further investigations in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 11 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 Toric Duality Now let us try to construct the transformations of the generators under toric duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As mentioned in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3, only fermionic nodes can be dualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' If the node ϝ is dualized, then we just need to add an adjoint loop to ϝ±1 if |ϝ±1| = 0 or remove the existing adjoint loop on ϝ ± 1 if |ϝ ± 1| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As a result, ς′ a = � −ςa, a = ϝ, ϝ + 1 ςa, otherwise , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='34) where the primed notation stands for the one after performing the duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, we have A′ ab = � � � � � � � −Aab, (a, b) = (ϝ ± 1, ϝ), (ϝ, ϝ ± 1) Aaa + 2Aaϝ, a = b = ϝ ± 1 Aab, otherwise (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='35) and M′ ab = � � � � � � � −Mab, a = ϝ − 1, ϝ, b = a + 1 −Mab, a = ϝ, ϝ + 1, b = a − 1 Mab, otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='36) Analogous to the rational case, the ke and kf commutation relations can be used to express higher e, f using lower modes9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The higher modes of k can in turn be obtained using the ef relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In fact, the relations involving higher modes can also be derived from those with lower modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, the toroidal BPS algebras for non-chiral quivers are finitely presented with the relations involving e0, e±1, f0, f±1, k0, k±1 (or equivalently, ψ±,0, ψ±,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Hence, it suffices to find the transformations for these modes10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We would like to mimic the isomorphisms for the rational case in [17], which was in turn found by virtue of the odd reflections of the underlying affine Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As all but three of the nodes are unaffected, we would expect the modes to be invariant for a ̸= ϝ, ϝ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, from their relations, we have C′ = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='37) Now, let us first consider the zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For a = ϝ, the k′ 0 modes should be determined only by k0 themselves, possibly with changes of minus signs (such as multiplication by −1), while the e0 and f0 modes should get swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In the rational case, the ψ′ 0 mode is a sum of ψ(a) 0 and ψ(ϝ) 0 for a = ϝ ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Here, our ansatz for ψ0 would still be a combination of ψ(a) 0 and ψ(ϝ) 0 , but we expect it to be a multiplication instead of addition as we are dealing with 9Here, by higher (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' lower) modes, we mean the modes with larger (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' smaller) absolute values |n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 10As pointed out in [17], there is a subtlety for the case xy = z2w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For one of the two toric phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' the one with only fermionic nodes, it seems that the Serre relations can not be fully recovered from the Serre relations for modes with n = 0, ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, we may still verify its transformation when using the currents as will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 12 – the trigonometric case (and hence addition for k0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' On the other hand, for e′(a) 0 , the ansatz would be a linear combination of e(a) 0 e(ϝ) 0 and e(ϝ) 0 e(a) 0 (and likewise for f′(a) 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' By computing the supercommutators [x, y} with x = e(ϝ) 0 e(a) 0 , e(a) 0 e(ϝ) 0 and y = f(a) 0 f(ϝ) 0 , f(ϝ) 0 f(a) 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='38) we find that for a = ϝ ± 1, ψ′(a) ±,0 = ψ(a) ±,0ψ(ϝ) ±,0, k′(a) 0 = k(a) 0 + k(ϝ) 0 , e′(a) 0 = e(ϝ) 0 e(a) 0 − (−1)|a|HAaϝ 1 e(a) 0 e(ϝ) 0 , f′(a) 0 = 1 HAaϝ 1 − H−Aaϝ 1 � f(a) 0 f(ϝ) 0 − (−1)|a|H−Aaϝ 1 f(ϝ) 0 f(a) 0 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='39) would verify the corresponding ef relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Likewise, checking the ef relation for a = ϝ, we have ψ′(ϝ) ±,0 = ψ(ϝ) ∓,0, k′(ϝ) 0 = −k(ϝ) 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='40) and e′(ϝ) 0 = f(ϝ) 0 , f′(ϝ) 0 = −e(ϝ) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, to be compatible with the ee and ff relations that contain modes with n = 0, ±1, we need to multiply them by some extra factors: e′(ϝ) 0 = ψ(ϝ) +,0f(ϝ) 0 = H−k(ϝ) 0 1 f(ϝ) 0 , f′(ϝ) 0 = −ψ(ϝ) −,0e(ϝ) 0 = −Hk(ϝ) 0 1 e(ϝ) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='41) Notice that they would still recover the transformations of the Chevalley generators under odd reflections in the limit β → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' One may check that these transformations are consistent with all the other relations involving zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Next, let us consider the modes with n = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' By considering the commutator of k′(b̸=ϝ) 1 and e′(a) 0 with b = a ± 1 (which is always possible since there are at least four nodes in the quiver), we find that for a = ϝ ± 1, e′(a) 1 = e(ϝ) 0 e(a) 1 − (−1)|a|HAaϝ 1 e(a) 1 e(ϝ) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='42) Likewise, f′(a) 1 = 1 HAaϝ 1 − H−Aaϝ 1 � f(a) 1 f(ϝ) 0 − (−1)|a|H−Aaϝ 1 f(ϝ) 0 f(a) 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='43) Again, computing [x, y} with x = e(ϝ) 0 e(a) 1 , e(a) 1 e(ϝ) 0 and y = f(a) 1 f(ϝ) 0 , f(ϝ) 0 f(a) 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='44) we find that ψ′(a) +,1 = ψ(ϝ) +,0ψ(a) +,1 − C1/2H−Maϝ 2 � HAaϝ 1 f(ϝ) 1 e(ϝ) 0 + H−Aaϝ 1 e(ϝ) 0 f(ϝ) 1 � ψ(a) +,0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='45) ψ′(a) −,1 = ψ(ϝ) −,0ψ(a) −,1 − C−1/2HMaϝ 2 � HAaϝ 1 e(ϝ) −1 f(ϝ) 0 + H−Aaϝ 1 f(ϝ) 0 e(ϝ) −1 � ψ(a) −,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='46) – 13 – In terms of the Heisenberg modes, we have k′(a) 1 = k(a) 1 − C1/2H−Maϝ 2 � HAaϝ 1 f(ϝ) 1 e(ϝ) 0 + H−Aaϝ 1 e(ϝ) 0 f(ϝ) 1 � Hk(ϝ) 0 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='47) k′(a) −1 = k(a) −1 − C−1/2HMaϝ 2 � HAaϝ 1 e(ϝ) −1 f(ϝ) 0 + H−Aaϝ 1 f(ϝ) 0 e(ϝ) −1 � H−k(ϝ) 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='48) By considering the commutation relations of k′(ϝ±1) 1 and e′(ϝ) 0 , we find that e′(ϝ) 1 = CH−2Maϝ 2 Hk(ϝ) 0 1 f(ϝ) 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='49) where a can either be ϝ + 1 or ϝ − 1 as Maϝ would be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Likewise, f′(ϝ) 1 = H−2Maϝ 2 � −C−1e(ϝ) 1 + C−1/2k(ϝ) 1 e(ϝ) 0 � Hk(ϝ) 0 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='50) e′(ϝ) −1 = H2Maϝ 2 � Cf(ϝ) −1 − C1/2k(ϝ) −1 f(ϝ) 0 � H−k(ϝ) 0 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='51) f′(ϝ) −1 = −C−1H2Maϝ 2 H−k(ϝ) 0 1 e(ϝ) −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='52) Using the ef relations, we get ψ′(ϝ) ±,1 = −H∓2Maϝ 2 � ψ(ϝ) ∓,0 �2 ψ(ϝ) ±,1, k′(ϝ) ±1 = −H∓2Maϝ 2 k(ϝ) ±1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='53) One may check that these transformations are consistent with all the other relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' From the above discussions, we may also derive the transformations in terms of currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' By applying the k±1 modes successively, it is not hard to see that e′(a)(U) = e(ϝ) 0 e(a)(U) − (−1)|a|HAaϝ 1 e(a)(U)e(ϝ) 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='54) f′(a)(U) = 1 HAaϝ 1 − H−Aaϝ 1 � f(a)(U)f(ϝ) 0 − (−1)|a|H−Aaϝ 1 f(ϝ) 0 f(a)(U) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='55) for a = ϝ ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then by considering their supercommutator, we find that each term contains some formal delta function with other terms being cancelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This yields ψ′(a) ± (U) =e(ϝ) 0 ψ(a) ± (U)f(ϝ) 0 − (−1)|a|HAaϝ 1 e(ϝ) 0 f(ϝ) 0 ψ(a) ± (U) − H−Aaϝ 1 ψ(a) ± (U)e(ϝ) 0 f(ϝ) 0 − f(ϝ) 0 ψ(a) ± (U)e(ϝ) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='56) It is less straightforward to write down the currents for ϝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Nevertheless, we can write some conjectural expressions by computing a few more higher modes and then verify them using the current relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The perturbative calculations show that e′(ϝ) 0 (U) = f(ϝ) >0 � C−1U � ψ (ϝ) + � C−1/2H2Maϝ 2 U � + f(ϝ) ≤0 (CU) ψ (ϝ) − � C1/2H2Maϝ 2 U � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='57) f′(ϝ) 0 (U) = −e(ϝ) ≥0 (CU) ψ (ϝ) + � C1/2H2Maϝ 2 U � − e(ϝ) <0 � C−1U � ψ (ϝ) − � C−1/2H2Maϝ 2 U � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='58) – 14 – where f(ϝ) >0 (U) = � n>0 f(ϝ) n U −n, f(ϝ) ≤0 (U) = � n≤0 f(ϝ) n U −n, e(ϝ) ≥0 (U) = � n≥0 e(ϝ) n U −n, e(ϝ) <0 (U) = � n<0 e(ϝ) n U −n, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='59) and ψ (ϝ) + (U) = � ψ(ϝ) −,0 �2 � � �ψ(ϝ) +,0 − ψ(ϝ) +,1 U − ψ(ϝ) +,2 − � ψ(ϝ) +,1 �2 ψ(ϝ) −,0 U 2 − ψ(ϝ) +,3 + � ψ(ϝ) +,1 �3 � ψ(ϝ) −,0 �2 U 3 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' � � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='60) ψ (ϝ) − (U) = � ψ(ϝ) +,0 �2 � ψ(ϝ) −,0 − ψ(ϝ) −,1U − � ψ(ϝ) −,2 − � ψ(ϝ) −,1 �2 ψ(ϝ) +,0 � U 2 − � ψ(ϝ) −,3 + � ψ(ϝ) −,1 �3 � ψ(ϝ) +,0 �2� U 3 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='61) In fact, we find that the perturbative expressions here coincide with the “inverse currents”,that is, ψ (ϝ) ± (U) = ψ(ϝ) ± (U)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='62) Then we have ψ′(ϝ) ± (U) = ψ(ϝ) ± � H2Maϝ 2 U �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='63) Indeed, one may verify these expressions using the current relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It is also worth noting that k′(ϝ) n = −H2nMaϝ 2 k(ϝ) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='64) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3 Higgsing As reviewed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2, the toric quiver gauge theories have nice features under the Higgs-Kibble mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It is then natural to wonder if their BPS algebras are also connected via blowing up/down the singularities, or more precisely, if there is a subalgebra structure for a higgsed theory from a parent theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As the higgsing process always merges the two neighbouring nodes, say a and a + 1, in the quiver for any toric CY without compact divisors, we expect the generators associated with other nodes (and the central element C) to be invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Of course, there is a relabelling for b > a + 1 as the number of nodes is reduced by one after higgsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For x′(a) (x = e, f, ψ, k), where the primed letters indicate the generators for the higgsed theory, it should be a combination of x(a) and x(a+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3, the parity should satisfy ��x′(a)�� = ��x(a)�� + ��x(a+1)��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, for the zero modes, a natural – 15 – candidate would be a combination of e(a) 0 e(a+1) 0 and e(a+1) 0 e(a) 0 (and likewise for f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Similar to the construction for toric duality, we find that e′(a) 0 = e(a+1) 0 e(a) 0 − (−1)|a||a+1|HAa,a+1 1 e(a) 0 e(a+1) 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='65) f′(a) 0 = 1 HAa,a+1 1 − H−Aa,a+1 1 � f(a) 0 f(a+1) 0 − (−1)|a||a+1|H−Aa,a+1 1 f(a+1) 0 f(a) 0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='66) ψ′(a) ±,0 = ψ(a) ±,0ψ(a+1) ±,0 , k′(a) 0 = k(a) 0 + k(a+1) 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='67) would give the expected subalgebra structure for the zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is precisely the trans- formation for a = ϝ ± 1 in the above discussions of toric duality with ϝ replaced by a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In fact, in the rational limit β → 0, this gives the surjection map of the Chevalley generators of the corresponding affine Lie superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, when we use k′(a−1) ±1 = k(a) ±1 or k′(a+1) ±1 = k(a+2) ±1 to get the higher modes from e′(a) 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' f′(a) 0 ), the expressions are not symmetric in e(a) and e(a+1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' f(a) and f(a+1)) any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Indeed, for instance, � k(a−1) 1 , e′(a) 0 � yields e′(a) 1 = e(a+1) 0 e(a) 1 − (−1)|a||a+1|HAa,a+1 1 e(a) 1 e(a+1) 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='68) while � k(a+2) 1 , e′(a) 0 � leads to e′(a) 1 = e(a+1) 1 e(a) 0 − (−1)|a||a+1|HAa,a+1 1 e(a) 0 e(a+1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='69) They are not equal to each other as can be seen from the ee relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Explicitly, e(a+1) 1 e(a) 0 −(−1)|a||a+1|HAa,a+1 1 e(a) 0 e(a+1) 1 = HMa,a+1 2 � e(a+1) 0 e(a) 1 − (−1)|a||a+1|HAa,a+1 1 e(a) 1 e(a+1) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='70) Due to the non-trivial factor HMa,a+1 2 , this transformation does not give the subalgebra struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Nevertheless, when H2 = 1, the quiver BPS algebras reduce to a one-parameter algebra, and the above two expressions for e′(a) 1 would coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, at least when h2 = 0, for non-chiral quivers11, the toroidal BPS algebra contains the ones for the higgsed theories as its subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The surjection for the generators associated with a and a + 1 are the same as the transformations for a = ϝ ± 1 under toric duality with ϝ replaced by a ∓ 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Of course, a + 1 (as well as a) can be either bosonic or fermionic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is also the case for the rational quiver Yangians, where the surjection map is most conveniently expressed in the J presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' See (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='5) in [17] (with conventions therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It is not clear whether higgsing would still lead to the subalgebra structure for generic h2, and if so, what the surjection map would be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Physically, the two parameters of the algebra are related to the Ω-background that is used to resolve the singular target space 11For C3/(Z2 × Z2) which can be higgsed to the suspended pinch point, this should also be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The discussions here do not cover C3, C × C2/Z2 and the conifold although we still expect this to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 12As a result, this gives two transformations, but they should essentially be the same up to a normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 16 – of the supersymmetric quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In particular, the scalars in the vector multiplets would also have non-zero VEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, the algebra structure under higgsing could be closely related to the localizations of the Higgs and Coulomb branches [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 4 Elliptic Algebras and Chiral Quivers Now, let us have a discussion on the remaining cases including the elliptic algebras for non- chiral quivers and the algebras for chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Unlike the rational and toroidal algebras for non-chiral quivers, it is more difficult to work with modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is due to the existence of q-Pochhammer symbols in the elliptic case while for chiral quivers, different CYs/quivers would have different “minimalistic” presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, we shall mainly consider the more unified current relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 Elliptic Algebras for Non-Chiral Quivers Given a generalized conifold xy = zMwN with M + N ≥ 3, the elliptic quiver algebra E has the relations ψ(a) ± (U)ψ(b) ± (V ) = ψ(b) ± (V )ψ(a) ± (U), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1) ψ(a) ± (U)ψ(b) ∓ (V ) = � UCV −1HAab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qU −1C−1V H−Aab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � U −1C−1V HAab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUCV −1H−Aab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � U −1CV HAab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUC−1V −1H−Aab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � UC−1V −1HAab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qU −1CV H−Aab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ ψ(b) ∓ (V )ψ(a) ± (U) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2) ψ(a) ± (U)e(b)(V ) = HAab 1 � U −1C∓ 1 2 V H−Aab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUC± 1 2 V −1HAab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � U −1C∓ 1 2 V HAab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUC± 1 2 V −1H−Aab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ e(b)(V )ψ(a) ± (U) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3) ψ(a) ± (U)f(b)(V ) = H−Aab 1 � U −1C± 1 2 V HAab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUC∓ 1 2 V −1H−Aab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � U −1C± 1 2 V H−Aab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUC∓ 1 2 V −1HAab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ f(b)(V )ψ(a) ± (U) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='4) e(a)(U)e(b)(V ) = (−1)|a||b|HAab 1 � U −1V H−Aab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUV −1HAab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � U −1V HAab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUV −1H−Aab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ e(b)(V )e(a)(U) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='5) – 17 – f(a)(U)f(b)(V ) = (−1)|a||b|H−Aab 1 � U −1V HAab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUV −1H−Aab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � U −1V H−Aab 1 H−Mab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUV −1HAab 1 HMab 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ f(b)(V )f(a)(U) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='6) � e(a)(U), f(b)(V ) � = −δab � δ � UV −1C−1� ψ(a) + � UC−1/2� − δ � UV −1C � ψ(a) − � V C−1/2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='7) Similar to the toroidal case, for any fermionic node ϝ, we have ψ(ϝ) ± (U)e(ϝ)(V ) = e(ϝ)(V )ψ(ϝ) ± (U), e(ϝ)(U)e(ϝ)(V ) = −e(ϝ)(V )e(ϝ)(U) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, when the central charge is trivial, that is, C = 1, ψ(a) + (U) commutes with ψ(b) − (V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 More on Mode Expansions Although we would like to work with the currents directly, it would still be helpful to have a look at their mode expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' There are infinitely many groups of relations as α can be any non-negative integer, but there are finitely many terms in each relation at each order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' At order q0, for instance, the ee relations read e(a) m+1,0e(b) n,0 − HAab 1 H−Mab 2 e(a) m,0e(b) n+1,0 = (−1)|a||b| � HAab 1 e(b) n,0e(a) m+1,0 − H−Mab 2 e(b) n+1,0e(a) m,0 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='8) which coincide with the ee relations for the toroidal algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In fact, all the relations at q0 are the same as those in the toroidal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, the elliptic subalgebra E0 at order q0 is isomorphic to the toroidal algebra T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is expected as the elliptic algebra E reduces to T in the limit q → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As another example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' let us also write the ψe relations at order q1 here: � HMab 2 U − HAab 1 V � �� −H−Aab 1 HMab 2 UV −1 − HAab 1 H−Mab 2 V U −1� ψ(a) ±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0 � C∓1/2U � e(b) 0 (V ) +ψ(a) ±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 � C∓1/2U � e(b) 0 (V ) + ψ(a) ±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0 � C∓1/2U � e(b) 1 (V ) � = � HAab 1 HMab 2 U − V � �� −HAab 1 HMab 2 UV −1 − H−Aab 1 H−Mab 2 V U −1� ψ(a) ±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0 � C∓1/2U � e(b) 0 (V ) +ψ(a) ±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 � C∓1/2U � e(b) 0 (V ) + e(b) 1 (V )ψ(a) ±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='0 � C∓1/2U �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='9) from which we can write the corresponding mode relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The other relations can be obtained in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For relations at higher orders of q, there would be more terms with larger ranges of modes in the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In general, at order qα, the ψ± � C∓1/2U � e(V ) relations read � HMab 2 U − HAab 1 V � α � γ=0 � α1,α2 α1+α2=α−γ Kγ(Aab)ψ(a) ±,α1 � C∓1/2U � e(b) α2(V ) = � HAab 1 HMab 2 U − V � α � γ=0 � α1,α2 α1+α2=α−γ Kγ(−Aab)e(b) α2(V )ψ(a) ±,α1 � C∓1/2U � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='10) – 18 – for some functions Kγ coming from the expansions of (the product of) the q-Pochhammer symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Here, we have suppressed the other indices and arguments in Kγ for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In particular, K0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The e(U)e(V ) relations have the same coefficients (with an extra sign factor (−1)|a||b|) while for the ψ± (C∓U) f(V ) and f(U)f(V ) relations, we simply have Aab ↔ −Aab on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We can then write the mode relations at each order of q from these current relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Heisenberg modes Similar to the discussions on the toroidal algebras above, as well as some elliptic deformed algebras in [23], we may expand the ψ± modes as ψ(a) + (U) = H−k(a) 0 1 exp � �� n̸=0 k(a) n U −n � � , ψ(a) − (U) = Hl(a) 0 1 exp � �� n̸=0 l(a) −nU n � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='11) For convenience, we shall still refer to the k and l modes as Heisenberg modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Notice that the sums are now over Z\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, ψ(a) +,n = H−k(a) 0 1 � � � � ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' � ri̸=0 r1+···+rm=n kr1kr2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' krm � � � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='12) ψ(a) −,n = H−l(a) 0 1 � � � � ∞ � m=0 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' � ri̸=0 r1+···+rm=n lr1lr2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' lrm � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='13) In particular, k(a) 0 and l(a) 0 are not equal to ψ(a) ±,0 (or ψ(a) ±,0,0) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Nevertheless, the Heisenberg modes may still play the role that raises or lowers the e, f modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' More explicitly, � k(a) r , k(b) s � = � l(a) r , l(b) s � = � k(a) 0 , l(b) s � = � k(a) r , l(b) 0 � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='14) � k(a) r̸=0, l(b) s � = δr+s,0 1 r 1 1 − qr � C−r − Cr� H−rMab 2 � HrAab 1 − qrH−rAab 1 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='15) � k(a) 0 , e(b) n � = −Aabe(b) n , � k(a) 0 , f(b) n � = Aabf(b) n , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='16) � l(a) 0 , e(b) n � = Aabe(b) n , � l(a) 0 , f(b) n � = −Aabf(b) n , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='17) � k(a) r̸=0, e(b) n � = 1 r 1 1 − qr C−r/2H−rMab 2 � HrAab 1 − H−rAab 1 � e(b) n+r, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='18) � k(a) r̸=0, f(b) n � = −1 r 1 1 − qr Cr/2H−rMab 2 � HrAab 1 − H−rAab 1 � f(b) n+r, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='19) � l(a) r̸=0, e(b) n � = 1 r 1 1 − qr C−r/2H−rMab 2 � HrAab 1 − H−rAab 1 � e(b) n−r, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='20) � l(a) r̸=0, f(b) n � = −1 r 1 1 − qr Cr/2H−rMab 2 � HrAab 1 − H−rAab 1 � f(b) n−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='21) However, the ef relations in terms of k and l would be quite different from those of the toroidal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is one of the difficulties when discussing toric duality for elliptic algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 19 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 Toric Duality Let us have a brief discussion on toric duality for the elliptic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In fact, as discussed in Appendix B, the dressed currents E(a)(u), F (a)(u) and Ψ(a) ± (u) introduced therein have the same relations as those of the toroidal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, the previous transformations should also apply to the elliptic cases using the dressed currents (with products replaced by correlators or normal orderings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, by comparing these relations with the ones using the “bare” generators at each order qα, we may write the correlators ⟨XY ⟩α in the expansion of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For instance, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='10), we have � Ψ(a) ± � C∓1/2U � E(b) (V ) � α = α � γ=0 � α1,α2 α1+α2=α−γ Kγ(Aab)ψ(a) ±,α1 � C∓1/2U � e(b) α2(V ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='22) � E(b) (V ) Ψ(a) ± � C∓1/2U �� α = α � γ=0 � α1,α2 α1+α2=α−γ Kγ(−Aab)e(b) α2(V )ψ(a) ±,α1 � C∓1/2U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='23) Nevertheless, let us still take a look at the original bare generators e, f, ψ± directly in the followings for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Suppose that the node ϝ is dualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then the currents associated to a ̸= ϝ, ϝ ± 1 (and hence C) should remain invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For a = ϝ ± 1, we expect the currents to have a combination of a and ϝ currents/modes similar to the ones in the toroidal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us recall that for the toroidal algebras, we have e′(a)(U) = � e(ϝ) 0 , e(a)(U) � H Aaϝ 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='24) where the deformed bracket is given by [x, y}q = xy −(−1)|x||y|qyx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Likewise, for the rational algebras, we have e′(a)(U) = � e(ϝ) 0 , e(a)(U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='25) As a result, each transformation is determined by its corresponding version of the bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, these are preciously the brackets that appear in their Serre relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, we propose that the elliptic version of the bracket is used here: e′(a)(U) = � e(ϝ)(V ), e(a)(U) � χ ���� V 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='26) where χ represents the elliptic deformed bracket as in Appendix B and V 0 indicates that we only take the terms of order V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' More explicitly, using the q-binomial theorem, we have e′(a)(U) = ∞ � n=0 � H2Aaϝ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q)n � qH−Aaϝ 1 HMaϝ 2 U �n � e(ϝ) −ne(a)(U) − (−1)|a|HAaϝ 1 H−2nMaϝ 2 U −2ne(a)(U)e(ϝ) n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='27) – 20 – Likewise, f′(a)(U) = ∞ � n=0 � H−2Aaϝ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � n (q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q)n � qHAaϝ 1 H−Maϝ 2 U −1�n HAaϝ 1 − H−Aaϝ 1 � f(a)(U)f(ϝ) n − (−1)|a|H−Aaϝ 1 H2nMaϝ 2 U 2nf(ϝ) −n f(a)(U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='28) For the node ϝ, we expect that ψ′ ± are still given by the inverse currents, that is, ψ′(ϝ) ± (U) = ψ(ϝ) ± � H2Maϝ 2 U �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='29) Analogously, it is natural to conjecture that e′(ϝ) and f′(ϝ) would have the same forms as in the toroidal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In other words e′ = f>0ψ−1 + + f≤0ψ−1 − , f′ = −e≥0ψ−1 + − e<0ψ−1 − , where we have omitted the different arguments in different factors for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Indeed, the inverse currents are consistent with the relations under toric duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For instance, the e′(a)e′(ϝ) relation contains e(a)(U)E(ϝ)F(ϝ) ± ψ(ϝ) ± � C∓1/2H2Maϝ 2 V �−1 =(−1)|a|U −1V HMaϝ 2 � UV −1H−Aaϝ 1 H−Maϝ 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � U −1V H−Aaϝ 1 HMaϝ 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qU −1V HAaϝ 1 HMaϝ 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qUV −1HAaϝ 1 H−Maϝ 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ F(ϝ) ± ψ(ϝ) ± � C∓1/2H2Maϝ 2 V �−1 e(a)(U)E(ϝ) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='30) where E(ϝ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' F(ϝ) ± ) sketchily indicates the factors containing only e(ϝ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' f(ϝ)) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The ellipsis stands for the extra terms coming from exchanging these factors which should be cancelled in the whole expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Recall that A′ aϝ = −Aaϝ and M′ aϝ = −Maϝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As we can see, this recovers the correct coefficient for the e′(a)e′(ϝ) relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Higgsing Similar to the rational and toroidal cases, the surjection (if it exists) induced from higgsing should leave the central element C and all but two (say, a and a + 1) currents invariant (with a possible relabelling of nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, due to the complication at higher orders of q, it is more difficult to write the currents associated to a′ in terms of those for a and a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Nevertheless, we may still conjecture that higgsing would also give subalgebras in the elliptic case, at least in certain one-parameter degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 Comments on Chiral Quivers We shall now make some comments on the cases for chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As mentioned in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1, only nodes with two arrows in and two arrows out will be dualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then all the possible cases are listed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, as we are now going to discuss, we will only consider the cases (a), (c) and (d) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the last two cases, (e) and (f), the quivers would remain the same after dualizing the red node (assuming that all the arrows added get integrated out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, their quiver – 21 – (a) (b) (c) (d) (e) (f) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1: The six possible configurations for the dualized node in the quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The node to be dualized is coloured red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The dashed nodes indicate that they can be connected to the remaining part of the quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' BPS algebras are trivially invariant, and we only need to focus on the remaining four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, for toric CYs with compact divisors, as the quiver nodes do not have adjoints (at least for all the known examples to our best knowledge), all the e and f modes/currents are fermionic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In particular, this means that (b) should be excluded as the node with two arrows (one in and one out) connected to the dualized node will get an adjoint loop that cannot be integrated out under duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Indeed, as far as we know, including the examples classified in [24–26], there is no case (b) appearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the remaining three cases, their quivers under toric duality are illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As a preliminary attempt of constructing the transformations, let us consider certain expressions similar to the cases for non-chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Of course, the central element C and the currents associated to the nodes that are not connected to the dualized node should be invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Suppose that the node ϝ is dualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As before, we expect e′(ϝ)(u) to be a combination of F(ϝ) ± (u)ψ(ϝ) ± (−u ∓ c/2)−1 for some F(ϝ) ± (u) (and likewise for f′(ϝ)(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For simplicity, let us just take e′(ϝ)(u) = f(ϝ)(u)ψ(ϝ) + (−u − c/2)−1 as an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Indeed, as ϝ has all its arrow(s) connected to a being reversed, e(a)(u)f(ϝ)(u)ψ(ϝ) + (−u − c/2)−1 would give the required prefactor from the e(a) � ψ(ϝ)�−1 relation while e(a)f(ϝ) would be responsible for the minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the nodes connected to ϝ, since they would remain fermionic after toric duality, we cannot multiply them by e(ϝ) or f(ϝ) as in the non-chiral quiver cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us first consider – 22 – (a) (d) (c) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2: How the arrows and hence the ζ factors would change under toric duality for (a), (c), (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The two types of dashed lines indicate the arrows connecting the orange nodes (before possible cancellations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' the cases (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Suppose that we take e′(a)(u) = � e(a)(−u)ψ(ϝ) + (−u − c/2), a ↠ ϝ or ϝ ↠ a e(a)(−u), a → ϝ or ϝ → a , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='31) where a → b and a ↠ b indicate the number of arrows from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then we have e′(a)(u)e′(ϝ)(v) = e′(a)(u)f(ϝ)(v)ψ(ϝ) + (−v − c/2)−1 = −φa⇐ϝ(−u, −v)−1f(ϝ)(v)ψ(ϝ) + (−v − c/2)−1e′(a)(u) = − (UV )− t 2 χaϝ φa⇐ϝ(v − u)−1f(ϝ)(v)ψ(ϝ) + (−v − c/2)−1e′(a)(u), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='32) which recovers the correct numbers of ζ in the relations as χ′ aϝ = −χaϝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For instance, if a ↠ ϝ in the original quiver, then we have ϝ ↠ a after toric duality, and φa⇐ϝ(v − u)−1 = 1 ζ � h1 aϝ − u + v � ζ � h2 aϝ − u + v �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='33) One may also check that the other e′e′ relations would also give the correct numbers of the ζ factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the case (d), we may choose e′(a)(u) = � e(a)(−u)ψ(ϝ) + (−u − c/2), a → ϝ e(a)(−u), ϝ → a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='34) However, only checking the numbers of ζ in the relations is not sufficient, and astute readers may have already found the following problems: – 23 – Recall that for the rational quiver Yangians, the equalities in the current relations are up to some umvn terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The transformations in terms of the currents may not incorporate these terms in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Whether the transformations in terms of currents would work or how corrections should be made would probably require more detailed delibrations on the relations of modes, which can be much more intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' From the transformations of e and f, we may obtain ψ′ ± from the ef relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, unlike the toroidal and elliptic algebras for non-chiral quivers discussed above, there would be terms that do not have formal delta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Although we could still in principle put them at the right orders of U, V (and q) in the mode expansions of ψ′ ±, there could be ambiguities in this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This subtlety should also be related to the ambiguities of multiplying ψ(ϝ) ± in the above transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Most importantly, when we check the ζ factors above, we have not taken the correct charge assignment for the dual algebra into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For instance, hi aϝ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='33) may not be the right charges for the arrows in the dualized quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In fact, by checking some examples, it is not hard to find that even if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='33) gives the correct charges, the arrows connecting the orange nodes do not have the required charges after the transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In fact, such transformations would only work when the two parameters h1,2 are both zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' One may consider possible shifts of the spectral parameters, such as e(a)(−u + ϵ1)ψ(ϝ)(−u − c/2 + ϵ2) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=', in the above transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, it would yield a set of homogeneous equations only with the trivial solution as there are more independent constraints than variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, the transformations for the dual algebras require a more careful construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Finding such maps may require more sophisticated ways, and we leave it to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Likewise, for higgsing, simple multiplications of the currents for the merged nodes would only give subalgebra structure in the trivial case with vanishing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, given a chiral quiver, it can be higgsed to either chiral or non-chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' There can also be more than one pair of nodes to be merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Although we still expect such surjection maps under higgsing (at least for one-parameter degeneracies), it could be very different from the above cases involving only non-chiral quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Heisenberg modes Similar to the discussions for non-chiral quivers, we may also take the mode expansions as ψ(a) + (U) = exp �� n∈Z k(a) n U −n � , ψ(a) − (U) = exp �� n∈Z l(a) −nU n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='35) We shall still refer to k and l as Heisenberg modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Notice that the conventions when writing k0 and l0 are slightly different from before, and the sums are over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Consider two nodes a and b in any chiral quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Suppose that there are |a → b| = r and |b → a| = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then � k(a) 0 , l(b) 0 � = log � C−r−s� = −(r + s)βc, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='36) – 24 – � k(a) 0 , k(b) 0 � = − � l(a) 0 , l(b) 0 � = log � Cr−s� = (r − s)βc, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='37) � k(a) m̸=0, k(b) n � = � l(a) m̸=0, l(b) n � = � k(a) 0 , l(b) n̸=0 � = � k(a) m̸=0, l(b) 0 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='38) Moreover, we have � k(a) 0 , e(b)(V ) � = � l(a) 0 , e(b)(V ) � = � � � � � � � log � HabV −(r−s)� e(b)(V ), r > s log � −HabV −(r−s)� e(b)(V ), r < s log ((−1)rHab) e(b)(V ), r = s , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='39) � k(a) 0 , f(b)(V ) � = � l(a) 0 , f(b)(V ) � = � � � � � � � − log � HabV −(r−s)� f(b)(V ), r > s − log � −HabV −(r−s)� f(b)(V ), r < s − log ((−1)rHab) f(b)(V ), r = s , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='40) where Hab = r� i=1 H1/2 ab,i s� j=1 H1/2 ba,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It would be more useful to write them as e± 1 r−s k(a) 0 e(b) n e∓ 1 r−s k(a) 0 = sgn(r, s)H ± 1 r−s ab e(b) n∓1 (r ̸= s), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='41) e± 1 r−s k(a) 0 f(b) n e∓ 1 r−s k(a) 0 = sgn(r, s)H ∓ 1 r−s ab f(b) n±1 (r ̸= s), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='42) ek(a) 0 e(b) n e−k(a) 0 = sgn(r, s)Habe(b) n (r = s), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='43) ek(a) 0 f(b) n e−k(a) 0 = sgn(r, s)Habf(b) n (r = s), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='44) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='45) and likewise for l(a) 0 , where we have defined sgn(r, s) = � � � � � � � 1, r > s (−1)r, r = s −1, r < s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='46) The remaining relations would be different for the toroidal and the elliptic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the toroidal algebras, we have � k(a) m , e(b) n � = 1 mC−m/2 � �� j Hm ba,j − � i H−m ab,i � � e(b) n+m (m > 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='47) � k(a) m , f(b) n � = − 1 mCm/2 � �� j Hm ba,j − � i H−m ab,i � � f(b) n+m (m > 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='48) � l(a) −m, e(b) n � = 1 mCm/2 � �� j Hm ba,j − � i H−m ab,i � � e(b) n+m (m > 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='49) – 25 – � l(a) −m, f(b) n � = − 1 mC−m/2 � �� j Hm ba,j − � i H−m ab,i � � f(b) n+m (m > 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='50) � k(a) m , e(b) n � = � k(a) m , f(b) n � = � l(a) −m, e(b) n � = � l(a) −m, f(b) n � = 0 (m < 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='51) � k(a) m , l(b) n � = δm+n,0 1 m � C−m − Cm� � �δm>0 � j Hm ba,j + δm<0 � i H−m ab,i � � (m ̸= 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='52) where δcond is 1 when the condition cond is satisfied and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Notice that we would only raise the e, f modes using the non-zero Heisenberg modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' If we take t = 1 in the balancing factor (UV ) t 2 χab for the toroidal algebras, then only km and l−m with m < 0 would lower the e, f modes while the other non-zero Heisenberg modes would commute with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This would also make certain changs in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='41)∼(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the elliptic algebras, we have � k(a) m , e(b) n � = 1 m 1 1 − qm C−m/2 � �� j Hm ba,j − � i H−m ab,i � � e(b) n+m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='53) � k(a) m , f(b) n � = − 1 m 1 1 − qm Cm/2 � �� j Hm ba,j − � i H−m ab,i � � f(b) n+m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='54) � l(a) −m, e(b) n � = 1 m 1 1 − qm Cm/2 � �� j Hm ba,j − � i H−m ab,i � � e(b) n+m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='55) � l(a) −m, f(b) n � = − 1 m 1 1 − qm C−m/2 � �� j Hm ba,j − � i H−m ab,i � � f(b) n+m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='56) � k(a) m , l(b) n � = δm+n,0 1 m 1 1 − qm � C−m − Cm� � �� i H−m ab,i − � j Hm ba,j � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='57) where m ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' If we take t = 1 in the balancing factor (UV ) t 2 χab, then 1/(1 − qm) would be changed to 1/ (q−m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 5 Discussions Let us mention some properties of the toric quivers that are not discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' They should be closely related to the truncations of the quiver BPS algebras, which could lead to important physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Specular duality There is another duality for toric quiver gauge theories known as the specular duality as proposed in [24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Many concepts and quantities enjoy nice properties – 26 – under such duality [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In general, specular duality does not preserve the mesonic moduli space (except self-dual cases) although the Hilbert series would agree up to some fugacity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Instead, it is a duality that preserves the master space [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, we do not expect the quiver BPS algebras to be isomorphic under specular duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, it exchanges the internal and external perfect matchings, which are associated to the internal and external points of the toric diagram respectively, of the dual brane tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As each arrow in the quiver can be written in terms of a product of some perfect match- ings, the arrows also have a one-to-one correspondence for specular dual theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It is then natural to wonder if the charge assignments would also follow this correspondence of the ar- rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' However, we have checked several examples and this is not the case, even for self-dual ones13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Nevertheless, as argued in [1], the perfect matchings can be used to determine cer- tain truncations of the quiver Yangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is because such truncations come from adding D4-branes to the divisors of the toric CY threefold, which correspond to the lattice points of the toric diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In [1], such truncations were only identified for external (or more pre- cisely, corner) perfect matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It could be possible that the truncations from D4-branes associated to internal points can be studied from the specular dual case, where the internal perfect matchings are mapped to the external ones14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Deformed VOAs The truncations of quiver BPS algebras are of particular interest since they are expected to be related to certain vertex operator algebras (VOAs), and hence im- plement the AGT (aka BPS/CFT) correspondence [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Indeed, the truncations of the rational algebras give rise to the (universal enveloping algebras of) rectangular W-algebras [17, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We expect that the truncations of the toroidal and even elliptic quiver BPS algebras would lead to deformations of the rational VOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In particular, the toroidal algebra for C3 is shown to be a q-deformation of the W1+∞-algebra in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We conjecture that there exist certain q-deformations of the WM|N×∞-algebras such that for toroidal BPS algebras T associated to the generalized conifolds, we have the following commutative diagram which would give the 5d AGT correspondence: T U(qWM|N×l) T�⊗T U(qWM|N×l1)�⊗U(qWM|N×l2) Φl Φl1 ⊗ Φl2 ∆ � ∆l1,l2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1) where Φl are some surjections and the hats denote the completions of the algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' On the BPS algebra side, this would require a detailed study on the so-called horizontal representa- tions of the algebras with non-trivial central element C so that we can get vertex operators from the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' On the VOA side, we need to find some suitable deformations of the 13For a self-dual quiver, an arrow would often be mapped to a different arrow in the quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 14Of course, there can also be external lattice points that are not at the corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, for non-reflexive polygons, specular duality can relate brane tilings on Riemann surfaces with higher genus [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 27 – WM|N×∞-algebras studied in [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It would also be helpful to know more about their free field realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Acknowledgement I am grateful to Ian Cheung, Yang-Hui He, Vishnu Jejjala, Jian-Rong Li and in particular, Deshuo Liu and Rak-Kyeong Seong for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The research is supported by a CSC scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' A Toric Quiver Gauge Theories In this appendix, we give a quick recap on some properties of 4d N = 1 quiver gauge theories from toric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' More details can be found in the references mentioned below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' See also [40–42] for reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 Toric Duality Given a quiver gauge theory with its associated brane tiling, one can study many of its salient features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us first consider quivers that are related by Seiberg duality [43] in the toric phase, which is also known as the toric duality [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In short, picking a node j in the quiver to dualize, we replace all the arrows connected to j with their conjugate (flavour) by reversing their orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then we add a meson, that is, an arrow from i to k, to the new quiver for each 2-path i → j → k in the original quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This process is depicted in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In cluster algebra, this is exactly the mutation i j k Contents 1 Seiberg Duality 1 1 Seiberg Duality In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 gauge theories [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We will phrase our discussion in the language of quivers, since all the theories considered in this paper are of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us consider dualizing a node j in the quiver, which does not have adjoint chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 The transformation of the gauge theory can be summarized in the following rules: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In physics, the arrows connected to the mutated node are usually referred to as flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The flavors transform by simply reversing their orientation, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='a) Replace every incoming arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j with the outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xij the incoming arrow, we replace it by the dual flavor ˜Xji 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='b) Replace every outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k with the incoming arrow k !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj This is the quiver implementation of the fact that the magnetic flavors are in the complex conjugate representations, of both the dualized gauge group and the spectator nodes, of the original flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Next we add mesons to the quiver, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' composite arrows, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For every 2-path i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k we add a new arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This meson Mik can be regarded as the composition of the flavors i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j and j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k of the original theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In other words, we generate all possible composite arrows consisting of incoming and outgoing chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- tions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' []).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2In our discussion, including the points that follow, we allow for the possibility of chiral fields connecting pairs of nodes in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 1 – Contents 1 Seiberg Duality 1 1 Seiberg Duality In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 gauge theories [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We will phrase our discussion in the language of quivers, since all the theories considered in this paper are of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us consider dualizing a node j in the quiver, which does not have adjoint chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 The transformation of the gauge theory can be summarized in the following rules: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In physics, the arrows connected to the mutated node are usually referred to as flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The flavors transform by simply reversing their orientation, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='a) Replace every incoming arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j with the outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xij the incoming arrow, we replace it by the dual flavor ˜Xji 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='b) Replace every outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k with the incoming arrow k !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj This is the quiver implementation of the fact that the magnetic flavors are in the complex conjugate representations, of both the dualized gauge group and the spectator nodes, of the original flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Next we add mesons to the quiver, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' composite arrows, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For every 2-path i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k we add a new arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This meson Mik can be regarded as the composition of the flavors i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j and j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k of the original theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In other words, we generate all possible composite arrows consisting of incoming and outgoing chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- tions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' []).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2In our discussion, including the points that follow, we allow for the possibility of chiral fields connecting pairs of nodes in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 1 – Contents 1 Seiberg Duality 1 1 Seiberg Duality In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 gauge theories [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We will phrase our discussion in the language of quivers, since all the theories considered in this paper are of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us consider dualizing a node j in the quiver, which does not have adjoint chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 The transformation of the gauge theory can be summarized in the following rules: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In physics, the arrows connected to the mutated node are usually referred to as flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The flavors transform by simply reversing their orientation, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='a) Replace every incoming arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j with the outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xij the incoming arrow, we replace it by the dual flavor ˜Xji 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='b) Replace every outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k with the incoming arrow k !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj This is the quiver implementation of the fact that the magnetic flavors are in the complex conjugate representations, of both the dualized gauge group and the spectator nodes, of the original flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Next we add mesons to the quiver, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' composite arrows, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For every 2-path i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k we add a new arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This meson Mik can be regarded as the composition of the flavors i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j and j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k of the original theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In other words, we generate all possible composite arrows consisting of incoming and outgoing chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- tions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' []).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2In our discussion, including the points that follow, we allow for the possibility of chiral fields connecting pairs of nodes in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 1 – Contents 1 Seiberg Duality 1 1 Seiberg Duality In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 gauge theories [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We will phrase our discussion in the language of quivers, since all the theories considered in this paper are of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us consider dualizing a node j in the quiver, which does not have adjoint chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 The transformation of the gauge theory can be summarized in the following rules: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In physics, the arrows connected to the mutated node are usually referred to as flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The flavors transform by simply reversing their orientation, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='a) Replace every incoming arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j with the outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xij the incoming arrow, we replace it by the dual flavor ˜Xji 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='b) Replace every outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k with the incoming arrow k !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj This is the quiver implementation of the fact that the magnetic flavors are in the complex conjugate representations, of both the dualized gauge group and the spectator nodes, of the original flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Next we add mesons to the quiver, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' composite arrows, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For every 2-path i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k we add a new arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This meson Mik can be regarded as the composition of the flavors i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j and j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k of the original theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In other words, we generate all possible composite arrows consisting of incoming and outgoing chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- tions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' []).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2In our discussion, including the points that follow, we allow for the possibility of chiral fields connecting pairs of nodes in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 1 – Contents 1 Seiberg Duality 1 1 Seiberg Duality In this section we review Seiberg duality, which is an IR equivalence between 4d N = 1 gauge theories [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We will phrase our discussion in the language of quivers, since all the theories considered in this paper are of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us consider dualizing a node j in the quiver, which does not have adjoint chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1 The transformation of the gauge theory can be summarized in the following rules: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In physics, the arrows connected to the mutated node are usually referred to as flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The flavors transform by simply reversing their orientation, namely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='a) Replace every incoming arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j with the outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xij the incoming arrow, we replace it by the dual flavor ˜Xji 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='b) Replace every outgoing arrow j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k with the incoming arrow k !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Calling Xjk the outgoing arrow, we replace it by the dual flavor ˜Xkj This is the quiver implementation of the fact that the magnetic flavors are in the complex conjugate representations, of both the dualized gauge group and the spectator nodes, of the original flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Next we add mesons to the quiver, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' composite arrows, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For every 2-path i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k we add a new arrow i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This meson Mik can be regarded as the composition of the flavors i !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' j and j !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k of the original theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In other words, we generate all possible composite arrows consisting of incoming and outgoing chiral fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 1Generalizations of Seiberg duality to gauge groups with adjoints are known, under certain condi- tions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' []).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2In our discussion, including the points that follow, we allow for the possibility of chiral fields connecting pairs of nodes in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 1 – i j k Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1: A sketch of how quivers transform under Seiberg duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Figure taken from [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' for quivers (without adjoint loops and 2-cycles) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The superpotential and the ranks of the gauge groups would be transformed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In particular, factors XijXjk should be replaced with Mik in the superpotential, and terms Mik � Xkj � Xji need to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Moreover, fields that acquire masses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=', quadratic terms in the superpotential, should be integrated out using their F-term relations [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As we shall consider toric quivers, only nodes with two arrows and two arrows out would be dualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In terms of brane tilings, this can be performed by the urban renewal [14, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2 The Higgs-Kibble Mechanism As studied in [13], higgsing of a theory corresponds to blowing down a compact 2-cycle to a point in the toric geometry while unhiggsing blows up a point to a compact 2-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The – 28 – (a) (b) 1 2 3 1 2 1 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2: The example of higgsing dP0 (aka KP2, C3/Z3 (1, 1, 1)) to C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' process of higgsing can also be nicely encoded in the toric diagrams and in the quivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' An example is depicted in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' By turning on a VEV of a bifundamental in the quiver at each step, some fields would acquire masses which should be integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Using their equations of motion, we can obtain the superpotential after higgsing similar to the process of performing toric duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In the above example, the superpotential changes as W =I1 12I3 23I2 31 + I1 12I1 23I3 31 + I2 23I1 31I3 12 − I2 12I3 23I1 31 − I1 12I2 23I3 31 − I2 23I2 31I3 12 ⟨X1 23⟩=1 −−−−−→I1 22I3 21I3 12I2 22 − I2 22I3 21I3 12I1 22 ⟨X3 12⟩=1 −−−−−→I1 11 � I3 11, I2 11 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1) where we have omitted the traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In terms of brane tilings, turning on a VEV of a bifundamental removes an edge, and integrating out massive fields corresponds to removing (and combining) certain nodes in the dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' An illustration can be found in [24, Figure 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3 Generalized Conifolds Recall that the toric diagram of any generalized conifold xy = zMwN is a trapezium on the lattice of height one with two horizontal lines of lengths M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The quiver (in any toric phase) can essentially be viewed as the “tripled” quiver15 of a Dynkin diagram associated to the underlying untwisted affine Lie superalgebra �slM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The quivers in different toric phases with different numbers of bosonic and fermionic nodes are encoded by the triangulations of the corresponding toric diagram [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Each simplex in a given triangulation corresponds to a sign ςa = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Together, they form a parity sequence ς = {ςa|a ∈ Z/(M + N)Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Overall, the numbers of plus and minus ones are given by M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' If two simplices are aligned side by side, then ςa = ςa+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' If they are aligned in the alternative way, then ςa and ςa+1 have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Some illustrations can be found in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Besides a pair of opposite arrows connecting each pair of nodes a and a+1, the quiver has a self-loop on each bosonic node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' If ςa = ςa+1, then the node a is bosonic/even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Otherwise, 15Here, by “tripled”, we mean that we first add an opposite arrow for each existing arrow in the Dynkin quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then we only add adjoint loops to the bosonic nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 29 – (a) (b) (c) Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3: Figure taken from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' We have (a) ς = {−1, +1}, (b) ς = {−1, −1} and (c) ς = {−1, −1, +1, +1, −1, +1, +1, +1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' it is fermionic/odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The superpotential is composed of terms � ςatr(Ia,aIa,a−1Ia−1,a − Ia,aIa,a+1Ia+1,a), ςa = ςa+1 ςatr(Ia,a+1Ia+1,aIa,a−1Ia−1,a), ςa = −ςa+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2) Following the above rule of toric duality, it is straightforward to see that we can only dualize fermionic nodes in the toric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This would just change the parity of the two nodes connected to the dualized node by adding or removing the adjoint loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Correspondingly, the Dynkin diagrams of the underlying affine Lie superalgebra are related by odd reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' A generalized conifold with a larger polygon can be higgsed to one with a smaller polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This can be decomposed into a sequence of higgsings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For each single higgsing, the leftmost or rightmost simplex is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In the quiver, we merge two adjacent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The two nodes can be either bosonic or fermionic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Suppose that the nodes a and a + 1 are merged, then |a′| = |a| + |a + 1|, where a′ denotes the corresponding node after higgsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us list how the Cartan matrices would change for the three possible cases: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' · · 2 −1 · · · · · −1 2 −1 · · · · · −1 2 −1 · · · −1 2 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 |a| = |a + 1| = 0 : a a + 1 a′ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' · · 2 −1 · · · · · −1 2 −1 · · · −1 2 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3) \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' · · 2 −1 · · · · · −1 2 −1 · · · · · −1 0 1 · · · 1 −2 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 |a| = 0, |a + 1| = 1 : a a + 1 a′ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' · · 2 −1 · · · · · −1 0 1 · · · 1 −2 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='4) – 30 – \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' · · 2 −1 · · · · · −1 0 1 · · · · 1 0 −1 · · · −1 2 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 |a| = |a + 1| = 1 : a a + 1 a′ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' · · 2 −1 · · · · · −1 2 −1 · · · −1 2 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='5) B Serre Relations Besides the relations listed in §2, the quiver BPS algebras also have Serre relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Here, we will only discuss the cases for non-chiral quivers with M + N ≥ 3, MN ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Although the Serre relations for general chiral quivers are still not known, examples can be found in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It is observed that the Serre relations (for either chiral or non-chiral quivers) are closely related to the superpotential of the theory [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For the rational algebras, we have Sym u1,u2 � e(a)(u1), � e(a)(u2), e(a±1)(v) �� = 0 (|a| = 0), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1) Sym u1,u2 � e(a)(u1), � e(a+1)(v1) � e(a)(u2), e(a−1)(v2) ��� = 0 (|a| = 1), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2) Sym u1,u2 � f(a)(u1), � f(a)(u2), f(a±1)(v) �� = 0 (|a| = 0), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3) Sym u1,u2 � f(a)(u1), � f(a+1)(v1) � f(a)(u2), f(a−1)(v2) ��� = 0 (|a| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='4) For the toroidal algebras, the Serre relations are Sym u1,u2 � e(a)(u1), � e(a)(u2), e(a±1)(v) � H1 � H1 = 0 (|a| = 0), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='5) Sym u1,u2 � e(a)(u1), � e(a+1)(v1) � e(a)(u2), e(a−1)(v2) � H1 � H1 � H1 = 0 (|a| = 1), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='6) Sym u1,u2 � f(a)(u1), � f(a)(u2), f(a±1)(v) � H−1 1 � H−1 1 = 0 (|a| = 0), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='7) Sym u1,u2 � f(a)(u1), � f(a+1)(v1) � f(a)(u2), f(a−1)(v2) � H−1 1 � H−1 1 � H−1 1 = 0 (|a| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='8) Here, the q-graded bracket is given by �x, y�q = xy − (−1)|x||y|q(x,y)yx, where (x, y) is the root pairing stemmed from the underlying affine Lie superalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For instance, the pairing of two simple roots gives the corresponding entry in the Cartan matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 31 – As we can see, both of the two types of the algebras have their versions of the brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, we would also like to use an “elliptic bracket” to write the Serre relations for the elliptic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Let us introduce the operators χa(u) and ξa(u) that commute with all e, f, ψ± generators in the elliptic algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' They have the following correlators: e⟨χa(u)χb(v)⟩ = � qHAab 1 H−Mab 2 U −1V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qH−Aab 1 H−Mab 2 U −1V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='9) e⟨ξa(u)ξb(v)⟩ = � qH−Aab 1 H−Mab 2 U −1V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ � qHAab 1 H−Mab 2 U −1V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' q � ∞ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='10) e⟨χa(u)ξb(v)⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='11) Then using the correlators of the “dressed” operators E(a)(u) = eχa(u)e(a)(u), F (a)(u) = eξa(u)f(a)(u), Ψ(a) ± (u) = eχa(u±c/2)eξa(u∓c/2)ψ(a) ± (u), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='12) the relations of the elliptic algebras can be written in the same forms as those of the toroidal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For instance, the ee relations of the elliptic algebras now become � HMab 2 U − HAab 1 V � � E(a)(u)E(b)(v) � = (−1)|a||b| � HAab 1 HMab 2 U − V � � E(b)(v)E(a)(u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='13) Therefore, the Serre relations of the elliptic algebras can simply be obtained by taking the ones of the toroidal algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then we replace the toroidal generators with the dressed elliptic generators and take the correlators of the whole expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' For brevity, we shall write them using the “elliptic brackets” as Sym u1,u2 � e(a)(u1), � e(a)(u2), e(a±1)(v) � χ � χ = 0 (|a| = 0), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='14) Sym u1,u2 � e(a)(u1), � e(a+1)(v1) � e(a)(u2), e(a−1)(v2) � χ � χ � χ = 0 (|a| = 1), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='15) Sym u1,u2 � f(a)(u1), � f(a)(u2), f(a±1)(v) � ξ � ξ = 0 (|a| = 0), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='16) Sym u1,u2 � f(a)(u1), � f(a+1)(v1) � f(a)(u2), f(a−1)(v2) � ξ � ξ � ξ = 0 (|a| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='17) C Conventions of Heisenberg Modes In the main context, we introduced the modes kr (and lr) for the ψ± currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Here, we mention some alternative convention to define these Heisenberg modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' It could be possible that this would be more convenient when considering certain aspects of the algebras such as their representations and the AGT correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 32 – Let us consider the toroidal algebras for non-chiral quivers as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' The other cases can be redefined in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' First, we rescale the e, f modes as e(a) n = � q − q−1�1/2 e(a) n , f(a) n = � q − q−1�1/2 f(a) n , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='1) where we have suggestively written q = exp(βh1) = H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Notice that this does not change the ee and ff relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Then the e0f0 relations (as well as the enf−n relations) would become � e(a) 0 , f(a) 0 � = δab qk(a) 0 − q−k(a) 0 q − q−1 = δab � k(a) 0 � q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='2) Here, [x]q = qx−q−x q−q−1 is the standard q-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' On the other hand, the k0en (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k0fn) relations remain the same as the ones for k0en (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k0fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' As we can see, the relations among the zero modes resemble the ones appeared in quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Likewise, we can write ψ(a) ± (U) = ψ(a) ±,0 exp � � q − q−1� ∞ � n=0 k(a) ±nU ∓n � (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='3) such that k(a) r = � q − q−1� k(a) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Therefore, ψ(a) ±,n = ψ(a) ±,0 n � m=1 � q − q−1�m m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' � r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=',rm>0 r1+···+rm=n k(a) ±r1k(a) ±,r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' k(a) ±,rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='4) The commutation relations involving k(a) r can be obtained with the substitutions HrAab 1 − H−rAab 1 → [rAab]q, C−r − Cr → C−r − Cr q − q−1 = −[rc/h1]q (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='5) in the relations for k(a) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Sometimes, it is also conventional to define the Heisenberg modes with signs inside the exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' In other words, we have exp � ± � n k±nU ∓n � in the expressions for ψ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' This is simply a redefinition of k−n → −k−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Li and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Yamazaki, “Quiver Yangian from Crystal Melting,” JHEP 11 (2020) 035, arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='08909 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Galakhov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Yamazaki, “Quiver Yangian and Supersymmetric Quantum Mechanics,” Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 396 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 2, (2022) 713–785, arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='07006 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Galakhov, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Li, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Yamazaki, “Shifted quiver Yangians and representations from BPS crystals,” JHEP 08 (2021) 146, arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='01230 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' – 33 – [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' Yamazaki, “Quiver Yangians and Crystal Melting: A Concise Summary,” in International Congress on Mathematical Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' 3, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content='14314 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} +page_content=' [5] D.' metadata={'source': 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+page_content=' – 36 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9AyT4oBgHgl3EQfxPl4/content/2301.00663v1.pdf'} diff --git a/ftFAT4oBgHgl3EQf8B6L/content/2301.08748v1.pdf b/ftFAT4oBgHgl3EQf8B6L/content/2301.08748v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4990ee76bc9ab5a8e7a386a988bd23d0e1b676fd --- /dev/null +++ b/ftFAT4oBgHgl3EQf8B6L/content/2301.08748v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ed32ac20d4f41f46aa157bac3da43b86eb23f870959728cc40823f415fe61aac +size 195472 diff --git a/ftFAT4oBgHgl3EQf8B6L/vector_store/index.faiss b/ftFAT4oBgHgl3EQf8B6L/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ed3500fd6a181e7a9684e034b10c663476d9b426 --- /dev/null +++ b/ftFAT4oBgHgl3EQf8B6L/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ce9a7d56ea629c346add4f232d898ccce0d11f982b99dd6f788da49021f13ff0 +size 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b/gNE1T4oBgHgl3EQfywXe/content/tmp_files/2301.03438v1.pdf.txt @@ -0,0 +1,2691 @@ +New error estimates of Lagrange-Galerkin methods for the +advection equation +Rodolfo Bermejoa, Jaime Carpiob, Laura Saavedrac +a) Dpto. Matem´atica Aplicada a la Ingenier´ıa Industral ETSII. Universidad Polit´ecnica de Madrid. +b) Dpto. Ingenier´ıa Energ´etica ETSII. Universidad Polit´ecnica de Madrid. +c) Dpto. Matem´atica Aplicada a la Ingenier´ıa Aeroespacial, ETSIAE. Univversidad Polit´ecnica de Madrid. +Abstract +We study in this paper new developments of the Lagrange-Galerkin method for the advection +equation. In the first part of the article we present a new improved error estimate of the conven- +tional Lagrange-Galerkin method. +In the second part, we introduce a new local projection stabi- +lized Lagrange-Galerkin method, whereas in the third part we introduce and analyze a discontinuity- +capturing Lagrange-Galerkin method. Also, attention has been paid to the influence of the quadrature +rules on the stability and accuracy of the methods via numerical experiments. +Keywords Advection equation, Lagrange-Galerkin, finite elements, local projection stabilization, +discontinuity capturing +Mathematics Subject Classification (2010) 65M12, 65M25, 65M60, 65M50 +1 +Introduction +We consider the Cauchy problem for the pure advection equation +� +� +� +� +� +∂c +∂t + u·∇c = 0, +x ∈ Rd, t > 0, +c(x, 0) = c0(x), +(1) +where c : Rd × [0, T] → R, u : Rd × [0, T] → Rd is a vector-valued function and c0(x) is a function of +compact support defined in a domain D0 ⊂ Rd. It is well known that the solution of this problem given +by the method of characteristics is of the form +c(X( · , t; t + τ), t + τ) = c(·, t), +X(x, t; t + τ) being the characteristic curves of the advection equation. In this paper, we present the +error analysis of three versions of the so called LG method applied to solve the Cauchy problem (1), +which represent numerical realizations of the theoretical method of characteristics in the framework of +H1-conforming finite elements. The first version, denoted in this paper with the name the conventional +LG method, consists basically on approximating the solution c(x, t) by the L2-projection onto the finite +element space; see, for instance, [19], [17] and [15]. The conventional LG method can be viewed as a +kind of high order upwind method that introduces artificial diffusion in the discrete formulation, thus +providing good stability properties to the numerical solution; however, as numerical experiments show, +this artificial diffusion is not high enough to suppress the oscillations that appear at discontinuities of the +exact solution. In order to alleviate this problem at discontinuities and extend the stability properties, we +shall study a local projection stabilized Lagrange-Galekin (LPS-LG) method and a discontinuity-capturing +Lagrange-Galerkin (DC-LG) method. +In the past, several authors have obtained different estimates for the L2-norm of the error of the +conventional LG method. For example, [19] calculates an estimate of the form O(hm+1/∆t), where m +denotes the degree of the polynomials of the H1-conforming finite element spaces, h being the largest +1 +arXiv:2301.03438v1 [math.NA] 9 Jan 2023 + +diameter of the elements of the spatial mesh and ∆t the size of the time step. The problem with this +estimate is that for fixed h the error becomes unbounded when ∆t → 0. [17] removes the ∆t−1 dependence +from the error estimate obtaining the new estimate O(hm), this estimate allows to prove convergence of +LG method for the advection equation when ∆t → 0 independently of h. [15] improves the estimate +of [19] calculating a new estimate O(hm+1/∆t1/2), which for ∆t = O(h) implies that the error of the +conventional LG method is of the same order as both the streamline-diffusion (SD) method formulated +in the framework of space-time finite elements continuous in space and discontinuous in time and the +characteristic streamline-diffusion (CSD) method, the latter method being a version of the SD method +that uses space-time meshes oriented along the characteristic curves of the advection equation. Later on, +[16] calculates a new estimate of the form O(min(1, ∥u∥L∞(Rd)d ∆t/h)hm+1/∆t), where ∥u∥L∞((0,T );Rd)d +denotes the supremum norm of the velocity vector u(x, t); then, considering that ∥u∥L∞(Rd)d ∆t/h is the +CFL number, we can say that for CFL numbers less than one, the error of LG methods is O(hm), whereas +for CFL numbers larger than one the error is O(hm+1/∆t). Numerical examples show that the latter +estimate provides a better description of the error behavior of the conventional LG method than the other +estimates do. In this paper, we revisit the results of the the above mentioned authors and calculate an +improved new error estimate of the form O(min(1, ∥u∥L∞(Rd)d ∆t1/2/h)hm+1/∆t1/2). Some numerical +examples will support the validity of this estimate. +Local projection stabilized methods have become quite popular for advection-diffusion-reaction equa- +tions, including Navier-Stokes equations, see [1], [2], [3], [9] and [20] just to cite a few, for they are +symmetric and introduce artificial diffusivity via a fluctuation operator acting on the small unresolved +scales. We prove that the LPS-LG method is stable in the L2-norm, and our error analysis shows that +the error of the LPS-LG method in the mesh dependent norm (to be defined below) +max +0≤tn≤T |||cn − cn +h||| = O(hβ + min(1, ∥u∥L∞(Rd)d ∆t1/2/h)hm+1/∆t1/2), +where β is a coefficient depending on m. However, despite the introduction of the artificial diffusivity, +numerical tests show that the LPS-LG method may exhibit an oscillatory behavior when the solution is +not sufficiently smooth. +The DC-LG method might be viewed as a version of the shock capturing CSD method [15] in which the +mesh alignment along the characteristic curves and the stream diffusion mechanism of the shock capturing +CSD are removed; in fact, one can consider that the DC-LG method is a reformulation, in the framework +of the conventional LG method, of the residual artificial viscosity method introduced in [18]. We prove +that DC-LG method is stable in the L2-norm, regardless the degree of the finite element spaces, and also +in the L∞-norm with linear finite elements, although numerical examples show L∞-norm stability with +quadratic polynomials in the presence of a strong discontinuity. For solutions sufficiently smooth, we are +able to prove that the error in the L2-norm is of the form O((hm+1 + Cεhα)/(∆t1/2)), α and Cε being +positive constants. +The theoretical analysis of LG methods presented in this paper are proven under the assumption that +the integrals +� +K φj(X(x, tn; tn−1))φi(x)dx , which appear in the formulation of the methods, are calculated +exactly; here, K is a generic element of the mesh, φi is the ith global basis function of the finite element +space and X(x, tn, tn−1) is the foot of the characteristic curve associated with the point x. Noting that the +integrand is the product of two piecewise continuous polynomial functions defined on two different meshes, +it may become very difficult to calculate such integrals exactly, so one has to resort to quadrature rules; +but as [17] and [13] show, the quadrature rules have to be of high order because otherwise the numerical +solution may become either inaccurate or unstable. Being aware of this fact, we shall test the validity +of our analysis of the LG methods studied in the paper by performing some benchmark numerical tests, +using symmetric Gaussian rules of different orders to assess the influence of the order of the quadrature +rules on the accuracy and stability. +The paper is organized as follows. We make a short presentation of the continuous problem in Section +2, and introduce the formulation and numerical analysis of the conventional LG method for the advection +equation in Section 3. Some numerical examples illustrating its performance are also reported in this +section. +Section 4 is devoted to the formulation, analysis and numerical performance of the LPS-LG +method. The DC-LG method is introduced in Section 5, studying its stability and convergence. We also +present in this section several numerical tests. Some concluding remarks are written in Section 6. +We introduce some notation about the functional spaces used in the paper. For s ≥ 0 real and real 1 ≤ +2 + +p ≤ ∞, W s,p(D) denotes the real Sobolev spaces defined on D for scalar real-valued functions. ∥·∥W s,p(D) +and |·|W s,p(D) denote the norm and semi-norm, respectively, of W s,p(D). When s = 0, W 0,p(D) := Lp(D). +For p = 2, the spaces W s,2(D) are denoted by Hs(D), which are real Hilbert spaces with inner product +(·, ·)s. For s = 0, H0(D) := L2(D), the inner product in L2(D) is denoted by (·, ·). H1 +0(D) is the space +of functions of H1(D) which vanish on the boundary ∂D in the sense of trace. H−1 denotes the dual +space of H1 +0(D). The corresponding spaces of real vector-valued functions, v : D → Rd are denoted by +W s,p(D)d := {v : D → Rd : vi ∈ W s,p(D), 1 ≤ i ≤ d}. Let X be a real Banach space (X, ∥·∥X), if v : +(0, T) → X is a strongly measurable function with values in X, we set ∥v∥Lp(0,t;X) = +�� t +0 ∥v(τ)∥p +X dτ +�1/p +for 1 ≤ p < ∞, and ∥v∥L∞(0,t;X) = ess sup +0<τ≤t +∥v(τ)∥X; when t = T, we shall write, unless otherwise stated, +∥v∥Lp(X). We shall also use the following discrete norms: +∥v∥lp(X) = +� +∆t +N +� +i=1 +∥v(τi)∥p +X +�1/p +, +∥v∥l∞(X) = max +1≤i≤N ∥v(τi)∥X , +corresponding to the time discrete space lp(X) ≡ lp(0, T; X), 1 ≤ p < ∞, defined as +lp(0, T; X) := +� +v : (0, t1, t2, . . . , tN = T) → X : +∥v∥lp(X) < ∞ +� +, +when p = ∞ +l∞(0, T; X) := +� +v : (0, t1, t2, . . . , tN = T) → X : +max +1≤i≤N ∥v(τi)∥X < ∞ +� +. +Finally, we shall also use the space of continuous functions such as Cr(D) that denotes the space of r- +times continuously differentiable functions on D, when r = 0 we write C(D) instead of C0(D); the space +Cr,1(D), r ≥ 0, of functions defined on the closure of D, r -times continuously differentiable and with +the rth derivative being Lipschitz continuous; and the space of continuous and bounded functions in time +with values in X denoted by C([0, T]; X). +Throughout this paper, C will denote a generic positive constant which is independent of both the +space and time discretization parameters h and ∆t respectively. C will have different values at different +places of appearance. In many places we shall use, without making any explicit statement, the Cauchy’s +inequality ab ≤ ϵ +2a2 + 1 +2ϵb2 (a, b > 0, ϵ > 0), and the discrete Gronwall inequality presented in [12]. +2 +The Cauchy problem for the advection equation +To introduce the LG method we consider the Cauchy problem for the first order linear hyperbolic equation +� +� +� +� +� +∂c +∂t + u·∇c = 0, +x ∈ Rd, t > 0, +c(x, 0) = c0(x), +(2) +where c : Rd × [0, T] → R, u : Rd × [0, T] → Rd is a vector-valued function and c0(x) is a function of +compact support defined in a domain D0 ⊂ Rd. Considering the characteristics curves of the first order +differential operator D/Dt := ∂/∂t + u·∇ which are the solution to the system of ordinary differential +equations +� +� +� +� +� +dX(x,s; t) +dt += u(X(x, s; t), t), +X(x, s; s) = x, +(3) +we can recast problem (2) as an ordinary differential equation along the characteristics curves, X(x,s; t), +of the form +� +� +� +� +� +Dc +Dt = 0 , X(x,s; t) ∈ Rd, t > 0, +c(X(x,0; 0), 0) = c0(x). +(4) +3 + +Assuming that u ∈ C([0, T], W 1,∞(Rd)d), so problem (3) has a unique solution, and c0(x) is sufficiently +smooth, we have that the solution of (4) is then given by +c(X(·, t; t + τ), t + τ) = c(·, t). +(5) +Concerning the solution t → X(x, s; t) to (3), the following regularity results are in order. +Lemma 1 Assume that u ∈ C([0, T], W k,∞(Rd)d), k ≥ 1. +Then for s, t ∈ [0, T], there exists a unique +solution t → X(x, s; t) of (3), such that X(x, s; t) ∈ W 1,∞(W k,∞(Rd)d). Furthermore, let the multi-index +α ∈ N d, then for all α, such that 1 ≤| α |≤ k, DαXi(x, s; t) ∈ C([0, T], L∞(Rd × [0, T])), 1 ≤ i ≤ d. +Next, we consider the mapping ϕt +s : Rd → Rd, defined by ϕt +s(x) = X(x, s; t), since X(X(x, s; t), t; s) = +x, then it follows that the mapping ϕs +t is the inverse of ϕt +s. The Jacobian determinant of this transformation +J(x, s; t) = det +�∂Xi(x, s; t) +∂xj +� +, 1 ≤ i, j ≤ d, +(6) +satisfies the equation +∂J(x, s; t) +∂t += J(x, s; t)div u(X(x, s; t), t). +(7) +It is easy to see that if Cu := ∥div u∥L∞(D×(0,T )), then +exp(−Cu |s − t|) ≤ J(x, s; t) ≤ exp(Cu |s − t|). +(8) +Moreover, for |t − s| sufficiently small it follows that +K1 | x − y |≤| X(x, s; t) − X(y, s; t) |≤ K2 | x − y |, +(9) +where K1 = (1− | s − t | · | ∇u |L∞(L∞(D)d×d)), and K2 = exp(| s − t | · | ∇u |L∞(L∞(D)d×d)). Here, +| a−b | denotes the Euclidean distance between the points a, b ∈ Rd. Hereafter, for the sake of simplicity, +we make the assumption div u = 0. An important consequence of this assumption is that J(x, s; t) = 1 +almost everywhere in R. However, we must remark that one can easily accommodate the proofs of our +results to the general case of div u ̸= 0. +3 +The conventional LG method for the advection equation +In the framework of finite elements, Douglas and Russell (1982) and Pironneau (1982) proposed the so +called conventional LG method as a time marching algorithm to approximate the solution of (2). +3.1 +Finite element formulation +The realization of this method requires the definition of a family of partitions Dh in a domain D ⊂ Rd +sufficiently large, such that given T > 0, D0 ⊂⊂ D and for all t ∈ [0, T] we can assume that c(x, t) = 0 +on the boundary ∂D. The partitions Dh generated in the closed region D := D ∪ ∂D are quasi-uniform +regular and composed of d-simplices K, the boundaries of which are denoted by ∂K. hK denotes the +diameter of K and the mesh parameter h := maxK hK. Moreover, we shall assume that u(x, t) is zero +on the boundary ∂D. To define the finite element spaces we use the reference simplex �K with vertices +{�xi}d+1 +i=1 , �K := +� +�x ∈ Rd : 0 ≤ �xi ≤ 1, 1 − �d +i=1 �xi ≥ 0 +� +, such that for each K ∈ Dh there is an invertible +affine mapping FK : �K → K, +FK(�x) = BK�x + bK, +BK ∈ L(Rd) and bK ∈ Rd. +The finite element spaces used in the formulation of the LG method are the following: +Wh := +� +vh ∈ C(D) : ∀K ∈ Dh, vh |K∈ Pm(K) +� +, and Vh = H1 +0(D) ∩ Wh, +4 + +with +Pm(K) = +� +p(x) : for x ∈ K, p(x) = �p ◦ F −1 +K (x), �p ∈ Pm( �K) +� +, +where Pm( �K) denotes the set of polynomials of degree ≤ m defined in �K. Next, we introduce some auxiliary +results concerning the approximation properties of the finite element spaces. For 0 < h < h0 < 1, there +exists a constant c1 independent of h such that for w ∈ Hq+1(D) ∩ H1 +0(D) and 1 ≤ q ≤ m, +inf +vh∈Vh {∥w − vh∥ + h ∥∇ (w − vh)∥} ≤ c1hq+1 |w|Hq+1(D) . +(10) +Since the partition Dh is quasi-uniformly regular, the following inverse inequality holds: for all wh ∈ Wh +and 0 ≤ k ≤ l ≤ 1, and 1 ≤ p ≤ q ≤ ∞, there exists a constant cinv independent of h such that, +∥wh∥W l,q(D) ≤ cinvhd/q−d/p+k−m ∥wh∥W k,p(D) . +(11) +Let Πh : C(D) → Wh be the Lagrange interpolation operator in Wh and let Ph : L2(D) → Vh be the +orthogonal L2-projector defined as +(w − Phw, vh) = 0 for all vh ∈ Vh, +(12) +then there are constants c2 and c3 independent of h, such that for 0 ≤ σ ≤ m, and 1 ≤ γ ≤ ∞, +∥w − Phw∥ + h ∥∇(w − Phw)∥ ≤ c2hσ+1 |w|Hσ+1(D) +(13) +and +∥w − Πhw∥Lγ(D) + h ∥∇(w − Πhw)∥Lγ(D) ≤ c3hσ+1 |w|W σ+1,γ(D) , +(14) +respectively [7]. It is worth noting that the estimate (14) and the inverse inequality are also valid when +the domain D is substituted by an element K. The following properties of the projector Ph are also used +in the paper: +Phvh = vh ∀vh ∈ Vh, +and (contractiveness) +∥Phv∥ ≤ ∥v∥ ∀v ∈ L2(D). +Let P := 0 = t0 < t1 < . . . < tN = T be a uniform partition of step length ∆t for the interval [0, T], +the finite element solution of (2) at time tn, denoted by cn +h ∈ Vh, is given by +cn +h = +NP +� +i=1 +Cn +i φi, +where Cn +i := cn +h(xi), xi being the ith mesh-point in Dh, NP denotes the number of mesh-points of the +partition Dh, and {φi}NP +i=1 is the set of global basis functions of Vh. The conventional LG method calculates +cn +h ∈ Vh as +cn +h(x) = Phcn−1 +h +◦ X(x, tn; tn−1), +(15) +or equivalently, for all i = 1, ..., NP , +� +D +cn +h(x)φi(x)dx = +� +D +cn−1 +h +◦ X(x, tn; tn−1)φi(x)dx, +(16) +where X(x, tn; tn−1) is the position at time instant tn−1 of the point that at time instant tn is located at +the point x. +Notations Let us introduce some shorthand notations in order to simplify the writing of the formulas +that will appear in the article. In the sequel, we sometimes use Xn,n−1 or if confusion may arise Xn,n−1(x), +to denote X(x, tn; tn−1). Also, let a(x, t) be a generic function defined in Rd×[0, T], then an(x) will denote +the value of a(x, t) at time instant tn, that is, an(x) = a(x, tn), whereas a∗n−1(x) denotes an−1◦Xn,n−1(x). +Hereafter, we assume that h ∈ (0, h0) and ∆t ∈ (0, ∆t0) with h0 < 1 and ∆t0 < 1. +5 + +3.2 +Analysis of the conventional LG method +We begin analyzing the L2-norm stability of the method. +Lemma 2 For all N ≥ 1, +��cN +h +��2 + +N +� +n=1 +��cn +h − c∗n−1 +h +��2 = +��c0 +h +��2 . +(17) +Proof. First of all, we show that for any function f ∗n−1(x) := f n−1(X(x, tn, tn−1)) ∈ L2(D) +��f ∗n−1�� = +��f n−1�� . +(18) +To see this is so, we make the change of variable y = Xn,n−1(x) and recall that the Jacobian determinant +of this transformation, J(x, tn; tn−1) = 1 a.e., then +� +D +��f ∗n−1(x) +��2 dx = +� +D +��f n−1(Xn,n−1(x)) +��2 dx = +� +D +��f n−1(y) +��2 J−1dy = +� +D +|f n(y)|2 dy. +Now, we notice that from (15) it follows that +� +cn +h − c∗n−1 +h +, cn +h +� += 0, +then using the elementary relation 2(a − b, a) = a2 + (a − b)2 − b2, a, b ∈ R, we obtain that +2 +� +cn +h − c∗n−1 +h +, cn +h +� += ∥cn +h∥2 + +��cn +h − c∗n−1 +h +��2 − +��c∗n−1 +h +��2 , +and by virtue of (18) +∥cn +h∥2 + +��cn +h − c∗n−1 +h +��2 − +��cn−1 +h +��2 = 0. +Summing this expression from n = 1 up to n = N it follows (17). +Remark 3 Following [15], we can interpret the term �N +n=1 +��cn +h − c∗n−1 +h +��2 as a measure of the numerical +dissipation of conventional LG methods. It is shown there that +N +� +n=1 +��cn +h − c∗n−1 +h +��2 ≤ C h4 +∆t +N +� +n=1 +∆t +��∆n−1 +h +c∗n−1 +h +��2 , +where ∆n−1 +h +: H1(D) → W ∗n−1 +h +:= +� +v∗n−1 +h +(x) = vn−1 +h +(Xn,n−1(x)) : vn−1 +h +(x) ∈ Wh +� +denotes the discrete +Laplacian operator. When ∆t = h this amounts to adding an artificial diffusion term to the continuous +advection equation of the form Ch3∆c; so, for sufficiently smooth solutions such an artificial diffusion is +not excessive, in particular if one compares with the usual upwind method that adds an artificial diffusion +therm of the form −Ch∆c, but it may be insufficient to eliminate the oscillations when the exact solution +is not smooth. To deal with the case of non smooth solutions we introduce the DC-LG method. +The remainder of this section is devoted to the analysis of the convergence. We have the following +result. +Theorem 4 Let c ∈ L∞(Hm+1(D) ∩ H1 +0(D)). Then there exists a constant C independent of ∆t, h, and +n, such that +∥c − ch∥l∞(L2(D)) ≤ +��e0�� + C min +� +1, +∆t1/2 ∥u∥L∞(W 1,∞(D)d)) +h +� +hm+1 +∆t1/2 |c|L∞(Hm+1(D)) . +(19) +Proof. The error en := cn − cn +h can be expressed as +en = (cn − Phcn) + (Phcn − cn +h) ≡ ρn + θn +h, +(20) +where θn +h ∈ Vh. Noting that Phρn = 0, then it follows that +∥en∥2 = (ρn + θn +h, ρn + θn +h) = ∥ρn∥2 + ∥θn +h∥2 . +6 + +By virtue of (13), ρn satisfies the bound +∥ρn∥Hl(D) ≤ c2hm+1−l |cn|Hm+1(D) +(0 ≤ l ≤ m + 1). +(21) +To estimate en we make use of (5), which implies that cn = c∗n−1, so that ∀vh ∈ Vh +(cn − c∗n−1, vh) = 0, +so, subtracting (15) from this equation it results the following error equation +� +en − e∗n−1, vh +� += 0, +(22) +where e∗n−1(x) = cn−1(Xn,n−1(x))−cn−1 +h +(Xn,n−1(x)) = ρn−1 � +Xn,n−1(x) +� ++θn−1 +h +� +Xn,n−1(x) +� +. Now, we +calculate an error estimate from (22). First, we notice that by virtue of (18) +��e∗n−1�� = +��en−1��, so this +property together with the elementary relation 2(a − b)a = a2 + (a − b)2 − b2, permits us to write +∥en∥2 + +��en − e∗n−1��2 − +��en−1��2 = 2 +� +en − e∗n−1, en� +. +(23) +Now, one needs to estimate the term +� +en − e∗n−1, en� +. For this purpose, we apply the argument of [15], +use (20) and set +(en − e∗n−1, en) = +� +en − e∗n−1, ρn� ++ +� +en − e∗n−1, θn +h +� +, +but by virtue of (22), +� +en − e∗n−1, θn +h +� += 0, then the only term we have to estimate is +� +en − e∗n−1, ρn� +. +Thus, by the Cauchy-Schwarz inequality +� +en − e∗n−1, ρn� +≤ 1 +4 +��en − e∗n−1��2 + ∥ρn∥2 ; +(24) +hence, one can write that +2 +� +en − e∗n−1, en� +≤ 1 +2 +��en − e∗n−1��2 + 2 ∥ρn∥2 . +Using this bound on the right hand side of (23) it follows that +∥en∥2 + 1 +2 +��en − e∗n−1��2 − +��en−1��2 ≤ 2 ∥ρn∥2 . +From this expression, summing from n = 1 up to n = N one readily obtains that +��eN��2 + 1 +2 +N +� +n=1 +��en − e∗n−1��2 ≤ 2T +∆t ∥ρ∥2 +l∞(L2(D)) + +��e0��2 . +Then using (21) yields +∥c − ch∥l∞(L2(D)) ≤ C +� hm+1 +∆t1/2 |c|L∞(Hm+1(D)) +� ++ +��e0�� . +(25) +For ∆t = O(h), the error is O(hm+1/2), which is of the same order as the streamline-diffusion method +[14] for the advection equation. However, this estimate does not allow the convergence of the method +when ∆t → 0 independently of h. To overcome this trouble, we apply the procedure of [16] to obtain an +error estimate valid for all ∆t. So, substituting en = ρn + θn +h, and e∗n−1 = ρ∗n−1 + θ∗n−1 +h +in (22) and +rearranging terms yields +� +θn +h − θ∗n−1 +h +, vh +� += −(ρn − ρn−1, vh) − (ρn−1 − ρ∗n−1, vh). +(26) +Letting vh = θn +h we bound each term of this equality as follows. First, we notice that +��θ∗n−1 +h +��2 = +��θn−1 +h +��2 +and consequently +∥θn +h∥2 + +��θn +h − θ∗n−1 +h +��2 − +��θn−1 +h +��2 = 2 +� +θn +h − θ∗n−1 +h +, θn +h +� +. +7 + +Second, since for each n, Phρn = 0, then it follows that (ρn − ρn−1, θn +h) = Ph(ρn − ρn−1) = 0. It remains +to bound the term 2(ρn−1 − ρ∗n−1, θn +h). To do so, we notice that +ρn−1 − ρ∗n−1 = +� tn +tn−1 +dρ(X(x, tn; t), tn−1) +dt +dt, +since dρ(X(x, tn; t), tn−1) +dt += u(X(x, tn; t), t)·∇Xρ(X(x, tn; t), tn−1), then by the Cauchy-Schwarz inequal- +ity we get +��ρn−1 − ρ∗n−1��2 ≤ ∆t +� tn +tn−1 +|u(X(x, tn; t), t) · ∇Xρ(X(x, tn; t), tn−1)|2 dt, +so, letting y = X(x, tn; t) and denoting by Jt,n the Jacobian determinant J(x, t; tn) := +� +∂X(x,t;tn) +∂x +� += 1, +it follows that +��ρn−1 − ρ∗n−1��2 ≤ ∆t +� +D +� tn +tn−1 +��u(y, t) · ∇ρn−1(y) +��2 � +Jt,n−1�−1 dtdy +≤ ∆t ∥u∥2 +L∞(L∞(D)d) +� tn +tn−1 +� +D +��∇ρn−1(y) +��2 dydt +≤ ∆t2 ∥u∥2 +L∞(L∞(D)d) +��∇ρn−1��2 . +Now, using the estimate (21) we can write that +��ρn−1 − ρ∗n−1��2 ≤ ∆t2C +� +∆t1/2 ∥u∥L∞(L∞(D)d) +h +�2 � hm+1 +∆t1/2 +�2 +|c|2 +L∞(Hm+1(D)) . +(27) +Then +2(ρn−1 − ρ∗n−1, θn +h) ≤ 2 +∆t +��ρn−1 − ρ∗n−1��2 + ∆t +2 ∥θn +h∥2 +≤ ∆tC +� +∆t1/2 ∥u∥L∞(L,∞(D)d) +h +�2 � hm+1 +∆t1/2 +�2 +|c|2 +L∞(0,T ;Hm+1(D)) + ∆t +2 ∥θn +h∥2 . +Collecting all these bounds we have that +∥θn +h∥2+ +��θn +h − θ∗n−1 +h +��2− +��θn−1 +h +��2 ≤ ∆tC +� +∆t1/2 ∥u∥L∞(L∞(D)d) +h +�2 � hm+1 +∆t1/2 +�2 +|c|2 +L∞(Hm+1(D))+ ∆t +2 ∥θn +h∥2 . +Now, summing from n = 1 up to n = N yields +��θN +h +��2 + +N +� +n=1 +��θn−1 +h +− θ∗n−1 +h +��2 ≤ +��θ0 +h +��2 + R2 + ∆t +2 +N +� +n=1 +∥θn +h∥2 , +where +R2 = C +� +∆t1/2 ∥u∥L∞(L∞(D)d) +h +�2 � hm+1 +∆t1/2 +�2 +|c|2 +L∞(Hm+1(D)) . +Since +��eN��2 = +��ρN��2 + +��θN +h +��2 and �N +n=1 ∥θn +h∥2 + +��ρN��2 ≤ �N +n=1 ∥en∥2, then it follows that +��eN��2 + +N +� +n=1 +��θn−1 +h +− θ∗n−1 +h +��2 ≤ +��e0��2 + R2 + ∆t +N +� +n=1 +∥en∥2 . +Applying Gronwall inequality yields +∥c − ch∥l∞(L2(D)) ≤ C1 +� +∆t1/2 ∥u∥L∞(L∞(D)d) +h +� � hm+1 +∆t1/2 +� +|c|L∞(0,T ;Hm+1(D)) + +��e0�� . +(28) +Since (25) is valid, then combining it with (28) yields the result (19). +8 + +3.3 +Numerical test with the conventional LG method +We study the behavior of the conventional LG method considering the rotating hump problem [13]. The +domain D := (−1, 1) × (−1, 1), the velocity field is u = 2π(−x2, x1), and the initial condition +c0(x) = +� +� +� +cos3( 3 +2πr), +r ≤ 1/3, +0 +otherwise, +(29) +where r2 = (x1 − 0.5)2 + x2 +2. Notice that the function cosp(3πr/2) ∈ Hp(D), p ≥ 1, then p = 3 allows +enough smoothness for the optimal estimate of the error when m = 1 and 2. We show in Figure 1 the +isolines of the L2-projected initial condition in a mesh with mesh parameter h = 0.05 and the cross section +of the exact initial condition c0(x) at x2 = 0. The purpose of this test is to see how the error behavior +−1 +−0.5 +0 +0.5 +1 +−1 +−0.5 +0 +0.5 +1 +−1 +−0.5 +0 +0.5 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +x1 +x2 = 0 +c0(x) +x2 +x1 +Figure 1: Initial condition of the rotating hump problem in a mesh with size h = 0.05 +fits Theorem 4. To do so, we shall mainly focus on the error as a function of the parameter ∆t. Since this +theorem is valid under the assumption that the integrals, +� +D +cn−1 +h +◦ Xn,n−1(x)φi(x)dx, +(30) +are calculated exactly, then we carry our goal out by using symmetric Gauss quadrature rules of different +orders of accuracy to evaluate such integrals; in doing so, we assess the influence of the order of the +quadrature rule on the accuracy and stability of the numerical solution. +Figure 2 shows the L2-norm of the error as a function of the time step ∆t in two meshes with h = 0.05 +and h = 0.025 respectively. The errors are calculated after one revolution, T = 1, of the hump using +quadrature rules for the Galerkin projection (30) of 7, 16, 25, and 42 points which are exact for polynomials +of degree 5, 8, 10, and 14 respectively, see [8]. Broken lines correspond to the error function of linear +polynomials (m = 1), and full lines to the error function of quadratic polynomials (m = 2). By inspection, +we notice the following items: (a) for quadrature rules of high order, i.e., quadrature rules of 16, 25, and 42 +points, there is a value ∆ts, such that for ∆t > ∆ts, the error grows with a rate tending toward O(∆t−1/2) +as ∆t decreases; the error tendency of the most accurate rule of 42 points is closer to O(∆t−1/2) than the +error tendency of the other two rules. On the other hand, for 0 < ∆t ≤ ∆ts, the error remains almost +constant and independent of ∆t. (b) For quadrature rules that are not sufficiently accurate, i.e., the +quadrature rule of 7 points, there is a value ∆tins ≫ ∆ts at which the error starts growing very fast as ∆t +decreases until it reaches a maximum or eventually the numerical solution may become extremely large at +∆t∗. For ∆ts < ∆t < ∆t∗ the error decreases and when 0 < ∆t ≤ ∆ts the error remains constant. This +strange behavior of the solution for the quadrature rule of 7 points, which illustrates the dependence of +the stability of the LG method upon the order of the quadrature rule, is a well known feature reported by +many authors, see for instance [17]; in our tests, we note that the instability with quadratic polynomials +9 + +10−5 +10−4 +10−3 +10−2 +10−1 +10−4 +10−3 +10−2 +10−1 +100 + + +#7 points +#16 points +#25 points +#42 points +eL2(T) +∆t +m = 2 +m = 1 +O(∆t−1/2) +LG +h = 0.05 +10−5 +10−4 +10−3 +10−2 +10−1 +10−4 +10−3 +10−2 +10−1 +100 + + +#7 points +#16 points +#25 points +#42 points +eL2(T) +∆t +O(∆t−1/2) +m = 1 +m = 2 +LG +h = 0.025 +Figure 2: L2-norm of the error of the conventional LG method in the rotating hump problem, for linear +m = 1 and quadratic m = 2 finite elements in two different meshes (h = 0.05 and h = 0.025). +sends the numerical solution to infinity in an interval of values of ∆t, whereas for linear polynomials the +numerical solution, though useless, remains bounded. +Other relevant results displayed in Figure 2 are the following: (c) provided that the integrals (30) are +evaluated with enough accuracy, the numerical solutions are stable either for large or very small values +of ∆t, and as Theorem 4 says, the error is O(hm+1/∆t1/2) in the first case and O(hm) in the second +one, with the particularity that in both cases the error does not depend very much upon the order of the +quadrature rule used to calculate (30) as long as such a rule is exact for polynomials of degree > 2(m+1). +(d) The error does not grow monotonically, though we notice that the higher the order of the quadrature +rule the smoother the growth of the error; however, we can not explain why the rule of order 8 (16 points) +gives for some values of ∆t smaller errors than the rule of order 10 (25 points). +4 +The LPS-LG method +To formulate the local projection stabilized Lagrange-Galerkin (LPS-LG) method we introduce additional +concepts. Besides the partition Dh, we consider another quasi-uniform regular partition Mh on D the +elements of which are termed macro-elements. Each macro-element M is decomposed into one or more +elements K of the partition Dh (the case Mh = Dh is allowed giving place to the so-called one-level LPS +approach). We assume that there exist positive constants γ1 and γ2 such that for all K ⊂ Dh and M ⊂ Mh, +γ1hM ≤ hK ≤ γ2hM. Next, we consider a discontinuous finite element space Gh associated with Mh and +set Gh(M) := {qh |M: qh ∈ Gh}. For each M, we use the local L2-projector πM : L2(M) → Gh(M) to +define the fluctuation operator κM := id − πM, where id := L2(M) → L2(M) is the identity operator. In +addition to the approximation properties (10)-(14), we make the following assumptions. +Assumption LPS1 Let s ∈ (0, . . . , m − 1) be the degree of the polynomials of the space Gh, the +fluctuation operator κM satisfies the approximation property +∥κMw∥L2(M) ≤ Chl +M ∥w∥Hl(M) , ∀w ∈ Hl(M), 0 ≤ l ≤ s + 1. +(31) +Let Ps(M) be the set of polynomials of degree at most s defined in M, then a sufficient condition for the +assumption LPS1 to hold is Ps(M) ⊂ Gh(M). We set Wh(M) := {wh |M: wh ∈ Wh, wh = 0 on D\M}. +Assumption LPS2 There is an interpolation operator jh : H1(D) → Wh, such that for all (w, qh) ∈ +H1(D) × Gh, +(w − jhw, qh) = 0, +(32) +and for all w ∈ Hl(D), with 1 ≤ l ≤ m + 1 and M ∈ Mh, +∥w − jhw∥L2(M) + hM ∥∇(w − jhw)∥L2(M) ≤ Chl +M ∥w∥Hl(Λ(M)) , +(33) +where Λ(M) denotes a neighborhood of M. +10 + +The existence of jh has been proven in Part III Chapter 3 of [20] for spaces Gh and Wh that satisfy +the following inf-sup condition : +inf +qh∈Gh(M) +sup +wh∈Wh(M) +(qh, wh)M +∥qh∥L2(M) ∥wh∥L2(M) +≥ β > 0, +where β is a constant independent of h. For simplicial meshes the spaces (Wh, Gh) are the following (see, +[20] for details): +let +P disc +m,h := {vh ∈ L2(D) : v |K= �v ◦ F −1 +K +∈ �Pm( �K) ∀K ∈ Dh} and +P disc +m,2h := {vh ∈ L2(D) : v |M= �v ◦ F −1 +M ∈ �Pm(� +M) ∀M ∈ Mh}, +where FM : � +M → M ∈ Mh is the bijective transformation and � +M is the reference element for the partition +Mh. The continuous finite element space Pm,h is defined as Pm,h := P disc +m,h ∩ H1(D). For the one-level +approach: +Wh = P + +m,h := Pm,h + spanK∈Mh{ΦK · Pm−1,h(K)}, and Gh = P disc +m−1,h, +(34) +here, ΦK denotes the mapped bubble function that vanishes on the boundary ∂K of the element. For +the two-level approach (the elements K ∈ Th are obtained from the elements M ∈ Mh by means of a +refinement criterium, see for instance [1] and [9]): +Wh = Pm,h, and Gh = P disc +m−1,2h. +(35) +Figure 3 illustrates these approaches for d = 2 and simplicial meshes. +T1 +T2 +T3 +M +Wh +Gh +T +Wh +Gh +T +Figure 3: Approximation and projection spaces. Left panel, the two-level approach with m = 2; right +panel, the one-level approach with m = 1. +The LPS Lagrange-Galerkin method calculates cn +h ∈ Vh as solution of the equation +(cn +h − c∗n−1 +h +, vh) + ∆tSh(cn +h, vh) = 0 ∀vh ∈ Vh, +(36) +where Sh(cn +h, vh) is the stabilization term given by the expression +Sh(cn +h, vh) = +� +M +τM(κM∇cn +h, κM∇vh)M, +(37) +here, (κM∇cn +h, κM∇vh)M := +� +M κM∇cn +h · κM∇vhdx and τM are element-wise constant coefficients that +depend on the diameter hM of the macro-elements, their optimal values are determined by the error +analysis. +Remark 5 For vh = cn +h the term Sh(cn +h, vh) can be written as a diffusion term of the form +Sh(cn +h, cn +h) = +� +M +τM ∥κM∇cn +h∥2 +L2(M) := νadd(cn +h) ∥∇cn +h∥2 , +where +νadd(cn +h) := +� +� +� +� +� +� +� +� +� +� +M τM ∥κM∇cn +h∥2 +L2(M) +∥∇cn +h∥2 +when ∥∇cn +h∥ ̸= 0, +0 +otherwise. +. +11 + +4.1 +Analysis of the LPS-LG method +We prove the stability of the LPS-LG method in the mesh dependent norm +|||vn||| = +� +�||vn||2 + ∆t +n +� +j=1 +Sh(vj, vj) +� +� +1/2 +, +(38) +where vj ∈ H1 +0(D) (j = 1, .., n), n being a positive integer. We have the following result. +Lemma 6 For all N ≥ 1 it holds +������cN +h +������2 + +N +� +i=1 +��ci +h − c∗i−1 +h +��2 ≤ ∥co +h∥2 . +(39) +Proof. Let vh = cn +h in (36), then it follows that +∥cn +h∥2 + +��cn +h − c∗n−1 +h +��2 − +��c∗n−1 +h +��2 + 2∆tSh(cn +h, cn +h) = 0. +Noting that by virtue of (18), +��c∗n−1 +h +��2 = +��cn−1 +h +��2, then summing from n = 1 up to n = N ≥ 1 yields +��cN +h +��2 + +N +� +n=1 +��cn +h − c∗n−1 +h +��2 + 2∆t +N +� +n=1 +Sh(cn +h, cn +h) = +��c0 +h +��2 . +Hence, (39) follows. +Next, we perform the error analysis. To do so, we again decompose the error function en := cn − cn +h as +en = (cn − Phcn) + (Phcn − cn +h) ≡ ρn + θn +h. +(40) +First, we calculate an estimate for +������ρL������. +Lemma 7 Let c ∈ L∞(Hm+1(D) ∩ H1 +0(D)). Then, for all 0 ≤ L ≤ N. it follows that there exists a +constant C independent of h, ∆t and L such that +������ρL������ ≤ C(h + τ 1/2 +max)hm ∥c∥L∞(Hm+1(D)) , +(41) +where τmax = maxM∈Mh(τM). +Proof. We recall that +������ρL������2 = +��ρL��2 + ∆t +L +� +n=0 +Sh(ρn, ρn). +So, by virtue of (13) it follows that for all L there is a constant independent of h, such that +��ρL�� ≤ Chm+1 |c|L∞(Hm+1(D)) . +Next, we estimate the term Sh(ρn, ρn). Making use of the triangle inequality, the contractiveness property +of the local L2-projector πM, and (13) we obtain that +Sh(ρn, ρn) = � +M τM ∥κM∇ρn∥2 +L2(M) ≤ 4 � +M τM ∥∇ρn∥2 +L2(M) +≤ Cτmaxh2m ∥c∥2 +L∞(Hm+1(D)) . +(42) +Hence, collecting these two estimates the result (41) follows. +We are ready to establish the convergence of the LPS-LG method. +12 + +Theorem 8 Under the assumptions of Theorem 4, there exists a constant C independent of h, ∆t and +L, such that for all L, 0 ≤ L ≤ N, +max +0≤L≤N +������eL������ ≤ +������e0������ + C3 +� +τ 1/2 +max(hm + hs+1) + hm+1 + min +� +1, +∆t1/2 ∥u∥L∞(L2(D)) +h +� � hm+1 +∆t1/2 +�� +. +(43) +Proof. From (5) with t = tn and τ = −∆t and (36) we obtain the error equation +(en − e∗n−1, vh) − ∆tSh(cn +h, vh) = 0. +(44) +Noting that cn +h = cn − en, we recast this equation as +(en − e∗n−1, vh) + ∆tSh(en, vh) = ∆tSh(cn, vh). +Next, setting vh = θn +h = en −ρn and observing that Sh(a+b, c) = Sh(a, c)+Sh(b, c) this equation becomes +(en − e∗n−1, en) + ∆tSh(en, en) = (en − e∗n−1, ρn) ++∆tSh(en, ρn) + ∆tSh(en, cn) − ∆tSh(cn, ρn) ≡ +4 +� +i=1 +Ti +(45) +We estimate the terms Ti of (45). Applying Cauchy-Schwarz inequality we have that +|T1| = +��(en − e∗n−1, ρn) +�� ≤ 1 +4 +��en − e∗n−1��2 + ∥ρn∥2 . +To estimate T2 we apply again Cauchy-Schwarz inequality and obtain +|T2| = ∆t |Sh(en, ρn)| ≤ ∆t (Sh(en, en))1/2 (Sh(ρn, ρn))1/2 +≤ (δ/2)∆tSh(en, en) + (2δ)−1∆tSh(ρn, ρn), +0 < δ < 1. +Similarly, we have that +|T3| ≤ (δ/2)∆tSh(en, en) + (2δ)−1∆tSh(cn, cn), +and +|T4| ≤ ∆t +2 Sh(cn, cn) + ∆t +2 Sh(ρn, ρn). +Substituting these estimates in (45) with δ = 1/2 and noting that +2(en − e∗n−1, en) = ∥en∥2 + +��en − e∗n−1��2 − +��en−1��2 , +we obtain that +∥en∥2 + 1 +2 +��en − e∗n−1��2 − +��en−1��2 + ∆tSh(en, en) +≤ 2 ∥ρn∥2 + 3∆t (Sh(cn, cn) + Sh(ρn, ρn)) . +Summing both terms of this inequality from n = 1 up to n = N yields +��eN��2 + 1 +2 +�N +n=1 +��en − e∗n−1��2 − +��e0��2 + ∆t �N +n=1 Sh(en, en) +≤ �N +n=1 ∥ρn∥2 + 5∆t �N +n=1 (Sh(cn, cn) + Sh(ρn, ρn)) . +Since for any non negative integer n, +∥ρn∥2 ≤ Ch2(m+1) |c|2 +L∞(Hm+1(D)) , +13 + +by virtue of assumption LPS1 +Sh(cn, cn) ≤ Cτmaxh2(s+1) |c|2 +L∞(Hs+2(D)) , +(46) +and observing that ∥κM∇ρn∥ ≤ 2 ∥∇ρn∥ because πM is contractive, then (see (42)) +Sh(ρn, ρn) ≤ Cτmaxh2m |c|2 +L∞(Hm+1(D)) , +(47) +we have that +��eN��2 + 1 +2 +�N +n=1 +��en − e∗n−1��2 − +��e0��2 + ∆t �N +n=1 Sh(en, en) +≤ C +� +τmax(h2m + h2(s+1)) + h2(m+1) +∆t +� +, +or equivalently +������eN������2 + 1 +2 +N +� +n=1 +��en − e∗n−1��2 ≤ +����e0����2 + C +� +τmax(h2m + h2(s+1)) + h2(m+1) +∆t +� +. +(48) +Hence, it follows that +max +0≤L≤N +������eL������ ≤ +������e0������ + C +� +τ 1/2 +max +� +hm + hs+1� ++ hm+1 +∆t1/2 +� +. +(49) +This estimate of the error depends on ∆t−1/2 so that, for any fixed h, is invalid when ∆t → 0 because +in this case the method does not converge. So, in order to get rid of the factor ∆t−1/2 we consider the +following approach. Starting with the error equation (44) and setting +vh = θn +h, en = ρn + θn +h, e∗n−1 = ρ∗n−1 + θ∗n−1 +h +and cn +h = cn − (ρn + θn +h), +we get +� +θn +h − θ∗n−1 +h +, θn +h +� ++ ∆tSh(θn +h, θn +h) = −(ρn − ρ∗n−1, θn +h) ++∆t (Sh(ρn, θn +h) − Sh(cn, θn +h)) . +Now, noticing that ρn − ρ∗n−1 = ρn − ρn−1 − +� +ρ∗n−1 − ρn−1� +and +� +ρn − ρn−1, θn +h +� += 0, we can write the +above equation as +1 +2 ∥θn +h∥2 + 1 +4 +��θn +h − θ∗n−1 +h +��2 − 1 +2 +��θn−1 +h +��2 + ∆tSh(θn +h, θn +h) +≤ −(ρn−1 − ρ∗n−1, θn +h) + ∆t (Sh(ρn, θn +h) − Sh(cn, θn +h)) ≡ �3 +i=1 Si. +(50) +We bound the terms Si on the right hand side of (50). Thus, by the Cauchy-Schwarz inequality we have +that +|S1| ≤ 1 +∆t +��ρn−1 − ρ∗n−1��2 + ∆t +4 ∥θn +h∥2 ; +since, see (27), +1 +∆t +��ρn−1 − ρ∗n−1��2 ≤ ∆tC +� +∆t1/2 ∥u∥L∞(Lθ(D)) +h +�2 � hm+1 +∆t1/2 +�2 +, +then +|S1| ≤ ∆tC +� +∆t1/2 ∥u∥L∞(Lθ(D)) +h +�2 � hm+1 +∆t1/2 +�2 ++ ∆t +4 ∥θn +h∥2 . +(51) +To bound the terms S2 and S3 we use the same technique as for the terms T1 and T2 above and obtain +|S2| = ∆t |Sh(ρn, θn +h)| ≤ (δ/2)∆tSh(θn +h, θn +h) + (2δ)−1∆tSh(ρn, ρn), +and +|S3| = ∆t |Sh(cn, θn +h)| ≤ (δ/2)∆tSh(θn +h, θn +h) + (2δ)−1∆tSh(cn, cn). +14 + +Setting δ = 1/2 and substituting these bounds in (50) yields +∥θn +h∥2 + 1 +2 +��θn +h − θ∗n−1 +h +��2 − +��θn−1 +h +��2 + ∆tSh(θn +h, θn +h) +≤ ∆tC +� +∆t1/2 ∥u∥L∞(Lθ(D)) +h +�2 � hm +∆t1/2 +�2 ++ 5∆t (Sh(ρn, ρn)) ++5∆t (Sh(cn, cn)) + ∆t +2 ∥θn +h∥2 . +Or equivalently, using (46) and (47), +∥θn +h∥2 + 1 +2 +��θn +h − θ∗n−1 +h +��2 − +��θn−1 +h +��2 + ∆tSh(θn +h, θn +h) +≤ ∆tC +� +τmax(h2m + h2(s+1)) + +� +∆t1/2∥u∥L∞(Lθ(D)) +h +�2 � +hm +∆t1/2 +�2 +� ++ ∆t +2 ∥θn +h∥2 . +Summing both sides of this inequality from n = 1 up to n = N and applying Gronwall inequality we +obtain that +������θN +h +������2 + 1 +2 +�N +n=1 +��θn +h − θ∗n−1 +h +��2 ≤ +��θ0 +h +��2 ++C +� +�τmax(h2m + h2(s+1)) +� +∆t1/2 ∥u∥L∞(Lθ(D)) +h +�2 � hm+1 +∆t1/2 +�2� +� . +(52) +Hence, +������θN +h +������ ≤ +������θ0 +h +������ + C +� +τ 1/2 +max(hm + hs+1) + +� +∆t1/2 ∥u∥L∞(Lθ(D)) +h +� � hm+1 +∆t1/2 +�� +. +(53) +Now, noting that +������eN������ ≤ +������θN +h +������ + +������ρN������ and +������ρN������ ≤ C(hm+1 + τ 1/2 +maxhm), +it follows from (53) that +������eN������ ≤ +������e0������ + C +� +τ 1/2 +max(hm + hs+1) + hm+1 + +� +∆t1/2 ∥u∥L∞(L2(D)) +h +� � hm+1 +∆t1/2 +�� +. +(54) +Thus, since both estimates (54) and (49) hold, then we can write that there exists a constant C such that +������eN������ ≤ +������e0������ + C +� +τ 1/2 +max(hm + hs+1) + hm+1 + min +� +1, +∆t1/2 ∥u∥L∞(L2(D)) +h +� � hm+1 +∆t1/2 +�� +. +4.2 +Numerical tests with the LPS-LG method +We run, under the same premises, the rotating hump problem defined in Section 3.3, although the mesh +is now composed of right triangles with legs of length h = 10−2. We show in the upper panel of Figure 4 +the L2-norm of the error as a function of ∆t for both the two-level LPS-LG method and the conventional +LG method for a mesh size h = +√ +2 × 10−2 in both cases. In these experiments we have calculated the +integrals (30) with a quadrature rule of 12 points, which is exact for polynomials of degree 6. The spaces +Wh and Gh of the LPS-LG method are those shown in Figure 3, whereas the finite element space for +the conventional LG method consists of piecewise quadratic polynomials defined on each one of the 3 +triangles that compose the macro-element. We observe that the LPS-LG method is more stable than +15 + +10-6 +10-5 +10-4 +10-3 +10-2 + t +10-4 +10-3 +10-2 +10-1 +100 +eL2(T) +LG +LPS-LG, =0.1h +LPS-LG, =1h +O( +t -1/2) +# 12 points +10-6 +10-5 +10-4 +10-3 +10-2 + t +10-4 +10-3 +10-2 +10-1 +100 +eL2(T) +LG +LPS-LG, =0.1h +O( +t -1/2) +# 16 points +Figure 4: L2-norm of the error in the rotating hump problem of the two-level LPS-LG method and the +conventional LG method . +the conventional LG, because the latter goes unstable whereas the LPS-LG method remains stable when +τM = τ = h for all M, but it becomes unstable, with an instability region along the ∆t-axis smaller than +the one of the LG method, when τM = τ = 0.1h for all M. The lower panel of the figure shows that by +increasing the order of the quadrature rule the LPS-LG method with τM = τ = 0.1h becomes stable. +Figure 5 displays the L2-norm of the error as a function of ∆t for the one level LPS-LG method, the +discrete spaces of which are shown in the right panel of Figure 3, and for the conventional LG method +with finite element space Wh = P + +1 . The mesh size of this experiment is h = +√ +2 × 10−2. The solid lines +represent the error for the LPS-LG method with τmax = τM = 0.1h and quadrature rules of 16 and 25 +points, respectively. The dashed lines correspond to the error of the conventional LG method. +We notice in these figures that for ∆t = O(h) and h such that min +� +1, +∆t1/2∥u∥L∞(L2(D)) +h +� += 1, the +solutions given by both the LPS-LG and LG methods are very similar, regardless the quadrature rule. +This fact agrees with the results of Theorem 4 and Theorem 8, because in this case the dominant term of +the error in the conventional LG method is O(hm+1/∆t1/2) = O(hm+1/2), and in the LPS-LG method the +dominant term of the error is O(hm+1/∆t1/2+τ 1/2 +max×(hm+hs+1)), so letting τ 1/2 +max = ch1/2 and s = m−1 one +16 + +has that the error is also O(hm+1/2). However, for ∆t small enough so that min +� +1, +∆t1/2∥u∥L∞(L2(D)) +h +� += +∆t1/2∥u∥L∞(L2(D)) +h +, the maximum error in the L2-norm for the conventional LG method is O(hm), whereas +the maximum error of the LPS-LG method in the mesh dependent norm is also O(hm); however, since +������eN������2 = ∥e∥2 + ∆t +N +� +n=1 +�� +M +τM(κM∇en, κM∇en)M +� +, +then the L2-error of the LPS-LG method is smaller than the L2-error of the conventional LG method, and +this is what we observe in Figure 5. + + +10−5 +10−4 +10−3 +10−2 +10−1 +10−4 +10−3 +10−2 +10−1 +100 +LPS-LG #16 points +LPS-LG #25 points +LG #16 points +LG #25 points +eL2(T) +∆t +O(∆t−1/2) +τ = 0.1h +Figure 5: L2-norm of the error in the rotating hump problem of the LPS-LG method (solid lines) and the +LG method (dashed lines). +5 +The DC-LG method +Numerical experiments show that when the analytical solution c(x, t) is not sufficiently smooth the LG +methods presented in the previous sections are not free from wiggles. Following the approach of [15], +where the so called shock-capturing characteristic streamline-diffusion method is developed, but scaling +the non linear dissipative term as in [18], we formulate a LG method that is stable in the maximum norm +with linear finite elements, although numerical experiments show that the method may also be stable with +quadratic elements; this stabilization is achieved by adding a non linear dissipative term on the left side +of the formulation (16), thus obtaining the so called discontinuity-capturing LG method. In this method, +we calculate cn +h as solution of +� +cn +h − c∗n−1 +h +, vh +� ++ ∆t +� +K +(εK(cn +h)∇cn +h, ∇vh)K = 0, +(55) +where (εK(cn +h)∇cn +h, ∇vh)K := +� +K εK(cn +h)∇cn +h · ∇vhdx and +εK(cn +h) := Cεhα +K|R(cn +h)||K ≡ Cεhα +K +��cn +h − c∗n−1 +h +�� +∆t +����� +K +. +(56) +Here, Cε < 1 is a user-defined positive constant, the coefficient α ∈ [1, 2) and |R(cn +h)||K denotes the +absolute value of the residual, restricted to the element K, generated by the discretization of the material +derivative along the characteristic curves. The existence of a solution of (55) can be proven making use of +Corollary 1.1 of Chapter IV of [10] as in [18]. Notice that the amount of artificial diffusion is externally +controlled by Cε, h and the parameter α, the latter must be less than 2 in order for the method to be +stable in the maximum norm when the finite element space is linear. +17 + +5.1 +Analysis of the DC-LG method +First, we study the stability of (55) in both the L2 norm and the L∞ norm. +Lemma 9 For all N ≥ 1, it holds +��cN +h +��2 + +N +� +n=1 +��cn +h − c∗n−1 +h +��2 + 2∆t +N +� +n=1 +� +K +���εK(cn +h)1/2∇cn +h +��� +2 +K ≤ +��c0 +h +��2 . +(57) +Proof. Letting vh = cn +h in (55) and taking into account that +��c∗n−1 +h +�� = +��cn−1 +h +��, it follows that +∥cn +h∥2 + +��cn +h − c∗n−1 +h +��2 − +��cn−1 +h +��2 + 2∆t +� +K +���εK(cn +h)1/2∇cn +h +��� +2 +K . +Hence, it follows that +��cN +h +��2 + +N +� +n=1 +��cn +h − c∗n−1 +h +��2 + 2∆t +N +� +n=1 +� +K +���εK(cn +h)1/2∇cn +h +��� +2 +K ≤ +��c0 +h +��2 . +Lemma 10 There is a constant C independent of h, ∆t, and n, but depending on the constant Cε, such +that for all n, +∥cn +h∥L∞(D) ≤ (1 + Ch +1 +2 (2−α) log( 1 +h)) +��c0 +h +�� +L∞(D) . +(58) +Noting that for p ≥ 1, +��c∗n−1 +h +�� +Lp(D) = +��cn−1 +h +�� +Lp(D), we can prove this lemma by using the the same +arguments as those employed to prove Lemma 6 in [18]. +See also the proof presented in [15] of the +stability in the maximum norm for the shock-capturing characteristic streamline-diffusion method. It is +worth remarking that maximum norm stability has only been proven for linear finite elements, because this +proof makes use of a result of [21], which says that there is a constant c independent of p = 2a, a = 1, 2 . . ., +such that for all wh ∈ Wh +� +Rd ∇wh · ∇Πh(wh)p−1dx = c +p2 +� +K +� +K +|∇wh| (wh)p−2 dx. +And this result has only been proven for linear finite elements. However, via numerical examples, we +have observed that the maximum norm stability also holds in cases where the solution exhibits strong +discontinuities for quadratic elements. +For the error analysis we have the following result. +Theorem 11 Let c ∈ L∞(Hm+1(D) ∩ H1 +0(D)) ∩ L∞(W 1,∞(D)). Then, there exists a constant C inde- +pendent of ∆t, h and n, but depending on |D|, |c|L∞(Hm+1(D)) and ∥∇c∥L∞(L∞(D)), such that +∥c − ch∥l∞(L2(D)) ≤ C(hm+1 + Cεhα) +∆t1/2 +. +(59) +Proof. The error equation is +� +en − e∗n−1, vh +� +− ∆t +� +K +(εK(cn +h)∇cn +h, ∇vh)K = 0 +(60) +Noting that cn +h = cn − en we have that +∆t � +K(εK(cn +h)∇cn +h, ∇vh)K = ∆t +� +K +(εK(cn +h)∇cn, ∇vh)K +−∆t +� +K +(εK(cn +h)∇en, ∇vh)K, +18 + +so we can write the error equation as +� +en − e∗n−1, vh +� ++ ∆t +� +K +(εK(cn +h)∇en, ∇vh)K = ∆t +� +K +(εK(cn +h)∇cn, ∇vh)K. +(61) +Now, setting in this equation vh = θn +h = en − ρn, where ρn = cn− Πhcn and θn +h = Πhcn − cn +h, we get +� +en − e∗n−1, en� ++ ∆t � +K(εK(cn +h)∇en, ∇en)K = +� +en − e∗n−1, ρn� ++∆t � +K(εK(cn +h)∇en, ∇ρn)K ++∆t � +K(εK(cn +h)∇en, ∇cn)K +−∆t � +K(εK(cn +h)∇cn, ∇ρn)K ≡ �4 +i=1 Ri. +(62) +We estimate the terms Ri on the right hand side. Thus, regarding R1 we apply the Cauchy-Schwarz +inequality to obtain that +|R1| = +��� +en − e∗n−1, ρn��� ≤ 1 +8 +��en − e∗n−1��2 + 2 ∥ρn∥2 . +As for the term R2, we use the same inequality to get +|R2| ≤ (δ/2)∆t +� +K +(εK(cn +h)∇en, ∇en)K + (2δ)−1∆t +� +K +(εK(cn +h)∇ρn, ∇ρn)K, 0 < δ < 1. +Similarly +|R3| ≤ (δ/2)∆t +� +K +(εK(cn +h)∇en, ∇en)K + (2δ)−1∆t +� +K +(εK(cn +h)∇cn, ∇cn)K, +and +|R4| ≤ 1 +2∆t +� +K +(εK(cn +h)∇cn, ∇cn)K + 1 +2∆t +� +K +(εK(cn +h)∇ρn, ∇ρn)K. +Substituting this estimates in (62) with δ = 1/2 yields +∥en∥2 + 3 +4 +��en − e∗n−1��2 − +��en−1��2 + ∆t � +K(εK(cn +h)∇en, ∇en)K +≤ 4 ∥ρn∥2 + 3∆t � +K(εK(cn +h)∇cn, ∇cn)K ++3∆t � +K(εK(cn +h)∇ρn, ∇ρn)K. +(63) +Next, we have to estimate the last two terms on the right hand side of this inequality. +∆t � +K(εK(cn +h)∇cn, ∇cn)K = Cε∆t +� +K +hα +K +� +K +��cn +h − c∗n−1 +h +�� +∆t +(∇cn)2 dK +≤ Cε ∥∇cn∥2 +L∞(D) +� +K +hα +K +� +K +��cn +h − c∗n−1 +h +�� dK +≤ Cε ∥∇cn∥2 +L∞(D) +� +K +hα +K +��cn +h − c∗n−1 +h +�� +L1(K) +≤ Cε ∥∇cn∥2 +L∞(D) hα ��cn +h − c∗n−1 +h +�� +L1(D) . +It remains to estimate the term +��cn +h − c∗n−1 +h +�� +L1(D). To do so, we observe that +cn +h − c∗n−1 +h += (cn − en) − +� +c∗n−1 − e∗n−1� += en − e∗n−1 +19 + +because cn = c∗n−1, then +��cn +h − c∗n−1 +h +�� +L1(D) = +��en − e∗n−1�� +L1(D) and by virtue of the Cauchy-Schwarz +inequality +��en − e∗n−1�� +L1(D) ≤ CD +��en − e∗n−1�� +L2(D), CD = C(|D|), |D| being the measure of D; hence +Cε ∥∇cn∥1 +L∞(D) hα ��cn +h − c∗n−1 +h +�� +L1(D) ≤ 1 +16 +��en − e∗n−1��2 +L2(D) + 4C2 +εh2αC2 +D ∥∇cn∥4 +L∞(D) . +Therefore, we can set that +∆t +� +K +(εK(cn +h)∇cn, ∇cn)K ≤ 1 +16 +��en − e∗n−1��2 +L2(D) + 4C2 +εh2αC2 +D ∥∇cn∥4 +L∞(D) . +(64) +We estimate now the term ∆t � +K(εK(cn +h)∇ρn, ∇ρn)K. To this end, we notice that +∆t +� +K +(εK(cn +h)∇ρn, ∇ρn)K = Cε +� +K +hα +K +� +K +��cn +h − c∗n−1 +h +�� (∇ρn)2 dK, +but by virtue of (14) we have that ∥∇ρn∥L∞(D) ≤ c3 ∥∇cn∥L∞(D), then we can write that +∆t +� +K +(εK(cn +h)∇ρn, ∇ρn)K ≤ c2 +3Cε ∥∇cn∥2 +L∞(D) +� +K +hα +K +� +K +��cn +h − c∗n−1 +h +�� dK; +so, arguing as we have just done for the term ∆t � +K(εK(cn +h)∇cn, ∇cn)K it follows that +∆t +� +K +(εK(cn +h)∇ρn, ∇ρn)K ≤ 1 +16 +��en − e∗n−1��2 +L2(D) + 4c4 +3C2 +εh2αC2 +D ∥∇cn∥4 +L∞(D) . +(65) +Substituting (64) and (65) in (63) and using the estimate (14) yields +∥en∥2 + 1 +2 +��en − e∗n−1��2 − +��en−1��2 + ∆t � +K(εK(cn +h)∇en, ∇en)K +≤ C +� +h2(m+1) + C2 +εh2α� +. +where C is a constant that depends on CD, |cn|Hm+1(D) and ∥∇cn∥L∞(D). Summing both terms of this +inequality from n = 1 up to n = N it follows that +��eN��2 + 1 +2 +�N +n=1 +��en − e∗n−1��2 + ∆t �N +n=1 +� +K(εK(cn +h)∇en, ∇en)K +≤ +C +∆t +� +h2(m+1) + C2 +εh2α� +, +or equivalently +��eN�� ≤ C +� +hm+1 + Cεhα� +∆t1/2 +. +Remark 12 This estimate depends on ∆t−1/2 so that for h fixed blows up as ∆t → 0. In contrast with +the previous LG methods, for the DC-LG method we have not been able to find an error estimate free +from the ∆t−1/2 dependence; however, based on numerical experiments and assuming that the maximum +norm stability holds, we may hypothesizes that there is a ∆tc, such that for ∆t ≤ ∆tc the error will not +increase, remaining nearly constant or decreasing very slowly. So, noting that cn +h − c∗n−1 +h += en − e∗n−1 +because cn = c∗n−1, we can argue that for ∆t ≤ ∆tc +max +n +max +K +��en − e∗n−1�� +∆t +����� +K += β, +β being a small constant that depends on m; hence, we can consider that εK(cn +h) is a constant, specifically, +for all K and n we set +ν := εK(cn +h) = Cεβhα. +20 + +Then, the error equation can be written now as +� +en − e∗n−1, vh +� +− ∆tν +� +K +(∇cn +h, ∇vh)K = 0. +(66) +So, as we have done above, we let +vh = θn +h, en = ρn + θn +h, e∗n−1 = ρ∗n−1 + θ∗n−1 +h +and cn +h = cn − (ρn + θn +h), +with ρn = cn − Phcn, and recast (66) as +� +θn +h − θ∗n−1 +h +, θn +h +� ++ ∆tν∇θn +h, ∇θn +h) = −(ρn − ρ∗n−1, θn +h) +−∆tν(∇ρn, ∇θn +h) + ∆tν(∇cn, ∇θn +h). +(67) +If we compares this equation with (26), we can consider that the artificial dissipation terms represent a +perturbation to the equation of the pure advection problem, so, we can expect that when ν → 0 (67) will +yield the same estimate as (26). To check that this is the case, we bound the terms −(ρn − ρ∗n−1, θn +h), +(∇ρn, ∇θn +h) and (∇cn, ∇θn +h) as we have done many times before and can easily arrive to the estimate +��θN +h +�� ≤ C +� +hm + ν1/2� +, +where the constant C depends on |c|L∞(0,T ;Hm+1(D)), and consequently, +��eN�� = O +� +hm + (Cεβhα)1/2� +. +So, if Cεβ is so small that (Cεβhα)1/2 ≤ hm, then +��eN�� = O(hm). +5.2 +Numerical tests with the DC-LG method +Since the method is designed to deal with discontinuous initial conditions, we shall perform two numerical +tests. The first one is again the hump problem to see wether the error behaves according to Theorem 11; +the second test uses as initial condition the so called “slotted” cylinder, this a typical initial condition to +study the ability of schemes to deal with strong discontinuities. +5.2.1 +The hump test +We run the test under the same conditions as the numerical test for the conventional LG method. We +show in Figures 6 and 7 the results for the meshes with mesh parameter h = 0.05 and h = 0.025 after one +revolution, and with the constants Cε and α of the expression for the artificial diffusivity (56) taking the +values Cε = 0.01, Cε = 0.1 and α = 3 +2. These results must be compared with those of Figure 2. +We notice the following facts: (a) For high order quadrature rules, the error of the DC-LG method +shows a similar, but smoother, behavior as the error of the conventional LG method, with the feature +that the higher the constant Cε or the coarser the mesh the smoother the profile of the error curves; this +is a consequence of the nonlinear artificial diffusivity that depends on both Cε and h. (b) For the low +order quadrature rule of 7 points, the DC-LG method loses accuracy for those values ∆t for which the +conventional LG method is inaccurate or even unstable; in fact, for Cε = 0.01 and h = 0.025 there is an +interval of values ∆t, which, roughly speaking, corresponds with those values for which the conventional +LG method with quadratic polynomials becomes unstable, in which the DC-LG method with quadratic +polynomials is less accurate than with linear polynomials. This can be explained because when both Cε +and h are low the artificial diffusivity is not sufficiently strong to prevent the instability. (c) Roughly +speaking, we can say that the higher the artificial viscosity the less sensitive the DC-LG method is to the +order of the quadrature rules, provided that such rules are exact for polynomials of degree 2m. (d) For +∆t = O(h) or ∆t = o(h2), all the quadrature rules give about the same solution. This means that in those +ranges of values ∆t it is not necessary the use of high order quadrature rules, just a rule which is exact for +polynomials of degree 2(m+1) would suffice. (e) Looking at the profiles of the error curves, we notice that +21 + +10−5 +10−4 +10−3 +10−2 +10−1 +10−4 +10−3 +10−2 +10−1 +100 + + +#7 points +#16 points +#25 points +#42 points +eL2(T) +∆t +m = 2 +m = 1 +O(∆t−1/2) +DC-LG +h = 0.05 +Cε = 0.01 +10−5 +10−4 +10−3 +10−2 +10−1 +10−4 +10−3 +10−2 +10−1 +100 + + +#7 points +#16 points +#25 points +#42 points +eL2(T) +∆t +m = 2 +O(∆t−1/2) +m = 1 +DC-LG +h = 0.05 +Cε = 0.1 +Figure 6: L2-error norm with the DC-LG method in the rotating hump problem for h = 0.05 +for high order quadrature rules the error behaves as Theorem 11 says, that is, there is a value ∆tc (in this +test, ∆tc = O(h−2)), such that for ∆t ≥ ∆tc the error is O(hm+1 + Cεhα)/∆t1/2. However, for ∆t < ∆tc, +the error does not grow and remains more or less constant, particularly as the artificial diffusivity is high +enough, see Figure 6. Finally, fixing the mesh and the parameter α, this test shows that as the constant +Cε becomes smaller and smaller, the DC-LG solution approaches the solution of conventional LG method. +10−5 +10−4 +10−3 +10−2 +10−1 +10−4 +10−3 +10−2 +10−1 +100 + + +#7 points +#16 points +#25 points +#42 points +eL2(T) +∆t +m = 2 +m = 1 +O(∆t−1/2) +DC-LG +h = 0.025 +Cε = 0.01 +10−5 +10−4 +10−3 +10−2 +10−1 +10−4 +10−3 +10−2 +10−1 +100 + + +#7 points +#16 points +#25 points +#42 points +eL2(T) +∆t +m = 2 +O(∆t−1/2) +m = 1 +DC-LG +h = 0.025 +Cε = 0.1 +Figure 7: L2-error norm with the DC-LG method in the rotating hump problem for h = 0.025 +5.2.2 +The slotted cylinder +Our second test is the so called slotted cylinder. The idea behind this test is to assess the ability of the +DC-LG method to deal with strong discontinuities; specifically, we wish to see how the scheme smears out +an initial condition that is strongly discontinuous. The domain D := [−1, 1] × [−1, 1], the velocity field u +is the same as in the previous tests, i.e., u = 2π(−x2, x1), and the initial condition is a cylinder of height +1 and radius 0.25 centered at (0.5,0), with a slot along the plane x = 0.5 of width 0.1 and depth 0.35. +The simulations are carried out with a time step ∆t = 0.01 in the mesh with mesh parameter h = 0.025, +and the numerical initial condition being computed by the L2-projection onto the finite element space Vh. +Although we are aware that this is not a good way to calculate the numerical initial condition because, +as we see in Figure 9, some overshoots and undershoots are generated by the L2-projection, we have left +it to test the capability of DC-LG method to suppress the wiggles; it is clear that the method is able to +kill them out after few time steps when the constant Cε of the artificial diffusion εh(cn +h) is Cε = 0.1. A +better approach to calculate the numerical initial condition would have been to perform L2-projection of +22 + +the exact initial condition with linear elements and lumped mass matrix, yielding this way a somewhat +smoother initial condition. The integration time T = 1. For the results, we have used the exact trajectories +and the integrals (30) have been calculated with the quadrature rule of 16 points. Based on the results of +the hump test, we know that for the values ∆t = 0.01 and h = 0.025 the solution is not sensitive to the +order of the quadrature rules used to approximate the integrals (30), provided that the rule is exact for +polynomials of degree > 2(m + 1). +0.1 +0.2 +0.3 +0.4 +0 +1 +0.8 +0.6 +0.4 +0.2 +0 +1 +0 +0.5 +exact +Cε = 0.1 +Cε = 0.01 +1 +0.8 +0.6 +0.4 +0.2 +0 +−0.3 +−0.5 +−0.4 +−0.2 +−0.1 +0.5 +0 +−0.5 +0.5 +1 +0 +x = 0.5 +y = 0 +y +x +y = 0 +x = 0.5 +m = 1 +0.5 +Cε = 0.01 +Cε = 0.1 +Figure 8: Slotted cylinder after one revolution for linear finite elements m = 1. +Upper panel: three +dimensional view of the solutions. Lower panel: the level lines (on the left) and cross sections (on the +right) that correspond with the figures of the upper panel. +We display in the upper panel of Figure 8 a three dimensional view of the cylinder after one revolution, +whereas in the low panel are represented the level lines (on the left) and cross sections (on the right) when +the constant of the artificial diffusion takes the values Cε = 0.1 and Cε = 0.01. This solution has been +calculated with linear polynomials (m = 1). We notice that the width of the upper face of the lobes and +the width of the “bridge” as well as the depth of the slot are reasonably well preserved for both constants +Cε = 0.1 and Cε = 0.01. It is worth remarking that the figures of the upper and middle panel with +Cε = 0.1 compare very well with those obtained in [15] and [11] applying the shock-capturing streamline- +diffusion method with a time step ∆t = 0.01 and the mesh size h = 0.01, which is 2.5 times smaller than +the one we use. It is clear that with the constant Cε = 0.1 the DC-LG method introduces a major degree +of smearing, and when Cε = 0.01 the method is not able to suppress the wiggles generated around the +discontinuities at the first time step. +Similar representations of the numerical solution calculated with quadratic polynomials (m=2) are +displayed in Figure 9. If we compare these graphs with those of Figure 8 one sees that it is clear the +improvement of the numerical solution calculated with quadratic polynomials; for instance, the slopes of +the cylinder sides, the width of the lobes of the upper face and the width of the “bridge” are much better +represented with quadratic elements than with linear elements. +23 + +Ce = 0.1 +0.5 +0.5.C = 0.010 +0 +1 +1 +0.5 +0.8 +0.6 +0.5 +0 +m= 1 +0.4 +0 +-0.50.5 +0 +0.2 +0 +-0.50.1 +0.2 +0.3 +0.4 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +exact +Cε = 0.1 +Cε = 0.01 +1 +0.8 +0.6 +0.4 +0.2 +0 +−0.5 +−0.4 +−0.3 +−0.2 +−0.1 +-0.5 +0 +0.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +x = 0.5 +y = 0 +y +x +m = 2 +y = 0 +x = 0.5 +0.5 +Cε = 0.01 +Cε = 0.1 +Figure 9: Slotted cylinder after one revolution for quadratic finite elements m = 2. Upper panel: three +dimensional view of the solutions. Lower panel: the level lines (on the left) and cross sections (on the +right) that correspond with the figures of the upper panel. +Finally, we represent in Figure 10 the time evolution of the maximum and minimum of the numerical +solutions obtained by the conventional LG method, and the DC-LG one (with Cε = 0.01 and Cε = 0.1). As +we commented above, our calculation of the numerical initial condition allows the generation of wiggles +at the first time step, in fact, the largest amplitude of such wiggles is 0.3. The DC-LG method with +Cε = 0.1 dissipates these wiggles as the solution progresses, such that the for m = 2 the dissipation is very +strong at the beginning, going very quickly the minimum to zero and the maximum to 1, as, on the other +hand, should be; however, when Cε = 0.01, the wiggles are also dissipated, but at a slower rate, with the +amplitudes of the minimum and maximum values decreasing somewhat oscillatorily, tending to −0.05 and +1.05 respectively. However, though we proof that linear polynomials are stable in the maximum norm, +the behavior of the maximum and minimum is not as good as that of quadratic elements; for instance, +when Cε = 0.1 the dissipation of the amplitude of the wiggles is slower and less strong than in the case of +quadratic elements, noting that the steady maximum and minimum are 1.03 and −0.03 respectively; when +Cε = 0.01 the maximum and minimum of DC-LG solution, though smaller in amplitude, exhibit a similar +oscillatory behavior as those of the conventional LG method. It is remarkable that both the maximum +and the minimum of the conventional LG method, either with m = 1 or m = 2, undergo dissipation at +the beginning of the calculations and then go on exhibiting an oscillatory behavior. +6 +Concluding remarks +1) We have obtained a new error estimate of the conventional LG method for the advection equation. +In contrast with previous estimates, ours is valid for all ∆t, no matter how small ∆t is, showing that +24 + +7 +Ce = 0.1 +0.5. +0.5.C = 0.010 +0 +0.8 +0.5 +0.6 +0.5 +m=2 +0.4 +0 +0 +-0.50.5 +0 +0.2 +0 +-0.5 + +1 +1.1 +1.2 +1.3 +1.4 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +−0.4 +−0.3 +−0.2 +−0.1 +0 +LG +DC-LG Cε = 0.01 +DC-LG Cε = 0.1 +min(ch) +max(ch) +m = 1 +t + + +1 +1.1 +1.2 +1.3 +1.4 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +−0.4 +−0.3 +−0.2 +−0.1 +0 +LG +DC-LG Cε = 0.01 +DC-LG Cε = 0.1 +max(ch) +m = 2 +t +min(ch) +Figure 10: Evolution with time of the maximum and minimum of the slotted cylinder during one revolution +for linear m = 1 and quadratic m = 2 finite elements +for ∆t ≤ Khp, p > 2, the error is O(hm), and for ∆t > Khp the error is O(hm+1/∆t1/2), here K = +(∥u∥L∞(L∞(D)d))−1/2. This error estimate has been obtained under the assumption that the integrals +� +K φj(Xh(x, tn+1, tn))φi(x)dx are calculated exactly. +2) To validate our theoretical result we perform +numerical tests using quadrature rules of different orders to evaluate those integrals and calculating exactly +the trajectories. We find that the higher the order of the quadrature rule the closer the error behavior +to the theoretical one. Other interesting finding is that for ∆t = O(h) and ∆t = O(h3), the error is +quite independent of the order of the quadrature rule as long as the rule calculates exactly polynomials +of degree ≥ 2(m + 1). 3) The LG approach is a natural way of introducing upwinding in the numerical +method, but the degree of upwinding is not strong enough if the initial condition lacks regularity. One way +of stabilizing the conventional LG method is using the so called local projection stabilization technique, +which is symmetric and acts on the small unresolved scales. We thus obtain the so called LPS-LG method +and estimate its error in a mesh dependent norm. +4) Neither the LPS-LG nor the conventional LG +methods are stable in the maximum norm, so they do not deal satisfactorily with strongly discontinuous +initial conditions. Following the idea of shock-capturing characteristic streamline-diffusion method of [15], +we have formulated the DC-LG method that is a residual stabilized LG method, which for linear finite +elements is stable in both the L2- and L∞-norms. This method has shown to be effective in preserving +the shape of the initial condition, in particular, when quadratic elements are used, though there is no +theoretical proof of the stability in the infinite norm for these elements. 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Johnson, A new approach to algorithms for convection problems which are based on exact transport ++ projection. Comput. Methods Appl. Mech. Engrg. 100: 45-62, 1992. +[16] K. W. Morton and E. S¨uli, Evolution-Galerkin methods and their supraconvergence. Numer. Math. +71: 331-355, 1995. +[17] K. W. Morton, A. Priestley and E. S¨uli, Stability of the Lagrange-Galerkin method with non-exact +integration. M2AN Math. Model. Numer. Anal. 22: 625-653, 1988. +[18] M. Nazarov, Convergence of a residual based artificial viscosity finite element method. Computers +and Mathematics with Applications 65: 616-636, 2013. +[19] O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equa- +tions. Numer. Math. 38: 309-332, 1982. +[20] H.-G Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differ- +ential Equations, Springer, Berlin, 2008. +[21] A. Szepessy, Convergence of a shock-capturing streamlone diffusion element method for a scalar +conservation law in two space dimensions. Mathematics of Computation 53: 527-545, 1989. +26 + diff --git a/gNE1T4oBgHgl3EQfywXe/content/tmp_files/load_file.txt b/gNE1T4oBgHgl3EQfywXe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8317b1c2cc8e39c2e3bae87c76d5c63cabc5367 --- /dev/null +++ b/gNE1T4oBgHgl3EQfywXe/content/tmp_files/load_file.txt @@ -0,0 +1,805 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf,len=804 +page_content='New error estimates of Lagrange-Galerkin methods for the advection equation Rodolfo Bermejoa, Jaime Carpiob, Laura Saavedrac a) Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Matem´atica Aplicada a la Ingenier´ıa Industral ETSII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Universidad Polit´ecnica de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' b) Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Ingenier´ıa Energ´etica ETSII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Universidad Polit´ecnica de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' c) Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Matem´atica Aplicada a la Ingenier´ıa Aeroespacial, ETSIAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Univversidad Polit´ecnica de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Abstract We study in this paper new developments of the Lagrange-Galerkin method for the advection equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In the first part of the article we present a new improved error estimate of the conven- tional Lagrange-Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In the second part, we introduce a new local projection stabi- lized Lagrange-Galerkin method, whereas in the third part we introduce and analyze a discontinuity- capturing Lagrange-Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Also, attention has been paid to the influence of the quadrature rules on the stability and accuracy of the methods via numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Keywords Advection equation, Lagrange-Galerkin, finite elements, local projection stabilization, discontinuity capturing Mathematics Subject Classification (2010) 65M12, 65M25, 65M60, 65M50 1 Introduction We consider the Cauchy problem for the pure advection equation � � � � � ∂c ∂t + u·∇c = 0, x ∈ Rd, t > 0, c(x, 0) = c0(x), (1) where c : Rd × [0, T] → R, u : Rd × [0, T] → Rd is a vector-valued function and c0(x) is a function of compact support defined in a domain D0 ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It is well known that the solution of this problem given by the method of characteristics is of the form c(X( · , t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t + τ), t + τ) = c(·, t), X(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t + τ) being the characteristic curves of the advection equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In this paper, we present the error analysis of three versions of the so called LG method applied to solve the Cauchy problem (1), which represent numerical realizations of the theoretical method of characteristics in the framework of H1-conforming finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The first version, denoted in this paper with the name the conventional LG method, consists basically on approximating the solution c(x, t) by the L2-projection onto the finite element space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' see, for instance, [19], [17] and [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The conventional LG method can be viewed as a kind of high order upwind method that introduces artificial diffusion in the discrete formulation, thus providing good stability properties to the numerical solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' however, as numerical experiments show, this artificial diffusion is not high enough to suppress the oscillations that appear at discontinuities of the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In order to alleviate this problem at discontinuities and extend the stability properties, we shall study a local projection stabilized Lagrange-Galekin (LPS-LG) method and a discontinuity-capturing Lagrange-Galerkin (DC-LG) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In the past, several authors have obtained different estimates for the L2-norm of the error of the conventional LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For example, [19] calculates an estimate of the form O(hm+1/∆t), where m denotes the degree of the polynomials of the H1-conforming finite element spaces, h being the largest 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='03438v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='NA] 9 Jan 2023 diameter of the elements of the spatial mesh and ∆t the size of the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The problem with this estimate is that for fixed h the error becomes unbounded when ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' [17] removes the ∆t−1 dependence from the error estimate obtaining the new estimate O(hm), this estimate allows to prove convergence of LG method for the advection equation when ∆t → 0 independently of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' [15] improves the estimate of [19] calculating a new estimate O(hm+1/∆t1/2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' which for ∆t = O(h) implies that the error of the conventional LG method is of the same order as both the streamline-diffusion (SD) method formulated in the framework of space-time finite elements continuous in space and discontinuous in time and the characteristic streamline-diffusion (CSD) method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' the latter method being a version of the SD method that uses space-time meshes oriented along the characteristic curves of the advection equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Later on, [16] calculates a new estimate of the form O(min(1, ∥u∥L∞(Rd)d ∆t/h)hm+1/∆t), where ∥u∥L∞((0,T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='Rd)d denotes the supremum norm of the velocity vector u(x, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' then, considering that ∥u∥L∞(Rd)d ∆t/h is the CFL number, we can say that for CFL numbers less than one, the error of LG methods is O(hm), whereas for CFL numbers larger than one the error is O(hm+1/∆t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Numerical examples show that the latter estimate provides a better description of the error behavior of the conventional LG method than the other estimates do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In this paper, we revisit the results of the the above mentioned authors and calculate an improved new error estimate of the form O(min(1, ∥u∥L∞(Rd)d ∆t1/2/h)hm+1/∆t1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Some numerical examples will support the validity of this estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Local projection stabilized methods have become quite popular for advection-diffusion-reaction equa- tions, including Navier-Stokes equations, see [1], [2], [3], [9] and [20] just to cite a few, for they are symmetric and introduce artificial diffusivity via a fluctuation operator acting on the small unresolved scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We prove that the LPS-LG method is stable in the L2-norm, and our error analysis shows that the error of the LPS-LG method in the mesh dependent norm (to be defined below) max 0≤tn≤T |||cn − cn h||| = O(hβ + min(1, ∥u∥L∞(Rd)d ∆t1/2/h)hm+1/∆t1/2), where β is a coefficient depending on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' However, despite the introduction of the artificial diffusivity, numerical tests show that the LPS-LG method may exhibit an oscillatory behavior when the solution is not sufficiently smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The DC-LG method might be viewed as a version of the shock capturing CSD method [15] in which the mesh alignment along the characteristic curves and the stream diffusion mechanism of the shock capturing CSD are removed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' in fact, one can consider that the DC-LG method is a reformulation, in the framework of the conventional LG method, of the residual artificial viscosity method introduced in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We prove that DC-LG method is stable in the L2-norm, regardless the degree of the finite element spaces, and also in the L∞-norm with linear finite elements, although numerical examples show L∞-norm stability with quadratic polynomials in the presence of a strong discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For solutions sufficiently smooth, we are able to prove that the error in the L2-norm is of the form O((hm+1 + Cεhα)/(∆t1/2)), α and Cε being positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The theoretical analysis of LG methods presented in this paper are proven under the assumption that the integrals � K φj(X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' tn−1))φi(x)dx , which appear in the formulation of the methods, are calculated exactly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' here, K is a generic element of the mesh, φi is the ith global basis function of the finite element space and X(x, tn, tn−1) is the foot of the characteristic curve associated with the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Noting that the integrand is the product of two piecewise continuous polynomial functions defined on two different meshes, it may become very difficult to calculate such integrals exactly, so one has to resort to quadrature rules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' but as [17] and [13] show, the quadrature rules have to be of high order because otherwise the numerical solution may become either inaccurate or unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Being aware of this fact, we shall test the validity of our analysis of the LG methods studied in the paper by performing some benchmark numerical tests, using symmetric Gaussian rules of different orders to assess the influence of the order of the quadrature rules on the accuracy and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We make a short presentation of the continuous problem in Section 2, and introduce the formulation and numerical analysis of the conventional LG method for the advection equation in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Some numerical examples illustrating its performance are also reported in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Section 4 is devoted to the formulation, analysis and numerical performance of the LPS-LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The DC-LG method is introduced in Section 5, studying its stability and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We also present in this section several numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Some concluding remarks are written in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We introduce some notation about the functional spaces used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For s ≥ 0 real and real 1 ≤ 2 p ≤ ∞, W s,p(D) denotes the real Sobolev spaces defined on D for scalar real-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' ∥·∥W s,p(D) and |·|W s,p(D) denote the norm and semi-norm, respectively, of W s,p(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' When s = 0, W 0,p(D) := Lp(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For p = 2, the spaces W s,2(D) are denoted by Hs(D), which are real Hilbert spaces with inner product (·, ·)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For s = 0, H0(D) := L2(D), the inner product in L2(D) is denoted by (·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' H1 0(D) is the space of functions of H1(D) which vanish on the boundary ∂D in the sense of trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' H−1 denotes the dual space of H1 0(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The corresponding spaces of real vector-valued functions, v : D → Rd are denoted by W s,p(D)d := {v : D → Rd : vi ∈ W s,p(D), 1 ≤ i ≤ d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Let X be a real Banach space (X, ∥·∥X), if v : (0, T) → X is a strongly measurable function with values in X, we set ∥v∥Lp(0,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='X) = �� t 0 ∥v(τ)∥p X dτ �1/p for 1 ≤ p < ∞, and ∥v∥L∞(0,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='X) = ess sup 0<τ≤t ∥v(τ)∥X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' when t = T, we shall write, unless otherwise stated, ∥v∥Lp(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We shall also use the following discrete norms: ∥v∥lp(X) = � ∆t N � i=1 ∥v(τi)∥p X �1/p , ∥v∥l∞(X) = max 1≤i≤N ∥v(τi)∥X , corresponding to the time discrete space lp(X) ≡ lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' X), 1 ≤ p < ∞, defined as lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' X) := � v : (0, t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' , tN = T) → X : ∥v∥lp(X) < ∞ � , when p = ∞ l∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' X) := � v : (0, t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' , tN = T) → X : max 1≤i≤N ∥v(τi)∥X < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Finally, we shall also use the space of continuous functions such as Cr(D) that denotes the space of r- times continuously differentiable functions on D, when r = 0 we write C(D) instead of C0(D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' the space Cr,1(D), r ≥ 0, of functions defined on the closure of D, r -times continuously differentiable and with the rth derivative being Lipschitz continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' and the space of continuous and bounded functions in time with values in X denoted by C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Throughout this paper, C will denote a generic positive constant which is independent of both the space and time discretization parameters h and ∆t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' C will have different values at different places of appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In many places we shall use, without making any explicit statement, the Cauchy’s inequality ab ≤ ϵ 2a2 + 1 2ϵb2 (a, b > 0, ϵ > 0), and the discrete Gronwall inequality presented in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 2 The Cauchy problem for the advection equation To introduce the LG method we consider the Cauchy problem for the first order linear hyperbolic equation � � � � � ∂c ∂t + u·∇c = 0, x ∈ Rd, t > 0, c(x, 0) = c0(x), (2) where c : Rd × [0, T] → R, u : Rd × [0, T] → Rd is a vector-valued function and c0(x) is a function of compact support defined in a domain D0 ⊂ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Considering the characteristics curves of the first order differential operator D/Dt := ∂/∂t + u·∇ which are the solution to the system of ordinary differential equations � � � � � dX(x,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) dt = u(X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), t), X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' s) = x, (3) we can recast problem (2) as an ordinary differential equation along the characteristics curves, X(x,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), of the form � � � � � Dc Dt = 0 , X(x,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) ∈ Rd, t > 0, c(X(x,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 0), 0) = c0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (4) 3 Assuming that u ∈ C([0, T], W 1,∞(Rd)d), so problem (3) has a unique solution, and c0(x) is sufficiently smooth, we have that the solution of (4) is then given by c(X(·, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t + τ), t + τ) = c(·, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (5) Concerning the solution t → X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) to (3), the following regularity results are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Lemma 1 Assume that u ∈ C([0, T], W k,∞(Rd)d), k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Then for s, t ∈ [0, T], there exists a unique solution t → X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) of (3), such that X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) ∈ W 1,∞(W k,∞(Rd)d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Furthermore, let the multi-index α ∈ N d, then for all α, such that 1 ≤| α |≤ k, DαXi(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) ∈ C([0, T], L∞(Rd × [0, T])), 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Next, we consider the mapping ϕt s : Rd → Rd, defined by ϕt s(x) = X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), since X(X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' s) = x, then it follows that the mapping ϕs t is the inverse of ϕt s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The Jacobian determinant of this transformation J(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) = det �∂Xi(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) ∂xj � , 1 ≤ i, j ≤ d, (6) satisfies the equation ∂J(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) ∂t = J(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t)div u(X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (7) It is easy to see that if Cu := ∥div u∥L∞(D×(0,T )), then exp(−Cu |s − t|) ≤ J(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) ≤ exp(Cu |s − t|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (8) Moreover, for |t − s| sufficiently small it follows that K1 | x − y |≤| X(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) − X(y, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) |≤ K2 | x − y |, (9) where K1 = (1− | s − t | · | ∇u |L∞(L∞(D)d×d)), and K2 = exp(| s − t | · | ∇u |L∞(L∞(D)d×d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Here, | a−b | denotes the Euclidean distance between the points a, b ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Hereafter, for the sake of simplicity, we make the assumption div u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' An important consequence of this assumption is that J(x, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) = 1 almost everywhere in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' However, we must remark that one can easily accommodate the proofs of our results to the general case of div u ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 3 The conventional LG method for the advection equation In the framework of finite elements, Douglas and Russell (1982) and Pironneau (1982) proposed the so called conventional LG method as a time marching algorithm to approximate the solution of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Finite element formulation The realization of this method requires the definition of a family of partitions Dh in a domain D ⊂ Rd sufficiently large, such that given T > 0, D0 ⊂⊂ D and for all t ∈ [0, T] we can assume that c(x, t) = 0 on the boundary ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The partitions Dh generated in the closed region D := D ∪ ∂D are quasi-uniform regular and composed of d-simplices K, the boundaries of which are denoted by ∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' hK denotes the diameter of K and the mesh parameter h := maxK hK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Moreover, we shall assume that u(x, t) is zero on the boundary ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To define the finite element spaces we use the reference simplex �K with vertices {�xi}d+1 i=1 , �K := � �x ∈ Rd : 0 ≤ �xi ≤ 1, 1 − �d i=1 �xi ≥ 0 � , such that for each K ∈ Dh there is an invertible affine mapping FK : �K → K, FK(�x) = BK�x + bK, BK ∈ L(Rd) and bK ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The finite element spaces used in the formulation of the LG method are the following: Wh := � vh ∈ C(D) : ∀K ∈ Dh, vh |K∈ Pm(K) � , and Vh = H1 0(D) ∩ Wh, 4 with Pm(K) = � p(x) : for x ∈ K, p(x) = �p ◦ F −1 K (x), �p ∈ Pm( �K) � , where Pm( �K) denotes the set of polynomials of degree ≤ m defined in �K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Next, we introduce some auxiliary results concerning the approximation properties of the finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For 0 < h < h0 < 1, there exists a constant c1 independent of h such that for w ∈ Hq+1(D) ∩ H1 0(D) and 1 ≤ q ≤ m, inf vh∈Vh {∥w − vh∥ + h ∥∇ (w − vh)∥} ≤ c1hq+1 |w|Hq+1(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (10) Since the partition Dh is quasi-uniformly regular, the following inverse inequality holds: for all wh ∈ Wh and 0 ≤ k ≤ l ≤ 1, and 1 ≤ p ≤ q ≤ ∞, there exists a constant cinv independent of h such that, ∥wh∥W l,q(D) ≤ cinvhd/q−d/p+k−m ∥wh∥W k,p(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (11) Let Πh : C(D) → Wh be the Lagrange interpolation operator in Wh and let Ph : L2(D) → Vh be the orthogonal L2-projector defined as (w − Phw, vh) = 0 for all vh ∈ Vh, (12) then there are constants c2 and c3 independent of h, such that for 0 ≤ σ ≤ m, and 1 ≤ γ ≤ ∞, ∥w − Phw∥ + h ∥∇(w − Phw)∥ ≤ c2hσ+1 |w|Hσ+1(D) (13) and ∥w − Πhw∥Lγ(D) + h ∥∇(w − Πhw)∥Lγ(D) ≤ c3hσ+1 |w|W σ+1,γ(D) , (14) respectively [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It is worth noting that the estimate (14) and the inverse inequality are also valid when the domain D is substituted by an element K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The following properties of the projector Ph are also used in the paper: Phvh = vh ∀vh ∈ Vh, and (contractiveness) ∥Phv∥ ≤ ∥v∥ ∀v ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Let P := 0 = t0 < t1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' < tN = T be a uniform partition of step length ∆t for the interval [0, T], the finite element solution of (2) at time tn, denoted by cn h ∈ Vh, is given by cn h = NP � i=1 Cn i φi, where Cn i := cn h(xi), xi being the ith mesh-point in Dh, NP denotes the number of mesh-points of the partition Dh, and {φi}NP i=1 is the set of global basis functions of Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The conventional LG method calculates cn h ∈ Vh as cn h(x) = Phcn−1 h X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' tn−1), (15) or equivalently, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=', NP , � D cn h(x)φi(x)dx = � D cn−1 h X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' tn−1)φi(x)dx, (16) where X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' tn−1) is the position at time instant tn−1 of the point that at time instant tn is located at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Notations Let us introduce some shorthand notations in order to simplify the writing of the formulas that will appear in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In the sequel, we sometimes use Xn,n−1 or if confusion may arise Xn,n−1(x), to denote X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' tn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Also, let a(x, t) be a generic function defined in Rd×[0, T], then an(x) will denote the value of a(x, t) at time instant tn, that is, an(x) = a(x, tn), whereas a∗n−1(x) denotes an−1◦Xn,n−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Hereafter, we assume that h ∈ (0, h0) and ∆t ∈ (0, ∆t0) with h0 < 1 and ∆t0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 Analysis of the conventional LG method We begin analyzing the L2-norm stability of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Lemma 2 For all N ≥ 1, ��cN h ��2 + N � n=1 ��cn h − c∗n−1 h ��2 = ��c0 h ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' First of all, we show that for any function f ∗n−1(x) := f n−1(X(x, tn, tn−1)) ∈ L2(D) ��f ∗n−1�� = ��f n−1�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (18) To see this is so, we make the change of variable y = Xn,n−1(x) and recall that the Jacobian determinant of this transformation, J(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' tn−1) = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=', then � D ��f ∗n−1(x) ��2 dx = � D ��f n−1(Xn,n−1(x)) ��2 dx = � D ��f n−1(y) ��2 J−1dy = � D |f n(y)|2 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Now, we notice that from (15) it follows that � cn h − c∗n−1 h , cn h � = 0, then using the elementary relation 2(a − b, a) = a2 + (a − b)2 − b2, a, b ∈ R, we obtain that 2 � cn h − c∗n−1 h , cn h � = ∥cn h∥2 + ��cn h − c∗n−1 h ��2 − ��c∗n−1 h ��2 , and by virtue of (18) ∥cn h∥2 + ��cn h − c∗n−1 h ��2 − ��cn−1 h ��2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Summing this expression from n = 1 up to n = N it follows (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Remark 3 Following [15], we can interpret the term �N n=1 ��cn h − c∗n−1 h ��2 as a measure of the numerical dissipation of conventional LG methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It is shown there that N � n=1 ��cn h − c∗n−1 h ��2 ≤ C h4 ∆t N � n=1 ∆t ��∆n−1 h c∗n−1 h ��2 , where ∆n−1 h : H1(D) → W ∗n−1 h := � v∗n−1 h (x) = vn−1 h (Xn,n−1(x)) : vn−1 h (x) ∈ Wh � denotes the discrete Laplacian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' When ∆t = h this amounts to adding an artificial diffusion term to the continuous advection equation of the form Ch3∆c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' so, for sufficiently smooth solutions such an artificial diffusion is not excessive, in particular if one compares with the usual upwind method that adds an artificial diffusion therm of the form −Ch∆c, but it may be insufficient to eliminate the oscillations when the exact solution is not smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To deal with the case of non smooth solutions we introduce the DC-LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The remainder of this section is devoted to the analysis of the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Theorem 4 Let c ∈ L∞(Hm+1(D) ∩ H1 0(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Then there exists a constant C independent of ∆t, h, and n, such that ∥c − ch∥l∞(L2(D)) ≤ ��e0�� + C min � 1, ∆t1/2 ∥u∥L∞(W 1,∞(D)d)) h � hm+1 ∆t1/2 |c|L∞(Hm+1(D)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The error en := cn − cn h can be expressed as en = (cn − Phcn) + (Phcn − cn h) ≡ ρn + θn h, (20) where θn h ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Noting that Phρn = 0, then it follows that ∥en∥2 = (ρn + θn h, ρn + θn h) = ∥ρn∥2 + ∥θn h∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 6 By virtue of (13), ρn satisfies the bound ∥ρn∥Hl(D) ≤ c2hm+1−l |cn|Hm+1(D) (0 ≤ l ≤ m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (21) To estimate en we make use of (5), which implies that cn = c∗n−1, so that ∀vh ∈ Vh (cn − c∗n−1, vh) = 0, so, subtracting (15) from this equation it results the following error equation � en − e∗n−1, vh � = 0, (22) where e∗n−1(x) = cn−1(Xn,n−1(x))−cn−1 h (Xn,n−1(x)) = ρn−1 � Xn,n−1(x) � +θn−1 h � Xn,n−1(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Now, we calculate an error estimate from (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' First, we notice that by virtue of (18) ��e∗n−1�� = ��en−1��, so this property together with the elementary relation 2(a − b)a = a2 + (a − b)2 − b2, permits us to write ∥en∥2 + ��en − e∗n−1��2 − ��en−1��2 = 2 � en − e∗n−1, en� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (23) Now, one needs to estimate the term � en − e∗n−1, en� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For this purpose, we apply the argument of [15], use (20) and set (en − e∗n−1, en) = � en − e∗n−1, ρn� + � en − e∗n−1, θn h � , but by virtue of (22), � en − e∗n−1, θn h � = 0, then the only term we have to estimate is � en − e∗n−1, ρn� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Thus, by the Cauchy-Schwarz inequality � en − e∗n−1, ρn� ≤ 1 4 ��en − e∗n−1��2 + ∥ρn∥2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (24) hence, one can write that 2 � en − e∗n−1, en� ≤ 1 2 ��en − e∗n−1��2 + 2 ∥ρn∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Using this bound on the right hand side of (23) it follows that ∥en∥2 + 1 2 ��en − e∗n−1��2 − ��en−1��2 ≤ 2 ∥ρn∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' From this expression, summing from n = 1 up to n = N one readily obtains that ��eN��2 + 1 2 N � n=1 ��en − e∗n−1��2 ≤ 2T ∆t ∥ρ∥2 l∞(L2(D)) + ��e0��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Then using (21) yields ∥c − ch∥l∞(L2(D)) ≤ C � hm+1 ∆t1/2 |c|L∞(Hm+1(D)) � + ��e0�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (25) For ∆t = O(h), the error is O(hm+1/2), which is of the same order as the streamline-diffusion method [14] for the advection equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' However, this estimate does not allow the convergence of the method when ∆t → 0 independently of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To overcome this trouble, we apply the procedure of [16] to obtain an error estimate valid for all ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' So, substituting en = ρn + θn h, and e∗n−1 = ρ∗n−1 + θ∗n−1 h in (22) and rearranging terms yields � θn h − θ∗n−1 h , vh � = −(ρn − ρn−1, vh) − (ρn−1 − ρ∗n−1, vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (26) Letting vh = θn h we bound each term of this equality as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' First, we notice that ��θ∗n−1 h ��2 = ��θn−1 h ��2 and consequently ∥θn h∥2 + ��θn h − θ∗n−1 h ��2 − ��θn−1 h ��2 = 2 � θn h − θ∗n−1 h , θn h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 7 Second, since for each n, Phρn = 0, then it follows that (ρn − ρn−1, θn h) = Ph(ρn − ρn−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It remains to bound the term 2(ρn−1 − ρ∗n−1, θn h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To do so, we notice that ρn−1 − ρ∗n−1 = � tn tn−1 dρ(X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), tn−1) dt dt, since dρ(X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), tn−1) dt = u(X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), t)·∇Xρ(X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), tn−1), then by the Cauchy-Schwarz inequal- ity we get ��ρn−1 − ρ∗n−1��2 ≤ ∆t � tn tn−1 |u(X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), t) · ∇Xρ(X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t), tn−1)|2 dt, so, letting y = X(x, tn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' t) and denoting by Jt,n the Jacobian determinant J(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' tn) := � ∂X(x,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='tn) ∂x � = 1, it follows that ��ρn−1 − ρ∗n−1��2 ≤ ∆t � D � tn tn−1 ��u(y, t) · ∇ρn−1(y) ��2 � Jt,n−1�−1 dtdy ≤ ∆t ∥u∥2 L∞(L∞(D)d) � tn tn−1 � D ��∇ρn−1(y) ��2 dydt ≤ ∆t2 ∥u∥2 L∞(L∞(D)d) ��∇ρn−1��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Now, using the estimate (21) we can write that ��ρn−1 − ρ∗n−1��2 ≤ ∆t2C � ∆t1/2 ∥u∥L∞(L∞(D)d) h �2 � hm+1 ∆t1/2 �2 |c|2 L∞(Hm+1(D)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (27) Then 2(ρn−1 − ρ∗n−1, θn h) ≤ 2 ∆t ��ρn−1 − ρ∗n−1��2 + ∆t 2 ∥θn h∥2 ≤ ∆tC � ∆t1/2 ∥u∥L∞(L,∞(D)d) h �2 � hm+1 ∆t1/2 �2 |c|2 L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='Hm+1(D)) + ∆t 2 ∥θn h∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Collecting all these bounds we have that ∥θn h∥2+ ��θn h − θ∗n−1 h ��2− ��θn−1 h ��2 ≤ ∆tC � ∆t1/2 ∥u∥L∞(L∞(D)d) h �2 � hm+1 ∆t1/2 �2 |c|2 L∞(Hm+1(D))+ ∆t 2 ∥θn h∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Now, summing from n = 1 up to n = N yields ��θN h ��2 + N � n=1 ��θn−1 h − θ∗n−1 h ��2 ≤ ��θ0 h ��2 + R2 + ∆t 2 N � n=1 ∥θn h∥2 , where R2 = C � ∆t1/2 ∥u∥L∞(L∞(D)d) h �2 � hm+1 ∆t1/2 �2 |c|2 L∞(Hm+1(D)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Since ��eN��2 = ��ρN��2 + ��θN h ��2 and �N n=1 ∥θn h∥2 + ��ρN��2 ≤ �N n=1 ∥en∥2, then it follows that ��eN��2 + N � n=1 ��θn−1 h − θ∗n−1 h ��2 ≤ ��e0��2 + R2 + ∆t N � n=1 ∥en∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Applying Gronwall inequality yields ∥c − ch∥l∞(L2(D)) ≤ C1 � ∆t1/2 ∥u∥L∞(L∞(D)d) h � � hm+1 ∆t1/2 � |c|L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='Hm+1(D)) + ��e0�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (28) Since (25) is valid, then combining it with (28) yields the result (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 Numerical test with the conventional LG method We study the behavior of the conventional LG method considering the rotating hump problem [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The domain D := (−1, 1) × (−1, 1), the velocity field is u = 2π(−x2, x1), and the initial condition c0(x) = � � � cos3( 3 2πr), r ≤ 1/3, 0 otherwise, (29) where r2 = (x1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5)2 + x2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Notice that the function cosp(3πr/2) ∈ Hp(D), p ≥ 1, then p = 3 allows enough smoothness for the optimal estimate of the error when m = 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We show in Figure 1 the isolines of the L2-projected initial condition in a mesh with mesh parameter h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 and the cross section of the exact initial condition c0(x) at x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The purpose of this test is to see how the error behavior −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 1 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 1 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 1 x1 x2 = 0 c0(x) x2 x1 Figure 1: Initial condition of the rotating hump problem in a mesh with size h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 fits Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To do so, we shall mainly focus on the error as a function of the parameter ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Since this theorem is valid under the assumption that the integrals, � D cn−1 h Xn,n−1(x)φi(x)dx, (30) are calculated exactly, then we carry our goal out by using symmetric Gauss quadrature rules of different orders of accuracy to evaluate such integrals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' in doing so, we assess the influence of the order of the quadrature rule on the accuracy and stability of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Figure 2 shows the L2-norm of the error as a function of the time step ∆t in two meshes with h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The errors are calculated after one revolution, T = 1, of the hump using quadrature rules for the Galerkin projection (30) of 7, 16, 25, and 42 points which are exact for polynomials of degree 5, 8, 10, and 14 respectively, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Broken lines correspond to the error function of linear polynomials (m = 1), and full lines to the error function of quadratic polynomials (m = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' By inspection, we notice the following items: (a) for quadrature rules of high order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=', quadrature rules of 16, 25, and 42 points, there is a value ∆ts, such that for ∆t > ∆ts, the error grows with a rate tending toward O(∆t−1/2) as ∆t decreases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' the error tendency of the most accurate rule of 42 points is closer to O(∆t−1/2) than the error tendency of the other two rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' On the other hand, for 0 < ∆t ≤ ∆ts, the error remains almost constant and independent of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (b) For quadrature rules that are not sufficiently accurate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=', the quadrature rule of 7 points, there is a value ∆tins ≫ ∆ts at which the error starts growing very fast as ∆t decreases until it reaches a maximum or eventually the numerical solution may become extremely large at ∆t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For ∆ts < ∆t < ∆t∗ the error decreases and when 0 < ∆t ≤ ∆ts the error remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' This strange behavior of the solution for the quadrature rule of 7 points, which illustrates the dependence of the stability of the LG method upon the order of the quadrature rule, is a well known feature reported by many authors, see for instance [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' in our tests, we note that the instability with quadratic polynomials 9 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 #7 points #16 points #25 points #42 points eL2(T) ∆t m = 2 m = 1 O(∆t−1/2) LG h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 #7 points #16 points #25 points #42 points eL2(T) ∆t O(∆t−1/2) m = 1 m = 2 LG h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025 Figure 2: L2-norm of the error of the conventional LG method in the rotating hump problem, for linear m = 1 and quadratic m = 2 finite elements in two different meshes (h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' sends the numerical solution to infinity in an interval of values of ∆t, whereas for linear polynomials the numerical solution, though useless, remains bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Other relevant results displayed in Figure 2 are the following: (c) provided that the integrals (30) are evaluated with enough accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' the numerical solutions are stable either for large or very small values of ∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' and as Theorem 4 says,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' the error is O(hm+1/∆t1/2) in the first case and O(hm) in the second one,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' with the particularity that in both cases the error does not depend very much upon the order of the quadrature rule used to calculate (30) as long as such a rule is exact for polynomials of degree > 2(m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (d) The error does not grow monotonically, though we notice that the higher the order of the quadrature rule the smoother the growth of the error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' however, we can not explain why the rule of order 8 (16 points) gives for some values of ∆t smaller errors than the rule of order 10 (25 points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 4 The LPS-LG method To formulate the local projection stabilized Lagrange-Galerkin (LPS-LG) method we introduce additional concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Besides the partition Dh, we consider another quasi-uniform regular partition Mh on D the elements of which are termed macro-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Each macro-element M is decomposed into one or more elements K of the partition Dh (the case Mh = Dh is allowed giving place to the so-called one-level LPS approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We assume that there exist positive constants γ1 and γ2 such that for all K ⊂ Dh and M ⊂ Mh, γ1hM ≤ hK ≤ γ2hM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Next, we consider a discontinuous finite element space Gh associated with Mh and set Gh(M) := {qh |M: qh ∈ Gh}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For each M, we use the local L2-projector πM : L2(M) → Gh(M) to define the fluctuation operator κM := id − πM, where id := L2(M) → L2(M) is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In addition to the approximation properties (10)-(14), we make the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Assumption LPS1 Let s ∈ (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' , m − 1) be the degree of the polynomials of the space Gh, the fluctuation operator κM satisfies the approximation property ∥κMw∥L2(M) ≤ Chl M ∥w∥Hl(M) , ∀w ∈ Hl(M), 0 ≤ l ≤ s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (31) Let Ps(M) be the set of polynomials of degree at most s defined in M, then a sufficient condition for the assumption LPS1 to hold is Ps(M) ⊂ Gh(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We set Wh(M) := {wh |M: wh ∈ Wh, wh = 0 on D\\M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Assumption LPS2 There is an interpolation operator jh : H1(D) → Wh, such that for all (w, qh) ∈ H1(D) × Gh, (w − jhw, qh) = 0, (32) and for all w ∈ Hl(D), with 1 ≤ l ≤ m + 1 and M ∈ Mh, ∥w − jhw∥L2(M) + hM ∥∇(w − jhw)∥L2(M) ≤ Chl M ∥w∥Hl(Λ(M)) , (33) where Λ(M) denotes a neighborhood of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 10 The existence of jh has been proven in Part III Chapter 3 of [20] for spaces Gh and Wh that satisfy the following inf-sup condition : inf qh∈Gh(M) sup wh∈Wh(M) (qh, wh)M ∥qh∥L2(M) ∥wh∥L2(M) ≥ β > 0, where β is a constant independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For simplicial meshes the spaces (Wh, Gh) are the following (see, [20] for details): let P disc m,h := {vh ∈ L2(D) : v |K= �v ◦ F −1 K ∈ �Pm( �K) ∀K ∈ Dh} and P disc m,2h := {vh ∈ L2(D) : v |M= �v ◦ F −1 M ∈ �Pm(� M) ∀M ∈ Mh}, where FM : � M → M ∈ Mh is the bijective transformation and � M is the reference element for the partition Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The continuous finite element space Pm,h is defined as Pm,h := P disc m,h ∩ H1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For the one-level approach: Wh = P + m,h := Pm,h + spanK∈Mh{ΦK · Pm−1,h(K)}, and Gh = P disc m−1,h, (34) here, ΦK denotes the mapped bubble function that vanishes on the boundary ∂K of the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For the two-level approach (the elements K ∈ Th are obtained from the elements M ∈ Mh by means of a refinement criterium, see for instance [1] and [9]): Wh = Pm,h, and Gh = P disc m−1,2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (35) Figure 3 illustrates these approaches for d = 2 and simplicial meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' T1 T2 T3 M Wh Gh T Wh Gh T Figure 3: Approximation and projection spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Left panel, the two-level approach with m = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' right panel, the one-level approach with m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The LPS Lagrange-Galerkin method calculates cn h ∈ Vh as solution of the equation (cn h − c∗n−1 h , vh) + ∆tSh(cn h, vh) = 0 ∀vh ∈ Vh, (36) where Sh(cn h, vh) is the stabilization term given by the expression Sh(cn h, vh) = � M τM(κM∇cn h, κM∇vh)M, (37) here, (κM∇cn h, κM∇vh)M := � M κM∇cn h · κM∇vhdx and τM are element-wise constant coefficients that depend on the diameter hM of the macro-elements, their optimal values are determined by the error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Remark 5 For vh = cn h the term Sh(cn h, vh) can be written as a diffusion term of the form Sh(cn h, cn h) = � M τM ∥κM∇cn h∥2 L2(M) := νadd(cn h) ∥∇cn h∥2 , where νadd(cn h) := � � � � � � � � � � M τM ∥κM∇cn h∥2 L2(M) ∥∇cn h∥2 when ∥∇cn h∥ ̸= 0, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Analysis of the LPS-LG method We prove the stability of the LPS-LG method in the mesh dependent norm |||vn||| = � �||vn||2 + ∆t n � j=1 Sh(vj, vj) � � 1/2 , (38) where vj ∈ H1 0(D) (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='., n), n being a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Lemma 6 For all N ≥ 1 it holds ������cN h ������2 + N � i=1 ��ci h − c∗i−1 h ��2 ≤ ∥co h∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (39) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Let vh = cn h in (36), then it follows that ∥cn h∥2 + ��cn h − c∗n−1 h ��2 − ��c∗n−1 h ��2 + 2∆tSh(cn h, cn h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Noting that by virtue of (18), ��c∗n−1 h ��2 = ��cn−1 h ��2, then summing from n = 1 up to n = N ≥ 1 yields ��cN h ��2 + N � n=1 ��cn h − c∗n−1 h ��2 + 2∆t N � n=1 Sh(cn h, cn h) = ��c0 h ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Hence, (39) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Next, we perform the error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To do so, we again decompose the error function en := cn − cn h as en = (cn − Phcn) + (Phcn − cn h) ≡ ρn + θn h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (40) First, we calculate an estimate for ������ρL������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Lemma 7 Let c ∈ L∞(Hm+1(D) ∩ H1 0(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Then, for all 0 ≤ L ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' it follows that there exists a constant C independent of h, ∆t and L such that ������ρL������ ≤ C(h + τ 1/2 max)hm ∥c∥L∞(Hm+1(D)) , (41) where τmax = maxM∈Mh(τM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We recall that ������ρL������2 = ��ρL��2 + ∆t L � n=0 Sh(ρn, ρn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' So, by virtue of (13) it follows that for all L there is a constant independent of h, such that ��ρL�� ≤ Chm+1 |c|L∞(Hm+1(D)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Next, we estimate the term Sh(ρn, ρn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Making use of the triangle inequality, the contractiveness property of the local L2-projector πM, and (13) we obtain that Sh(ρn, ρn) = � M τM ∥κM∇ρn∥2 L2(M) ≤ 4 � M τM ∥∇ρn∥2 L2(M) ≤ Cτmaxh2m ∥c∥2 L∞(Hm+1(D)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (42) Hence, collecting these two estimates the result (41) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We are ready to establish the convergence of the LPS-LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 12 Theorem 8 Under the assumptions of Theorem 4, there exists a constant C independent of h, ∆t and L, such that for all L, 0 ≤ L ≤ N, max 0≤L≤N ������eL������ ≤ ������e0������ + C3 � τ 1/2 max(hm + hs+1) + hm+1 + min � 1, ∆t1/2 ∥u∥L∞(L2(D)) h � � hm+1 ∆t1/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (43) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' From (5) with t = tn and τ = −∆t and (36) we obtain the error equation (en − e∗n−1, vh) − ∆tSh(cn h, vh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (44) Noting that cn h = cn − en, we recast this equation as (en − e∗n−1, vh) + ∆tSh(en, vh) = ∆tSh(cn, vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Next, setting vh = θn h = en −ρn and observing that Sh(a+b, c) = Sh(a, c)+Sh(b, c) this equation becomes (en − e∗n−1, en) + ∆tSh(en, en) = (en − e∗n−1, ρn) +∆tSh(en, ρn) + ∆tSh(en, cn) − ∆tSh(cn, ρn) ≡ 4 � i=1 Ti (45) We estimate the terms Ti of (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Applying Cauchy-Schwarz inequality we have that |T1| = ��(en − e∗n−1, ρn) �� ≤ 1 4 ��en − e∗n−1��2 + ∥ρn∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To estimate T2 we apply again Cauchy-Schwarz inequality and obtain |T2| = ∆t |Sh(en, ρn)| ≤ ∆t (Sh(en, en))1/2 (Sh(ρn, ρn))1/2 ≤ (δ/2)∆tSh(en, en) + (2δ)−1∆tSh(ρn, ρn), 0 < δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Similarly, we have that |T3| ≤ (δ/2)∆tSh(en, en) + (2δ)−1∆tSh(cn, cn), and |T4| ≤ ∆t 2 Sh(cn, cn) + ∆t 2 Sh(ρn, ρn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Substituting these estimates in (45) with δ = 1/2 and noting that 2(en − e∗n−1, en) = ∥en∥2 + ��en − e∗n−1��2 − ��en−1��2 , we obtain that ∥en∥2 + 1 2 ��en − e∗n−1��2 − ��en−1��2 + ∆tSh(en, en) ≤ 2 ∥ρn∥2 + 3∆t (Sh(cn, cn) + Sh(ρn, ρn)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Summing both terms of this inequality from n = 1 up to n = N yields ��eN��2 + 1 2 �N n=1 ��en − e∗n−1��2 − ��e0��2 + ∆t �N n=1 Sh(en, en) ≤ �N n=1 ∥ρn∥2 + 5∆t �N n=1 (Sh(cn, cn) + Sh(ρn, ρn)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Since for any non negative integer n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' ∥ρn∥2 ≤ Ch2(m+1) |c|2 L∞(Hm+1(D)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 13 by virtue of assumption LPS1 Sh(cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' cn) ≤ Cτmaxh2(s+1) |c|2 L∞(Hs+2(D)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (46) and observing that ∥κM∇ρn∥ ≤ 2 ∥∇ρn∥ because πM is contractive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' then (see (42)) Sh(ρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' ρn) ≤ Cτmaxh2m |c|2 L∞(Hm+1(D)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (47) we have that ��eN��2 + 1 2 �N n=1 ��en − e∗n−1��2 − ��e0��2 + ∆t �N n=1 Sh(en,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' en) ≤ C � τmax(h2m + h2(s+1)) + h2(m+1) ∆t � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' or equivalently ������eN������2 + 1 2 N � n=1 ��en − e∗n−1��2 ≤ ����e0����2 + C � τmax(h2m + h2(s+1)) + h2(m+1) ∆t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (48) Hence, it follows that max 0≤L≤N ������eL������ ≤ ������e0������ + C � τ 1/2 max � hm + hs+1� + hm+1 ∆t1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (49) This estimate of the error depends on ∆t−1/2 so that, for any fixed h, is invalid when ∆t → 0 because in this case the method does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' So, in order to get rid of the factor ∆t−1/2 we consider the following approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Starting with the error equation (44) and setting vh = θn h, en = ρn + θn h, e∗n−1 = ρ∗n−1 + θ∗n−1 h and cn h = cn − (ρn + θn h), we get � θn h − θ∗n−1 h , θn h � + ∆tSh(θn h, θn h) = −(ρn − ρ∗n−1, θn h) +∆t (Sh(ρn, θn h) − Sh(cn, θn h)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Now, noticing that ρn − ρ∗n−1 = ρn − ρn−1 − � ρ∗n−1 − ρn−1� and � ρn − ρn−1, θn h � = 0, we can write the above equation as 1 2 ∥θn h∥2 + 1 4 ��θn h − θ∗n−1 h ��2 − 1 2 ��θn−1 h ��2 + ∆tSh(θn h, θn h) ≤ −(ρn−1 − ρ∗n−1, θn h) + ∆t (Sh(ρn, θn h) − Sh(cn, θn h)) ≡ �3 i=1 Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (50) We bound the terms Si on the right hand side of (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Thus, by the Cauchy-Schwarz inequality we have that |S1| ≤ 1 ∆t ��ρn−1 − ρ∗n−1��2 + ∆t 4 ∥θn h∥2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' since, see (27), 1 ∆t ��ρn−1 − ρ∗n−1��2 ≤ ∆tC � ∆t1/2 ∥u∥L∞(Lθ(D)) h �2 � hm+1 ∆t1/2 �2 , then |S1| ≤ ∆tC � ∆t1/2 ∥u∥L∞(Lθ(D)) h �2 � hm+1 ∆t1/2 �2 + ∆t 4 ∥θn h∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (51) To bound the terms S2 and S3 we use the same technique as for the terms T1 and T2 above and obtain |S2| = ∆t |Sh(ρn, θn h)| ≤ (δ/2)∆tSh(θn h, θn h) + (2δ)−1∆tSh(ρn, ρn), and |S3| = ∆t |Sh(cn, θn h)| ≤ (δ/2)∆tSh(θn h, θn h) + (2δ)−1∆tSh(cn, cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 14 Setting δ = 1/2 and substituting these bounds in (50) yields ∥θn h∥2 + 1 2 ��θn h − θ∗n−1 h ��2 − ��θn−1 h ��2 + ∆tSh(θn h, θn h) ≤ ∆tC � ∆t1/2 ∥u∥L∞(Lθ(D)) h �2 � hm ∆t1/2 �2 + 5∆t (Sh(ρn, ρn)) +5∆t (Sh(cn, cn)) + ∆t 2 ∥θn h∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Or equivalently, using (46) and (47), ∥θn h∥2 + 1 2 ��θn h − θ∗n−1 h ��2 − ��θn−1 h ��2 + ∆tSh(θn h, θn h) ≤ ∆tC � τmax(h2m + h2(s+1)) + � ∆t1/2∥u∥L∞(Lθ(D)) h �2 � hm ∆t1/2 �2 � + ∆t 2 ∥θn h∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Summing both sides of this inequality from n = 1 up to n = N and applying Gronwall inequality we obtain that ������θN h ������2 + 1 2 �N n=1 ��θn h − θ∗n−1 h ��2 ≤ ��θ0 h ��2 +C � �τmax(h2m + h2(s+1)) � ∆t1/2 ∥u∥L∞(Lθ(D)) h �2 � hm+1 ∆t1/2 �2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (52) Hence, ������θN h ������ ≤ ������θ0 h ������ + C � τ 1/2 max(hm + hs+1) + � ∆t1/2 ∥u∥L∞(Lθ(D)) h � � hm+1 ∆t1/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (53) Now, noting that ������eN������ ≤ ������θN h ������ + ������ρN������ and ������ρN������ ≤ C(hm+1 + τ 1/2 maxhm), it follows from (53) that ������eN������ ≤ ������e0������ + C � τ 1/2 max(hm + hs+1) + hm+1 + � ∆t1/2 ∥u∥L∞(L2(D)) h � � hm+1 ∆t1/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (54) Thus, since both estimates (54) and (49) hold, then we can write that there exists a constant C such that ������eN������ ≤ ������e0������ + C � τ 1/2 max(hm + hs+1) + hm+1 + min � 1, ∆t1/2 ∥u∥L∞(L2(D)) h � � hm+1 ∆t1/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 Numerical tests with the LPS-LG method We run, under the same premises, the rotating hump problem defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3, although the mesh is now composed of right triangles with legs of length h = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We show in the upper panel of Figure 4 the L2-norm of the error as a function of ∆t for both the two-level LPS-LG method and the conventional LG method for a mesh size h = √ 2 × 10−2 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In these experiments we have calculated the integrals (30) with a quadrature rule of 12 points, which is exact for polynomials of degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The spaces Wh and Gh of the LPS-LG method are those shown in Figure 3, whereas the finite element space for the conventional LG method consists of piecewise quadratic polynomials defined on each one of the 3 triangles that compose the macro-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We observe that the LPS-LG method is more stable than 15 10-6 10-5 10-4 10-3 10-2 t 10-4 10-3 10-2 10-1 100 eL2(T) LG LPS-LG, =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1h LPS-LG, =1h O( t -1/2) # 12 points 10-6 10-5 10-4 10-3 10-2 t 10-4 10-3 10-2 10-1 100 eL2(T) LG LPS-LG, =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1h O( t -1/2) # 16 points Figure 4: L2-norm of the error in the rotating hump problem of the two-level LPS-LG method and the conventional LG method .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' the conventional LG, because the latter goes unstable whereas the LPS-LG method remains stable when τM = τ = h for all M, but it becomes unstable, with an instability region along the ∆t-axis smaller than the one of the LG method, when τM = τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1h for all M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The lower panel of the figure shows that by increasing the order of the quadrature rule the LPS-LG method with τM = τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1h becomes stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Figure 5 displays the L2-norm of the error as a function of ∆t for the one level LPS-LG method, the discrete spaces of which are shown in the right panel of Figure 3, and for the conventional LG method with finite element space Wh = P + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The mesh size of this experiment is h = √ 2 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The solid lines represent the error for the LPS-LG method with τmax = τM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1h and quadrature rules of 16 and 25 points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The dashed lines correspond to the error of the conventional LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We notice in these figures that for ∆t = O(h) and h such that min � 1, ∆t1/2∥u∥L∞(L2(D)) h � = 1, the solutions given by both the LPS-LG and LG methods are very similar, regardless the quadrature rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' This fact agrees with the results of Theorem 4 and Theorem 8, because in this case the dominant term of the error in the conventional LG method is O(hm+1/∆t1/2) = O(hm+1/2), and in the LPS-LG method the dominant term of the error is O(hm+1/∆t1/2+τ 1/2 max×(hm+hs+1)), so letting τ 1/2 max = ch1/2 and s = m−1 one 16 has that the error is also O(hm+1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' However, for ∆t small enough so that min � 1, ∆t1/2∥u∥L∞(L2(D)) h � = ∆t1/2∥u∥L∞(L2(D)) h , the maximum error in the L2-norm for the conventional LG method is O(hm), whereas the maximum error of the LPS-LG method in the mesh dependent norm is also O(hm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' however, since ������eN������2 = ∥e∥2 + ∆t N � n=1 �� M τM(κM∇en, κM∇en)M � , then the L2-error of the LPS-LG method is smaller than the L2-error of the conventional LG method, and this is what we observe in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 LPS-LG #16 points LPS-LG #25 points LG #16 points LG #25 points eL2(T) ∆t O(∆t−1/2) τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1h Figure 5: L2-norm of the error in the rotating hump problem of the LPS-LG method (solid lines) and the LG method (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 5 The DC-LG method Numerical experiments show that when the analytical solution c(x, t) is not sufficiently smooth the LG methods presented in the previous sections are not free from wiggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Following the approach of [15], where the so called shock-capturing characteristic streamline-diffusion method is developed, but scaling the non linear dissipative term as in [18], we formulate a LG method that is stable in the maximum norm with linear finite elements, although numerical experiments show that the method may also be stable with quadratic elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' this stabilization is achieved by adding a non linear dissipative term on the left side of the formulation (16), thus obtaining the so called discontinuity-capturing LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In this method, we calculate cn h as solution of � cn h − c∗n−1 h , vh � + ∆t � K (εK(cn h)∇cn h, ∇vh)K = 0, (55) where (εK(cn h)∇cn h, ∇vh)K := � K εK(cn h)∇cn h · ∇vhdx and εK(cn h) := Cεhα K|R(cn h)||K ≡ Cεhα K ��cn h − c∗n−1 h �� ∆t ����� K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (56) Here, Cε < 1 is a user-defined positive constant, the coefficient α ∈ [1, 2) and |R(cn h)||K denotes the absolute value of the residual, restricted to the element K, generated by the discretization of the material derivative along the characteristic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The existence of a solution of (55) can be proven making use of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 of Chapter IV of [10] as in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Notice that the amount of artificial diffusion is externally controlled by Cε, h and the parameter α, the latter must be less than 2 in order for the method to be stable in the maximum norm when the finite element space is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Analysis of the DC-LG method First, we study the stability of (55) in both the L2 norm and the L∞ norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Lemma 9 For all N ≥ 1, it holds ��cN h ��2 + N � n=1 ��cn h − c∗n−1 h ��2 + 2∆t N � n=1 � K ���εK(cn h)1/2∇cn h ��� 2 K ≤ ��c0 h ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (57) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Letting vh = cn h in (55) and taking into account that ��c∗n−1 h �� = ��cn−1 h ��, it follows that ∥cn h∥2 + ��cn h − c∗n−1 h ��2 − ��cn−1 h ��2 + 2∆t � K ���εK(cn h)1/2∇cn h ��� 2 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Hence, it follows that ��cN h ��2 + N � n=1 ��cn h − c∗n−1 h ��2 + 2∆t N � n=1 � K ���εK(cn h)1/2∇cn h ��� 2 K ≤ ��c0 h ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Lemma 10 There is a constant C independent of h, ∆t, and n, but depending on the constant Cε, such that for all n, ∥cn h∥L∞(D) ≤ (1 + Ch 1 2 (2−α) log( 1 h)) ��c0 h �� L∞(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (58) Noting that for p ≥ 1, ��c∗n−1 h �� Lp(D) = ��cn−1 h �� Lp(D), we can prove this lemma by using the the same arguments as those employed to prove Lemma 6 in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' See also the proof presented in [15] of the stability in the maximum norm for the shock-capturing characteristic streamline-diffusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It is worth remarking that maximum norm stability has only been proven for linear finite elements, because this proof makes use of a result of [21], which says that there is a constant c independent of p = 2a, a = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=', such that for all wh ∈ Wh � Rd ∇wh · ∇Πh(wh)p−1dx = c p2 � K � K |∇wh| (wh)p−2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' And this result has only been proven for linear finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' However, via numerical examples, we have observed that the maximum norm stability also holds in cases where the solution exhibits strong discontinuities for quadratic elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For the error analysis we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Theorem 11 Let c ∈ L∞(Hm+1(D) ∩ H1 0(D)) ∩ L∞(W 1,∞(D)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Then, there exists a constant C inde- pendent of ∆t, h and n, but depending on |D|, |c|L∞(Hm+1(D)) and ∥∇c∥L∞(L∞(D)), such that ∥c − ch∥l∞(L2(D)) ≤ C(hm+1 + Cεhα) ∆t1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (59) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The error equation is � en − e∗n−1, vh � − ∆t � K (εK(cn h)∇cn h, ∇vh)K = 0 (60) Noting that cn h = cn − en we have that ∆t � K(εK(cn h)∇cn h, ∇vh)K = ∆t � K (εK(cn h)∇cn, ∇vh)K −∆t � K (εK(cn h)∇en, ∇vh)K, 18 so we can write the error equation as � en − e∗n−1, vh � + ∆t � K (εK(cn h)∇en, ∇vh)K = ∆t � K (εK(cn h)∇cn, ∇vh)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (61) Now, setting in this equation vh = θn h = en − ρn, where ρn = cn− Πhcn and θn h = Πhcn − cn h, we get � en − e∗n−1, en� + ∆t � K(εK(cn h)∇en, ∇en)K = � en − e∗n−1, ρn� +∆t � K(εK(cn h)∇en, ∇ρn)K +∆t � K(εK(cn h)∇en, ∇cn)K −∆t � K(εK(cn h)∇cn, ∇ρn)K ≡ �4 i=1 Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (62) We estimate the terms Ri on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Thus, regarding R1 we apply the Cauchy-Schwarz inequality to obtain that |R1| = ��� en − e∗n−1, ρn��� ≤ 1 8 ��en − e∗n−1��2 + 2 ∥ρn∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' As for the term R2, we use the same inequality to get |R2| ≤ (δ/2)∆t � K (εK(cn h)∇en, ∇en)K + (2δ)−1∆t � K (εK(cn h)∇ρn, ∇ρn)K, 0 < δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Similarly |R3| ≤ (δ/2)∆t � K (εK(cn h)∇en, ∇en)K + (2δ)−1∆t � K (εK(cn h)∇cn, ∇cn)K, and |R4| ≤ 1 2∆t � K (εK(cn h)∇cn, ∇cn)K + 1 2∆t � K (εK(cn h)∇ρn, ∇ρn)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Substituting this estimates in (62) with δ = 1/2 yields ∥en∥2 + 3 4 ��en − e∗n−1��2 − ��en−1��2 + ∆t � K(εK(cn h)∇en, ∇en)K ≤ 4 ∥ρn∥2 + 3∆t � K(εK(cn h)∇cn, ∇cn)K +3∆t � K(εK(cn h)∇ρn, ∇ρn)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (63) Next, we have to estimate the last two terms on the right hand side of this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' ∆t � K(εK(cn h)∇cn, ∇cn)K = Cε∆t � K hα K � K ��cn h − c∗n−1 h �� ∆t (∇cn)2 dK ≤ Cε ∥∇cn∥2 L∞(D) � K hα K � K ��cn h − c∗n−1 h �� dK ≤ Cε ∥∇cn∥2 L∞(D) � K hα K ��cn h − c∗n−1 h �� L1(K) ≤ Cε ∥∇cn∥2 L∞(D) hα ��cn h − c∗n−1 h �� L1(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It remains to estimate the term ��cn h − c∗n−1 h �� L1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To do so, we observe that cn h − c∗n−1 h = (cn − en) − � c∗n−1 − e∗n−1� = en − e∗n−1 19 because cn = c∗n−1, then ��cn h − c∗n−1 h �� L1(D) = ��en − e∗n−1�� L1(D) and by virtue of the Cauchy-Schwarz inequality ��en − e∗n−1�� L1(D) ≤ CD ��en − e∗n−1�� L2(D), CD = C(|D|), |D| being the measure of D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' hence Cε ∥∇cn∥1 L∞(D) hα ��cn h − c∗n−1 h �� L1(D) ≤ 1 16 ��en − e∗n−1��2 L2(D) + 4C2 εh2αC2 D ∥∇cn∥4 L∞(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Therefore, we can set that ∆t � K (εK(cn h)∇cn, ∇cn)K ≤ 1 16 ��en − e∗n−1��2 L2(D) + 4C2 εh2αC2 D ∥∇cn∥4 L∞(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (64) We estimate now the term ∆t � K(εK(cn h)∇ρn, ∇ρn)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To this end, we notice that ∆t � K (εK(cn h)∇ρn, ∇ρn)K = Cε � K hα K � K ��cn h − c∗n−1 h �� (∇ρn)2 dK, but by virtue of (14) we have that ∥∇ρn∥L∞(D) ≤ c3 ∥∇cn∥L∞(D), then we can write that ∆t � K (εK(cn h)∇ρn, ∇ρn)K ≤ c2 3Cε ∥∇cn∥2 L∞(D) � K hα K � K ��cn h − c∗n−1 h �� dK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' so, arguing as we have just done for the term ∆t � K(εK(cn h)∇cn, ∇cn)K it follows that ∆t � K (εK(cn h)∇ρn, ∇ρn)K ≤ 1 16 ��en − e∗n−1��2 L2(D) + 4c4 3C2 εh2αC2 D ∥∇cn∥4 L∞(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (65) Substituting (64) and (65) in (63) and using the estimate (14) yields ∥en∥2 + 1 2 ��en − e∗n−1��2 − ��en−1��2 + ∆t � K(εK(cn h)∇en, ∇en)K ≤ C � h2(m+1) + C2 εh2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' where C is a constant that depends on CD, |cn|Hm+1(D) and ∥∇cn∥L∞(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Summing both terms of this inequality from n = 1 up to n = N it follows that ��eN��2 + 1 2 �N n=1 ��en − e∗n−1��2 + ∆t �N n=1 � K(εK(cn h)∇en, ∇en)K ≤ C ∆t � h2(m+1) + C2 εh2α� , or equivalently ��eN�� ≤ C � hm+1 + Cεhα� ∆t1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Remark 12 This estimate depends on ∆t−1/2 so that for h fixed blows up as ∆t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In contrast with the previous LG methods, for the DC-LG method we have not been able to find an error estimate free from the ∆t−1/2 dependence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' however, based on numerical experiments and assuming that the maximum norm stability holds, we may hypothesizes that there is a ∆tc, such that for ∆t ≤ ∆tc the error will not increase, remaining nearly constant or decreasing very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' So, noting that cn h − c∗n−1 h = en − e∗n−1 because cn = c∗n−1, we can argue that for ∆t ≤ ∆tc max n max K ��en − e∗n−1�� ∆t ����� K = β, β being a small constant that depends on m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' hence, we can consider that εK(cn h) is a constant, specifically, for all K and n we set ν := εK(cn h) = Cεβhα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 20 Then, the error equation can be written now as � en − e∗n−1, vh � − ∆tν � K (∇cn h, ∇vh)K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (66) So, as we have done above, we let vh = θn h, en = ρn + θn h, e∗n−1 = ρ∗n−1 + θ∗n−1 h and cn h = cn − (ρn + θn h), with ρn = cn − Phcn, and recast (66) as � θn h − θ∗n−1 h , θn h � + ∆tν∇θn h, ∇θn h) = −(ρn − ρ∗n−1, θn h) −∆tν(∇ρn, ∇θn h) + ∆tν(∇cn, ∇θn h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (67) If we compares this equation with (26), we can consider that the artificial dissipation terms represent a perturbation to the equation of the pure advection problem, so, we can expect that when ν → 0 (67) will yield the same estimate as (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' To check that this is the case, we bound the terms −(ρn − ρ∗n−1, θn h), (∇ρn, ∇θn h) and (∇cn, ∇θn h) as we have done many times before and can easily arrive to the estimate ��θN h �� ≤ C � hm + ν1/2� , where the constant C depends on |c|L∞(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='Hm+1(D)), and consequently, ��eN�� = O � hm + (Cεβhα)1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' So, if Cεβ is so small that (Cεβhα)1/2 ≤ hm, then ��eN�� = O(hm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 Numerical tests with the DC-LG method Since the method is designed to deal with discontinuous initial conditions, we shall perform two numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The first one is again the hump problem to see wether the error behaves according to Theorem 11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' the second test uses as initial condition the so called “slotted” cylinder, this a typical initial condition to study the ability of schemes to deal with strong discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 The hump test We run the test under the same conditions as the numerical test for the conventional LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We show in Figures 6 and 7 the results for the meshes with mesh parameter h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025 after one revolution, and with the constants Cε and α of the expression for the artificial diffusivity (56) taking the values Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01, Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 and α = 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' These results must be compared with those of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We notice the following facts: (a) For high order quadrature rules, the error of the DC-LG method shows a similar, but smoother, behavior as the error of the conventional LG method, with the feature that the higher the constant Cε or the coarser the mesh the smoother the profile of the error curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' this is a consequence of the nonlinear artificial diffusivity that depends on both Cε and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (b) For the low order quadrature rule of 7 points, the DC-LG method loses accuracy for those values ∆t for which the conventional LG method is inaccurate or even unstable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' in fact, for Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025 there is an interval of values ∆t, which, roughly speaking, corresponds with those values for which the conventional LG method with quadratic polynomials becomes unstable, in which the DC-LG method with quadratic polynomials is less accurate than with linear polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' This can be explained because when both Cε and h are low the artificial diffusivity is not sufficiently strong to prevent the instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (c) Roughly speaking, we can say that the higher the artificial viscosity the less sensitive the DC-LG method is to the order of the quadrature rules, provided that such rules are exact for polynomials of degree 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (d) For ∆t = O(h) or ∆t = o(h2), all the quadrature rules give about the same solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' This means that in those ranges of values ∆t it is not necessary the use of high order quadrature rules, just a rule which is exact for polynomials of degree 2(m+1) would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' (e) Looking at the profiles of the error curves, we notice that 21 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 #7 points #16 points #25 points #42 points eL2(T) ∆t m = 2 m = 1 O(∆t−1/2) DC-LG h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 #7 points #16 points #25 points #42 points eL2(T) ∆t m = 2 O(∆t−1/2) m = 1 DC-LG h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Figure 6: L2-error norm with the DC-LG method in the rotating hump problem for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 for high order quadrature rules the error behaves as Theorem 11 says, that is, there is a value ∆tc (in this test, ∆tc = O(h−2)), such that for ∆t ≥ ∆tc the error is O(hm+1 + Cεhα)/∆t1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' However, for ∆t < ∆tc, the error does not grow and remains more or less constant, particularly as the artificial diffusivity is high enough, see Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Finally, fixing the mesh and the parameter α, this test shows that as the constant Cε becomes smaller and smaller, the DC-LG solution approaches the solution of conventional LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 #7 points #16 points #25 points #42 points eL2(T) ∆t m = 2 m = 1 O(∆t−1/2) DC-LG h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 10−5 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 100 #7 points #16 points #25 points #42 points eL2(T) ∆t m = 2 O(∆t−1/2) m = 1 DC-LG h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Figure 7: L2-error norm with the DC-LG method in the rotating hump problem for h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 The slotted cylinder Our second test is the so called slotted cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The idea behind this test is to assess the ability of the DC-LG method to deal with strong discontinuities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' specifically, we wish to see how the scheme smears out an initial condition that is strongly discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The domain D := [−1, 1] × [−1, 1], the velocity field u is the same as in the previous tests, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=', u = 2π(−x2, x1), and the initial condition is a cylinder of height 1 and radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='25 centered at (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5,0), with a slot along the plane x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 and depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The simulations are carried out with a time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 in the mesh with mesh parameter h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025, and the numerical initial condition being computed by the L2-projection onto the finite element space Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Although we are aware that this is not a good way to calculate the numerical initial condition because, as we see in Figure 9, some overshoots and undershoots are generated by the L2-projection, we have left it to test the capability of DC-LG method to suppress the wiggles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' it is clear that the method is able to kill them out after few time steps when the constant Cε of the artificial diffusion εh(cn h) is Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' A better approach to calculate the numerical initial condition would have been to perform L2-projection of 22 the exact initial condition with linear elements and lumped mass matrix, yielding this way a somewhat smoother initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The integration time T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' For the results, we have used the exact trajectories and the integrals (30) have been calculated with the quadrature rule of 16 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Based on the results of the hump test, we know that for the values ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='025 the solution is not sensitive to the order of the quadrature rules used to approximate the integrals (30), provided that the rule is exact for polynomials of degree > 2(m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 exact Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 1 0 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 y = 0 y x y = 0 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 m = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Figure 8: Slotted cylinder after one revolution for linear finite elements m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Upper panel: three dimensional view of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Lower panel: the level lines (on the left) and cross sections (on the right) that correspond with the figures of the upper panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We display in the upper panel of Figure 8 a three dimensional view of the cylinder after one revolution, whereas in the low panel are represented the level lines (on the left) and cross sections (on the right) when the constant of the artificial diffusion takes the values Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 and Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' This solution has been calculated with linear polynomials (m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We notice that the width of the upper face of the lobes and the width of the “bridge” as well as the depth of the slot are reasonably well preserved for both constants Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 and Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It is worth remarking that the figures of the upper and middle panel with Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 compare very well with those obtained in [15] and [11] applying the shock-capturing streamline- diffusion method with a time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 and the mesh size h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 times smaller than the one we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It is clear that with the constant Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 the DC-LG method introduces a major degree of smearing, and when Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 the method is not able to suppress the wiggles generated around the discontinuities at the first time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Similar representations of the numerical solution calculated with quadratic polynomials (m=2) are displayed in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' If we compare these graphs with those of Figure 8 one sees that it is clear the improvement of the numerical solution calculated with quadratic polynomials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' for instance, the slopes of the cylinder sides, the width of the lobes of the upper face and the width of the “bridge” are much better represented with quadratic elements than with linear elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 23 Ce = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='010 0 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 m= 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 1 exact Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 1 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 y = 0 y x m = 2 y = 0 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 Figure 9: Slotted cylinder after one revolution for quadratic finite elements m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Upper panel: three dimensional view of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Lower panel: the level lines (on the left) and cross sections (on the right) that correspond with the figures of the upper panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Finally, we represent in Figure 10 the time evolution of the maximum and minimum of the numerical solutions obtained by the conventional LG method, and the DC-LG one (with Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 and Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' As we commented above, our calculation of the numerical initial condition allows the generation of wiggles at the first time step, in fact, the largest amplitude of such wiggles is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' The DC-LG method with Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 dissipates these wiggles as the solution progresses, such that the for m = 2 the dissipation is very strong at the beginning, going very quickly the minimum to zero and the maximum to 1, as, on the other hand, should be;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' however, when Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01, the wiggles are also dissipated, but at a slower rate, with the amplitudes of the minimum and maximum values decreasing somewhat oscillatorily, tending to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='05 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' However, though we proof that linear polynomials are stable in the maximum norm, the behavior of the maximum and minimum is not as good as that of quadratic elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' for instance, when Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 the dissipation of the amplitude of the wiggles is slower and less strong than in the case of quadratic elements, noting that the steady maximum and minimum are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='03 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='03 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' when Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 the maximum and minimum of DC-LG solution, though smaller in amplitude, exhibit a similar oscillatory behavior as those of the conventional LG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' It is remarkable that both the maximum and the minimum of the conventional LG method, either with m = 1 or m = 2, undergo dissipation at the beginning of the calculations and then go on exhibiting an oscillatory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 6 Concluding remarks 1) We have obtained a new error estimate of the conventional LG method for the advection equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' In contrast with previous estimates, ours is valid for all ∆t, no matter how small ∆t is, showing that 24 7 Ce = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='010 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 m=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='9 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0 LG DC-LG Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 DC-LG Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 min(ch) max(ch) m = 1 t 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='9 1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 0 LG DC-LG Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='01 DC-LG Cε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content='1 max(ch) m = 2 t min(ch) Figure 10: Evolution with time of the maximum and minimum of the slotted cylinder during one revolution for linear m = 1 and quadratic m = 2 finite elements for ∆t ≤ Khp, p > 2, the error is O(hm), and for ∆t > Khp the error is O(hm+1/∆t1/2), here K = (∥u∥L∞(L∞(D)d))−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' This error estimate has been obtained under the assumption that the integrals � K φj(Xh(x, tn+1, tn))φi(x)dx are calculated exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 2) To validate our theoretical result we perform numerical tests using quadrature rules of different orders to evaluate those integrals and calculating exactly the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We find that the higher the order of the quadrature rule the closer the error behavior to the theoretical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Other interesting finding is that for ∆t = O(h) and ∆t = O(h3), the error is quite independent of the order of the quadrature rule as long as the rule calculates exactly polynomials of degree ≥ 2(m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 3) The LG approach is a natural way of introducing upwinding in the numerical method, but the degree of upwinding is not strong enough if the initial condition lacks regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' One way of stabilizing the conventional LG method is using the so called local projection stabilization technique, which is symmetric and acts on the small unresolved scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' We thus obtain the so called LPS-LG method and estimate its error in a mesh dependent norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' 4) Neither the LPS-LG nor the conventional LG methods are stable in the maximum norm, so they do not deal satisfactorily with strongly discontinuous initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Following the idea of shock-capturing characteristic streamline-diffusion method of [15], we have formulated the DC-LG method that is a residual stabilized LG method, which for linear finite elements is stable in both the L2- and L∞-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' This method has shown to be effective in preserving the shape of the initial condition, in particular, when quadratic elements are used, though there is no theoretical proof of the stability in the infinite norm for these elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Finally, we must say that this dependence of the error behavior on the CFL number of the LG methods is not exclusive for the pure advection problem, it can also be proven for advection-dominated and NS problems, see [5], [4] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Acknowledgements This research has been partially funded by grant PGC-2018-097565-B100 of Ministerio de Ciencia, Inno- vaci´on y Universidades of Spain and of the European Regional Development Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} +page_content=' Braack and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE1T4oBgHgl3EQfywXe/content/2301.03438v1.pdf'} 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a/h9FLT4oBgHgl3EQfaS_3/content/tmp_files/2301.12074v1.pdf.txt b/h9FLT4oBgHgl3EQfaS_3/content/tmp_files/2301.12074v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a643285d5260dcd11d5ed377c7da47df84236685 --- /dev/null +++ b/h9FLT4oBgHgl3EQfaS_3/content/tmp_files/2301.12074v1.pdf.txt @@ -0,0 +1,838 @@ +Comparing Intrinsic Gender Bias Evaluation Measures +without using Human Annotated Examples +Masahiro Kaneko1 +Danushka Bollegala2,3∗ +Naoaki Okazaki1 +1Tokyo Institute of Technology +2University of Liverpool +3Amazon +masahiro.kaneko@nlp.c.titech.ac.jp +danushka@liverpool.ac.uk +okazaki@c.titech.ac.jp +Abstract +Numerous types of social biases have been +identified in pre-trained language models +(PLMs), and various intrinsic bias evaluation +measures have been proposed for quantifying +those social biases. Prior works have relied on +human annotated examples to compare exist- +ing intrinsic bias evaluation measures. How- +ever, this approach is not easily adaptable to +different languages nor amenable to large scale +evaluations due to the costs and difficulties +when recruiting human annotators. To over- +come this limitation, we propose a method to +compare intrinsic gender bias evaluation mea- +sures without relying on human-annotated ex- +amples. Specifically, we create multiple bias- +controlled versions of PLMs using varying +amounts of male vs. +female gendered sen- +tences, mined automatically from an unanno- +tated corpus using gender-related word lists. +Next, each bias-controlled PLM is evaluated +using an intrinsic bias evaluation measure, and +the rank correlation between the computed +bias scores and the gender proportions used +to fine-tune the PLMs is computed. Experi- +ments on multiple corpora and PLMs repeat- +edly show that the correlations reported by our +proposed method that does not require human +annotated examples are comparable to those +computed using human annotated examples in +prior work. +1 +Introduction +Pre-trained language models (PLMs) trained on +large datasets have reported impressive perfor- +mance improvements in various NLP tasks (Devlin +et al., 2019; Lan et al., 2019) greatly. However, +these PLMs also demonstrate significantly worry- +ing levels of social biases (Bolukbasi et al., 2016; +Kurita et al., 2019). To address this issue, numerous +∗Danushka Bollegala holds concurrent appointments as +a Professor at University of Liverpool and as an Amazon +Scholar. This paper describes work performed at the Univer- +sity of Liverpool and is not associated with Amazon. +intrinsic bias evaluation measures for PLMs have +been proposed (Nangia et al., 2020; Dhamala et al., +2021; Nadeem et al., 2021; Kaneko and Bollegala, +2022; Zhou et al., 2022), which are also used for +comparing debiasing methods for PLMs (Webster +et al., 2020; Kaneko and Bollegala, 2021a; Schick +et al., 2021). +Existing bias evaluation methods use different +criteria such as pseudo likelihood (Kaneko and Bol- +legala, 2022), cosine similarity (Caliskan et al., +2017; May et al., 2019), inner-product (Ethayarajh +et al., 2019) etc. Moreover, current bias evalua- +tion methods require manually-annotated datasets +containing stereotypical and antistereotypical ex- +amples that express different types of social biases +(Nangia et al., 2020; Nadeem et al., 2021). There- +fore, we consider that it is important to compare +the differences in existing bias evaluation measures +proposed for PLMs (Orgad and Belinkov, 2022; +Dev et al., 2021; Kaneko et al., 2022a) to under- +stand their relative strengths and weaknesses. +To objectively compare the existing bias evalu- +ation measures, Kaneko and Bollegala (2022) cal- +culated the rank correlation between the number +of human annotators who labelled an example to +be stereotypically biased towards a protected at- +tribute in Crowds-Pairs (CP), and the bias score for +that example returned by an intrinsic bias evalua- +tion measure (Nangia et al., 2020; Nadeem et al., +2021). However, due to the costs and difficulties in +recruiting human annotators, this approach cannot +be easily adapted to different languages, accommo- +date large-scale evaluations, or compare evaluation +metrics that do not use human-annotated data. +We propose a method to compare intrinsic bias +evaluation measures without using human anno- +tated examples. Figure 1 outlines the intuition be- +hind our proposed method. First, we train bias- +controlled versions of PLMs obtained via fine- +tuning a PLM on male and female gendered sen- +tences, automatically mined from an unannotated +arXiv:2301.12074v1 [cs.CL] 28 Jan 2023 + +Figure 1: Overview of our proposed method. +We +first create bias-controlled PLMs by fine-tuning a PLM +on male and female gendered sentences that are auto- +matically mined from unannotated corpora. Next, we +measure the rank correlation between the scores re- +ported by an intrinsic bias evaluation measure and the +male/female bias rates (r) used to fine-tune the PLMs. +corpus using a gender-related word list. We de- +fine rate of bias (r) as the ratio between male and +female gendered sentences in a training sample +used to fine-tune a PLM. A PLM fine-tuned mostly +on male sentences is likely to generate sentences +containing mostly male words, while a PLM fine- +tuned on female sentences is likely to generate sen- +tences containing mostly female words (Kaneko +and Bollegala, 2022; Kaneko et al., 2022c). There- +fore, an accurate intrinsic bias evaluation measure +is expected to return a score indicating a bias to- +wards the male gender for a male bias-controlled +PLM, while it is expected to return a score indicat- +ing a bias towards the female gender for a female +bias-controlled PLM. We then compute the rank +correlation between (a) the rate of biases in the +bias-controlled PLMs, and (b) the bias scores re- +turned by an intrinsic evaluation measure for the +corresponding PLMs, as a measure of accuracy of +the bias evaluation measure. +Our experiments with multiple corpora and +PLMs show that the correlations reported by our +proposed method, which does not require human +annotated examples, are comparable to those com- +puted using human annotated examples in previ- +ous studies. Furthermore, by examining the out- +put probabilities of the PLM, we verify that the +proposed method, which fine-tunes bias-controlled +PLMs with varying amounts of male vs. female +sentences, is indeed able to control biases associ- +ated with male and female gender directions. +2 +Bias-controlled Fine-Tuning +The imbalance of gender words in the training data +affects the gender bias of a PLM fine-tuned us- +ing that data (Kaneko and Bollegala, 2022; Kaneko +et al., 2022c). Using this fact, we propose a method +to learn bias-controlled versions of PLMs that ex- +press different levels of known gender biases. Let +us first assume that we are given a list of female +gender related words Vf (e.g. she, woman, female), +and a separate list of male gender related words +Vm (e.g. he, man, male). Next, we select sen- +tences that contain either at least one of female or +male words from an unannotated set of sentences +D. Sentences that contain both male and female +words are excluded here. Let us denote the set of +sentences extracted for a female or a male word w +by Φ(w). Moreover, let Df = � +w∈Vf Φ(w) and +Dm = � +w∈Vm Φ(w) be the sets of sentences con- +taining respectively female and male words. We ap- +propriately downsample Df and Dm to have equal +numbers of sentences N (i.e. |Df| = |Dm| = N). +Next, we create training datasets Dr by varying +the rate of bias, r (∈ [0, 1]), by randomly sampling +a subset Sr(Dm) of Nr sentences from Df and +a subset S1−r(Df) of N(1 − r) sentences from +Dm such that Dr = Sr(Dm) ∪ S1−r(Df). When +r = 0, Dr consists of only female sentences (i.e. +Dr ⊆ Df), and when r = 1, it consists of only +male sentences (i.e. Dr ⊆ Dm). To obtain mul- +tiple bias-controlled PLMs at different levels of +gender biases, we fine-tune a given PLM on differ- +ent datasets, Dr, sampled with different values of +r. We use a given intrinsic bias evaluation measure +to separately evaluate each bias-controlled PLM. +Finally, we measure the agreement between the +bias scores reported by the intrinsic bias evaluation +measure under consideration and the correspond- +ing rates of biases of those PLMs using Pearson’s +rank correlation coefficient. +3 +Experiments +3.1 +Settings +In our experiments, we used the female words +she, woman, female, her, wife, mother, girl, sister, +daughter, girlfriend as Vf, and male words he, man, +male, him, his, husband, father, boy, brother, son, +boyfriend as Vm. We sampled 2M sentences each +representing male and female genders from News +crawl 2021 corpus (news)1 and BookCorpus (Zhu +1https://data.statmt.org/news-crawl/en/ + +Bias Controled +Fine-tuning +Corpus +Evaluation +Male +Female +1.0 +0.0 +PLM ++7.8 +0.9 +0.1 +PLM ++7.0 +Bias +... +Eval +0.1 +0.9 +PLM +-6.2 +0.0 +1.0 +PLM +-6.9 +Bias +Intrinsic +Rank Correlation +Rate r +Bias ScoreMeasure +BERT +ALBERT +news +book +HA +news +book +HA +TBS +0.14 +0.09 +- +0.25 +0.14 +- +SSS +0.22 +0.22 +0.45 +0.31 +0.22 +0.53 +CPS +0.30 +0.27 +0.57 +0.37 +0.22 +0.48 +AUL +0.37 +0.32 +0.68 +0.55 +0.36 +0.56 +AULA +0.42 +0.34 +0.71 +0.60 +0.42 +0.57 +Table 1: Peason correlation between biased PLM order +and each bias scores. News and book represent the cor- +pus used for biasing, respectively. HA is AUC value +of method using human annotation (Kaneko and Bolle- +gala, 2021a). +et al., 2015) (books) for training bias-controlled +PLMs and a separate 100K sentences as devel- +opment data. +We used BERT2 (Devlin et al., +2019) and ALBERT3 (Lan et al., 2019) as the +PLMs. We fine-tune PLMs with masked language +model learning. +We use publicly available +Transformer library4 to fine-tuning PLMs, and +all hyperparameters are set to their default values. +We trained 11 bias-controlled PLMs for r in +{0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0} +on four Tesla V100 GPUs. +3.2 +Intrinsic Bias Evaluation Measures +We compare five previously proposed intrin- +sic gender bias evaluation measures in this pa- +per: Template-Based Score (TBS; Kurita et al., +2019), StereoSet Score (SSS; Nadeem et al., +2021), CrowS-Pairs Score (CPS; Nangia et al., +2020), All Unmasked Likelihood (AUL; Kaneko +and Bollegala, 2022), and AUL with Attention +weights (AULA; Kaneko and Bollegala, 2022). +Further details of these measures are given in the +Appendix. +Note that TBS uses templates for evalua- +tion and cannot be used with human-annotated +stereotypical/anti-stereotypical sentences. On the +other hand, SSS, CPS, AUL, and AULA all require +human-annotated sentences that express social bi- +ases. +3.3 +Comparing Intrinsic Gender Bias +Evaluation Measures +We compare the proposed method and Kaneko and +Bollegala (2022)’s method using CP dataset, which +has human annotations, and show the effectiveness +2https://huggingface.co/bert-base-uncased +3https://huggingface.co/albert-base-v2 +4https://github.com/huggingface/transformers/ +tree/v4.22.2 +of the proposed method. In addition, we will use +several PLMs and corpora to analyze the trends of +the proposed method. Table 1 shows the correla- +tion results of the proposed method for TBS, SSS, +CPS, AUL, and AULA when fine-tuning BERT and +ALBERT on news or book corpora, respectively. +HA is the AUC value of the Kaneko and Bollegala +(2022)’s method using human annotations. Since +TBS uses templates, it cannot be evaluated using +HA. +For BERT, the proposed method induces the +same order among measures (i.e. AULA > AUL > +CPS > SSS) as done by HA in both news and book. +For ALBERT, only the rankings of SSS and CPS +differ between the proposed method and HA. These +results show that the proposed method and the ex- +isting method that use human annotations rank the +intrinsic gender bias evaluation measures in almost +the same order.5 It can be seen that the values of +the correlation coefficients vary depending on the +PLM and corpus. For example, ALBERT has a +maximum correlation of 0.60, while BERT has a +maximum correlation of only 0.42. +A major limitation of human annotation-based +evaluation is that it cannot be used to compare TBS +that does not human annotated examples against +other intrinsic bias evaluation measures. However, +our proposed method does not have this limitation +and can be used to compare TBS against other +bias evaluation measures. As it can be seen from +Table 1, TBS consistently reports the lowest corre- +lations, indicating that it is not an accurate intrin- +sic gender bias evaluation measure. This finding +agrees with Kaneko et al. (2022a), who highlighted +the inadequacy of templates as a method for evalu- +ating social biases. +3.4 +Bias-controlled PLMs +To verify that the proposed method can indeed con- +trol the bias of a PLM, we investigate the variation +of the output probabilities of the PLMs fine-tuned +with different r. Specifically, we investigate the +output probabilities of masked he and she in the +input text “[MASK] is a/an [Occupation].” for +the bias-controlled PLMs. For [Occupation], we +use gender- and stereotype-neutral occupational +words6 (e.g. writer, musician) from the word list +created by Bolukbasi et al. (2016). When r in- +5Because of the different methods of measuring correla- +tions, it is not possible to compare the magnitude of values +between the proposed and existing methods. +6https://github.com/tolga-b/debiaswe + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +rate of bias +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +probability +bert (he) +bert (she) +albert (he) +albert (she) +Figure 2: Average output probabilities for “[MASK] +is a/an [Occupation]” produced by the bias-controlled +BERT and ALBERT PLMs fine-tuned with different r +on the news dataset. +creases, a PLM will be fine-tuned with increasing +amounts of male sentences. Therefore, if the av- +erage probability of he increases with r, it would +imply that the PLMs are correctly bias-controlled +by the proposed method. +Figure 2 shows that the average output probabili- +ties of he and she when r is incremented in step size +of 0.1. When r = 1 the PLM predicts he with fairly +high probability and when r = 0 the PLM predicts +she with fairly high probability. Furthermore, when +r = 0.5, the probability of he and she is almost 0.5. +Original BERT (without fine-tuning) returns 0.48 +and 0.28, respectively for he and she, while the +corresponding probabilities returned by ALBERT +are respectively 0.64 and 0.22. Both the original +BERT and ALBERT predict relatively larger out- +put probabilities for he, indicating that they are +male-biased, without performing any bias-control. +From these results, it can be seen that the output +probabilities of he and she fluctuate according to r, +and the proposed method can control the bias of the +PLM. On the other hand, when r is less than 0.2 or +greater than 0.8, the output probabilities of she and +he are greater than the proportion in the data set, +respectively. Therefore, finer increments of r may +make it difficult to control bias more finely when r +is small or large. +To illustrate how bias-controlled PLMs produced +by the proposed method for different rates of biases +(r) predict the probabilities of gender pronouns, we +consider the masked sentence “[MASK] doesn’t +have time for the family due to work obligations.” +selected from the CP dataset. Here, He and She +0.0 +0.5 +1.0 +he +and +dad +also +but +(a) r = 1.0 +0.0 +0.5 +1.0 +he +she +mum +and +it +(b) r = 0.7 +0.0 +0.5 +1.0 +she +he +it +and +wife +(c) r = 0.5 +0.0 +0.5 +1.0 +she +he +and +but +it +(d) r = 0.3 +0.0 +0.5 +1.0 +she +mum +mother +kim +woman +(e) r = 0.0 +0.0 +0.5 +1.0 +he +she +mom +dad +it +(f) original (without fine-tuning) +Figure 3: Top 5 words with BERT output probability +for “[MASK] doesn’t have time for family due to work +obligations.”. Blue and red represent masculine and +feminine words, respectively. +are unmodified tokens. Figure 3 shows the proba- +bilities of the tokens predicted for the [MASK] by +the different bias-controlled PLMs. We see that the +original BERT model predicts both he and she with +approximately equal probabilities. However, when +r is gradually increased from 0 to 1, we see that +the probability of he increases, while that of she +decreases, demonstrating that the proposed method +correctly learns bias-controlled PLMs. +4 +Conclusion +We proposed a method to compare intrinsic gen- +der bias evaluation measures using an unannotated +corpus and gender-related word lists. Experiments + +show that the correlations computed by the pro- +posed method for existing bias evaluation measures +agrees with the prior evaluations conducted using +human annotations. +5 +Limitations +In this paper, we limited our investigation to En- +glish PLMs. However, as reported in a lot of previ- +ous work, social biases are language independent +and omnipresent in PLMs trained for many lan- +guages (Kaneko et al., 2022c; Lewis and Lupyan, +2020; Liang et al., 2020; Zhao et al., 2020). We +plan to extend this study to cover non-English +PLMs in the future. +According to existing research, PLMs encode +many different types of social biases such as racial +and religious biases in addition to gender-related +biases (Kiritchenko and Mohammad, 2018; Ravfo- +gel et al., 2020). On the other hand, in this paper, +we focused on only gender bias. Extending the +proposed method to handle other types of social +biases beyond gender bias is beyond the scope of +the current short paper and is deferred to future +work. +Furthermore, discriminatory bias is learned in +word embeddings as well as PLMs (Bolukbasi +et al., 2016; Brunet et al., 2019; Kaneko and Bol- +legala, 2019, 2020, 2021b; Kaneko et al., 2022b). +Therefore, it may be possible to make it applicable +to word embeddings as well. +6 +Ethical Considerations +Our goal in this paper was to compare the pre- +viously proposed and widely-used intrinsic bias +evaluation measures of gender bias in pre-trained +PLMs. Although we used a broad range of existing +datasets that are annotated for social biases, we did +not annotate nor release new datasets as part of this +research. Moreover, we fine-tune a large number +of bias-controlled PLMs for evaluation purposes +that demonstrates varying levels of gender biases. +However, these PLMs are not supposed to be used +in downstream tasks other than for evaluation pur- +poses. +Even with the highly correlated bias evaluation +measure in our proposed method, the bias of the +PLM may not be sufficiently evaluated. There- +fore, we consider that it important to select intrinsic +gender bias evaluation measures carefully and not +purely based on correlation coefficients computed +by the proposed method alone. +There are various discussions on how to define +social bias in PLMs (Blodgett et al., 2021). Since +the proposed method can use any method as the +bias-controlled fine-tuning of the PLMs, the bias- +controlled fine-tuning can be selected according to +the definition of social bias. +Acknowledgements +This paper is based on results obtained from a +project, JPNP18002, commissioned by the New +Energy and Industrial Technology Development +Organization (NEDO). +References +Su Lin Blodgett, Gilsinia Lopez, Alexandra Olteanu, +Robert Sim, and Hanna Wallach. 2021. Stereotyp- +ing Norwegian salmon: An inventory of pitfalls in +fairness benchmark datasets. In Proceedings of the +59th Annual Meeting of the Association for Compu- +tational Linguistics and the 11th International Joint +Conference on Natural Language Processing (Vol- +ume 1: Long Papers), pages 1004–1015, Online. 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Self-diagnosis and self-debiasing: A proposal +for reducing corpus-based bias in NLP. +Transac- +tions of the Association for Computational Linguis- +tics, 9:1408–1424. +Kellie Webster, Xuezhi Wang, Ian Tenney, Alex Beu- +tel, Emily Pitler, Ellie Pavlick, Jilin Chen, Ed Chi, +and Slav Petrov. 2020. +Measuring and reducing +gendered correlations in pre-trained models. arXiv +preprint arXiv:2010.06032. +Jieyu Zhao, Subhabrata Mukherjee, Saghar Hosseini, +Kai-Wei Chang, and Ahmed Hassan Awadallah. +2020. Gender bias in multilingual embeddings and +cross-lingual transfer. +In Proceedings of the 58th +Annual Meeting of the Association for Computa- +tional Linguistics, pages 2896–2907, Online. Asso- +ciation for Computational Linguistics. +Yi Zhou, Masahiro Kaneko, and Danushka Bollegala. +2022. Sense embeddings are also biased – evaluat- +ing social biases in static and contextualised sense +embeddings. +In Proceedings of the 60th Annual +Meeting of the Association for Computational Lin- +guistics (Volume 1: Long Papers), pages 1924–1935, +Dublin, Ireland. Association for Computational Lin- +guistics. +Yukun Zhu, Ryan Kiros, Rich Zemel, Ruslan Salakhut- +dinov, Raquel Urtasun, Antonio Torralba, and Sanja +Fidler. 2015. Aligning books and movies: Towards +story-like visual explanations by watching movies +and reading books. In Proceedings of the IEEE inter- +national conference on computer vision, pages 19– +27. +A +Intrinsic Bias Evaluation Measures +TBS +Kurita et al. (2019) proposed template- +based bias evaluation measure. The log-odds of +the likelihood of a template sentence masked with +a gender word (e.g. “[MASK] is a programmer”) +and the likelihood of a gender word masked with +an occupation word (e.g. “[MASK] is a [MASK]”) +are calculated for male and female words, respec- +tively. TBA then calculates the difference between +them as the bias score. +SSS +SSS (Nadeem et al., 2021) uses stereotypi- +cal and anti-stereotypical sentence pairs (e.g. “She +is a nurse” and “He is a nurse”) to evaluate bias +in PLMs. Calculate the likelihood of masked modi- +fied tokens (e.g. She, He) given unmasked unmodi- +fied tokens (e.g. is, a, nurse) for each stereotypical +and anti-stereotypical sentence. The bias score is +calculated by dividing the number of sentences for +which the total likelihood is higher for stereotypical +sentences compared to anti-stereotypical sentences +by the total number of data. +CPS +CPS (Nangia et al., 2020) also uses stereo- +typical and anti-stereotypical sentence pairs. On +the other hand, calculate the likelihood of masked +unmodified tokens given unmasked modified to- +kens for each stereotypical and anti-stereotypical +sentence. The bias score is calculated by dividing +the number of sentences for which the total like- +lihood is higher for stereotypical sentences com- +pared to anti-stereotypical sentences by the total +number of data. As with SSS, the bias score is +calculated using the sum of the likelihoods of the +stereotyped and anti-stereotyped sentences. +AUL and AULA +AUL and AULA (Kaneko and +Bollegala, 2022) also uses stereotypical and anti- +stereotypical sentence pairs, but they calculate the +likelihood of unmasked unmodified tokens and +modified tokens for each stereotypical and anti- +stereotypical sentence. As with SSS and CPS, the +bias score is calculated using the sum of the likeli- +hoods of the stereotyped and anti-stereotyped sen- +tences. AULA calculates the likelihood of the en- +tire sentence by weighting and averaging with the +attention weights to prioritize the likelihood of im- +portant words. + diff --git a/h9FLT4oBgHgl3EQfaS_3/content/tmp_files/load_file.txt b/h9FLT4oBgHgl3EQfaS_3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec047e8ae8e91abf72cfb90c376954aa4d606d4c --- /dev/null +++ b/h9FLT4oBgHgl3EQfaS_3/content/tmp_files/load_file.txt @@ -0,0 +1,493 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf,len=492 +page_content='Comparing Intrinsic Gender Bias Evaluation Measures without using Human Annotated Examples Masahiro Kaneko1 Danushka Bollegala2,3∗ Naoaki Okazaki1 1Tokyo Institute of Technology 2University of Liverpool 3Amazon masahiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='kaneko@nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='titech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='jp danushka@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='uk okazaki@c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='titech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='jp Abstract Numerous types of social biases have been identified in pre-trained language models (PLMs), and various intrinsic bias evaluation measures have been proposed for quantifying those social biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Prior works have relied on human annotated examples to compare exist- ing intrinsic bias evaluation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' How- ever, this approach is not easily adaptable to different languages nor amenable to large scale evaluations due to the costs and difficulties when recruiting human annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' To over- come this limitation, we propose a method to compare intrinsic gender bias evaluation mea- sures without relying on human-annotated ex- amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Specifically, we create multiple bias- controlled versions of PLMs using varying amounts of male vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' female gendered sen- tences, mined automatically from an unanno- tated corpus using gender-related word lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Next, each bias-controlled PLM is evaluated using an intrinsic bias evaluation measure, and the rank correlation between the computed bias scores and the gender proportions used to fine-tune the PLMs is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Experi- ments on multiple corpora and PLMs repeat- edly show that the correlations reported by our proposed method that does not require human annotated examples are comparable to those computed using human annotated examples in prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 1 Introduction Pre-trained language models (PLMs) trained on large datasets have reported impressive perfor- mance improvements in various NLP tasks (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019) greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' However, these PLMs also demonstrate significantly worry- ing levels of social biases (Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kurita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' To address this issue, numerous ∗Danushka Bollegala holds concurrent appointments as a Professor at University of Liverpool and as an Amazon Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' This paper describes work performed at the Univer- sity of Liverpool and is not associated with Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' intrinsic bias evaluation measures for PLMs have been proposed (Nangia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Dhamala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko and Bollegala, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2022), which are also used for comparing debiasing methods for PLMs (Webster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko and Bollegala, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Schick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Existing bias evaluation methods use different criteria such as pseudo likelihood (Kaneko and Bol- legala, 2022), cosine similarity (Caliskan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' May et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019), inner-product (Ethayarajh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Moreover, current bias evalua- tion methods require manually-annotated datasets containing stereotypical and antistereotypical ex- amples that express different types of social biases (Nangia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' There- fore, we consider that it is important to compare the differences in existing bias evaluation measures proposed for PLMs (Orgad and Belinkov, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Dev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2022a) to under- stand their relative strengths and weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' To objectively compare the existing bias evalu- ation measures, Kaneko and Bollegala (2022) cal- culated the rank correlation between the number of human annotators who labelled an example to be stereotypically biased towards a protected at- tribute in Crowds-Pairs (CP), and the bias score for that example returned by an intrinsic bias evalua- tion measure (Nangia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' However, due to the costs and difficulties in recruiting human annotators, this approach cannot be easily adapted to different languages, accommo- date large-scale evaluations, or compare evaluation metrics that do not use human-annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We propose a method to compare intrinsic bias evaluation measures without using human anno- tated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Figure 1 outlines the intuition be- hind our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' First, we train bias- controlled versions of PLMs obtained via fine- tuning a PLM on male and female gendered sen- tences, automatically mined from an unannotated arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='12074v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='CL] 28 Jan 2023 Figure 1: Overview of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We first create bias-controlled PLMs by fine-tuning a PLM on male and female gendered sentences that are auto- matically mined from unannotated corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Next, we measure the rank correlation between the scores re- ported by an intrinsic bias evaluation measure and the male/female bias rates (r) used to fine-tune the PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' corpus using a gender-related word list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We de- fine rate of bias (r) as the ratio between male and female gendered sentences in a training sample used to fine-tune a PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' A PLM fine-tuned mostly on male sentences is likely to generate sentences containing mostly male words, while a PLM fine- tuned on female sentences is likely to generate sen- tences containing mostly female words (Kaneko and Bollegala, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2022c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' There- fore, an accurate intrinsic bias evaluation measure is expected to return a score indicating a bias to- wards the male gender for a male bias-controlled PLM, while it is expected to return a score indicat- ing a bias towards the female gender for a female bias-controlled PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We then compute the rank correlation between (a) the rate of biases in the bias-controlled PLMs, and (b) the bias scores re- turned by an intrinsic evaluation measure for the corresponding PLMs, as a measure of accuracy of the bias evaluation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Our experiments with multiple corpora and PLMs show that the correlations reported by our proposed method, which does not require human annotated examples, are comparable to those com- puted using human annotated examples in previ- ous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Furthermore, by examining the out- put probabilities of the PLM, we verify that the proposed method, which fine-tunes bias-controlled PLMs with varying amounts of male vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' female sentences, is indeed able to control biases associ- ated with male and female gender directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 2 Bias-controlled Fine-Tuning The imbalance of gender words in the training data affects the gender bias of a PLM fine-tuned us- ing that data (Kaneko and Bollegala, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2022c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Using this fact, we propose a method to learn bias-controlled versions of PLMs that ex- press different levels of known gender biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Let us first assume that we are given a list of female gender related words Vf (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' she, woman, female), and a separate list of male gender related words Vm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' he, man, male).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Next, we select sen- tences that contain either at least one of female or male words from an unannotated set of sentences D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Sentences that contain both male and female words are excluded here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Let us denote the set of sentences extracted for a female or a male word w by Φ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Moreover, let Df = � w∈Vf Φ(w) and Dm = � w∈Vm Φ(w) be the sets of sentences con- taining respectively female and male words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We ap- propriately downsample Df and Dm to have equal numbers of sentences N (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' |Df| = |Dm| = N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Next, we create training datasets Dr by varying the rate of bias, r (∈ [0, 1]), by randomly sampling a subset Sr(Dm) of Nr sentences from Df and a subset S1−r(Df) of N(1 − r) sentences from Dm such that Dr = Sr(Dm) ∪ S1−r(Df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' When r = 0, Dr consists of only female sentences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Dr ⊆ Df), and when r = 1, it consists of only male sentences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Dr ⊆ Dm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' To obtain mul- tiple bias-controlled PLMs at different levels of gender biases, we fine-tune a given PLM on differ- ent datasets, Dr, sampled with different values of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We use a given intrinsic bias evaluation measure to separately evaluate each bias-controlled PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Finally, we measure the agreement between the bias scores reported by the intrinsic bias evaluation measure under consideration and the correspond- ing rates of biases of those PLMs using Pearson’s rank correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 3 Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='1 Settings In our experiments, we used the female words she, woman, female, her, wife, mother, girl, sister, daughter, girlfriend as Vf, and male words he, man, male, him, his, husband, father, boy, brother, son, boyfriend as Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We sampled 2M sentences each representing male and female genders from News crawl 2021 corpus (news)1 and BookCorpus (Zhu 1https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='statmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='org/news-crawl/en/ Bias Controled Fine-tuning Corpus Evaluation Male Female 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 PLM +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='1 PLM +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 Bias .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Eval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='9 PLM 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 PLM 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='9 Bias Intrinsic Rank Correlation Rate r Bias ScoreMeasure BERT ALBERT news book HA news book HA TBS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='14 SSS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='53 CPS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='48 AUL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='56 AULA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='57 Table 1: Peason correlation between biased PLM order and each bias scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' News and book represent the cor- pus used for biasing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' HA is AUC value of method using human annotation (Kaneko and Bolle- gala, 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2015) (books) for training bias-controlled PLMs and a separate 100K sentences as devel- opment data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We used BERT2 (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019) and ALBERT3 (Lan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019) as the PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We fine-tune PLMs with masked language model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We use publicly available Transformer library4 to fine-tuning PLMs, and all hyperparameters are set to their default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We trained 11 bias-controlled PLMs for r in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0} on four Tesla V100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='2 Intrinsic Bias Evaluation Measures We compare five previously proposed intrin- sic gender bias evaluation measures in this pa- per: Template-Based Score (TBS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kurita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019), StereoSet Score (SSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021), CrowS-Pairs Score (CPS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Nangia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020), All Unmasked Likelihood (AUL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko and Bollegala, 2022), and AUL with Attention weights (AULA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko and Bollegala, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Further details of these measures are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Note that TBS uses templates for evalua- tion and cannot be used with human-annotated stereotypical/anti-stereotypical sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' On the other hand, SSS, CPS, AUL, and AULA all require human-annotated sentences that express social bi- ases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='3 Comparing Intrinsic Gender Bias Evaluation Measures We compare the proposed method and Kaneko and Bollegala (2022)’s method using CP dataset, which has human annotations, and show the effectiveness 2https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='co/bert-base-uncased 3https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='co/albert-base-v2 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='com/huggingface/transformers/ tree/v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='2 of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' In addition, we will use several PLMs and corpora to analyze the trends of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Table 1 shows the correla- tion results of the proposed method for TBS, SSS, CPS, AUL, and AULA when fine-tuning BERT and ALBERT on news or book corpora, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' HA is the AUC value of the Kaneko and Bollegala (2022)’s method using human annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Since TBS uses templates, it cannot be evaluated using HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' For BERT, the proposed method induces the same order among measures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' AULA > AUL > CPS > SSS) as done by HA in both news and book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' For ALBERT, only the rankings of SSS and CPS differ between the proposed method and HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' These results show that the proposed method and the ex- isting method that use human annotations rank the intrinsic gender bias evaluation measures in almost the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5 It can be seen that the values of the correlation coefficients vary depending on the PLM and corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' For example, ALBERT has a maximum correlation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='60, while BERT has a maximum correlation of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' A major limitation of human annotation-based evaluation is that it cannot be used to compare TBS that does not human annotated examples against other intrinsic bias evaluation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' However, our proposed method does not have this limitation and can be used to compare TBS against other bias evaluation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' As it can be seen from Table 1, TBS consistently reports the lowest corre- lations, indicating that it is not an accurate intrin- sic gender bias evaluation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' This finding agrees with Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' (2022a), who highlighted the inadequacy of templates as a method for evalu- ating social biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='4 Bias-controlled PLMs To verify that the proposed method can indeed con- trol the bias of a PLM, we investigate the variation of the output probabilities of the PLMs fine-tuned with different r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Specifically, we investigate the output probabilities of masked he and she in the input text “[MASK] is a/an [Occupation].” for the bias-controlled PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' For [Occupation], we use gender- and stereotype-neutral occupational words6 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' writer, musician) from the word list created by Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' When r in- 5Because of the different methods of measuring correla- tions, it is not possible to compare the magnitude of values between the proposed and existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='com/tolga-b/debiaswe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 rate of bias 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 probability bert (he) bert (she) albert (he) albert (she) Figure 2: Average output probabilities for “[MASK] is a/an [Occupation]” produced by the bias-controlled BERT and ALBERT PLMs fine-tuned with different r on the news dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' creases, a PLM will be fine-tuned with increasing amounts of male sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Therefore, if the av- erage probability of he increases with r, it would imply that the PLMs are correctly bias-controlled by the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Figure 2 shows that the average output probabili- ties of he and she when r is incremented in step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' When r = 1 the PLM predicts he with fairly high probability and when r = 0 the PLM predicts she with fairly high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Furthermore, when r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5, the probability of he and she is almost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Original BERT (without fine-tuning) returns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='48 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='28, respectively for he and she, while the corresponding probabilities returned by ALBERT are respectively 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='64 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Both the original BERT and ALBERT predict relatively larger out- put probabilities for he, indicating that they are male-biased, without performing any bias-control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' From these results, it can be seen that the output probabilities of he and she fluctuate according to r, and the proposed method can control the bias of the PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' On the other hand, when r is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='2 or greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='8, the output probabilities of she and he are greater than the proportion in the data set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Therefore, finer increments of r may make it difficult to control bias more finely when r is small or large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' To illustrate how bias-controlled PLMs produced by the proposed method for different rates of biases (r) predict the probabilities of gender pronouns, we consider the masked sentence “[MASK] doesn’t have time for the family due to work obligations.” selected from the CP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Here, He and She 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 he and dad also but (a) r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 he she mum and it (b) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 she he it and wife (c) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 she he and but it (d) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 she mum mother kim woman (e) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='0 he she mom dad it (f) original (without fine-tuning) Figure 3: Top 5 words with BERT output probability for “[MASK] doesn’t have time for family due to work obligations.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Blue and red represent masculine and feminine words, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' are unmodified tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Figure 3 shows the proba- bilities of the tokens predicted for the [MASK] by the different bias-controlled PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We see that the original BERT model predicts both he and she with approximately equal probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' However, when r is gradually increased from 0 to 1, we see that the probability of he increases, while that of she decreases, demonstrating that the proposed method correctly learns bias-controlled PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 4 Conclusion We proposed a method to compare intrinsic gen- der bias evaluation measures using an unannotated corpus and gender-related word lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Experiments show that the correlations computed by the pro- posed method for existing bias evaluation measures agrees with the prior evaluations conducted using human annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 5 Limitations In this paper, we limited our investigation to En- glish PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' However, as reported in a lot of previ- ous work, social biases are language independent and omnipresent in PLMs trained for many lan- guages (Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2022c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Lewis and Lupyan, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' We plan to extend this study to cover non-English PLMs in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' According to existing research, PLMs encode many different types of social biases such as racial and religious biases in addition to gender-related biases (Kiritchenko and Mohammad, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Ravfo- gel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' On the other hand, in this paper, we focused on only gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Extending the proposed method to handle other types of social biases beyond gender bias is beyond the scope of the current short paper and is deferred to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Furthermore, discriminatory bias is learned in word embeddings as well as PLMs (Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Brunet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko and Bol- legala, 2019, 2020, 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Therefore, it may be possible to make it applicable to word embeddings as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 6 Ethical Considerations Our goal in this paper was to compare the pre- viously proposed and widely-used intrinsic bias evaluation measures of gender bias in pre-trained PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Although we used a broad range of existing datasets that are annotated for social biases, we did not annotate nor release new datasets as part of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Moreover, we fine-tune a large number of bias-controlled PLMs for evaluation purposes that demonstrates varying levels of gender biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' However, these PLMs are not supposed to be used in downstream tasks other than for evaluation pur- poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Even with the highly correlated bias evaluation measure in our proposed method, the bias of the PLM may not be sufficiently evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' There- fore, we consider that it important to select intrinsic gender bias evaluation measures carefully and not purely based on correlation coefficients computed by the proposed method alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' There are various discussions on how to define social bias in PLMs (Blodgett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Since the proposed method can use any method as the bias-controlled fine-tuning of the PLMs, the bias- controlled fine-tuning can be selected according to the definition of social bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Acknowledgements This paper is based on results obtained from a project, JPNP18002, commissioned by the New Energy and Industrial Technology Development Organization (NEDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' References Su Lin Blodgett, Gilsinia Lopez, Alexandra Olteanu, Robert Sim, and Hanna Wallach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 2021.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Asso- ciation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Yi Zhou, Masahiro Kaneko, and Danushka Bollegala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Sense embeddings are also biased – evaluat- ing social biases in static and contextualised sense embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' In Proceedings of the 60th Annual Meeting of the Association for Computational Lin- guistics (Volume 1: Long Papers), pages 1924–1935, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Association for Computational Lin- guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Yukun Zhu, Ryan Kiros, Rich Zemel, Ruslan Salakhut- dinov, Raquel Urtasun, Antonio Torralba, and Sanja Fidler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Aligning books and movies: Towards story-like visual explanations by watching movies and reading books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' In Proceedings of the IEEE inter- national conference on computer vision, pages 19– 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' A Intrinsic Bias Evaluation Measures TBS Kurita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' (2019) proposed template- based bias evaluation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' The log-odds of the likelihood of a template sentence masked with a gender word (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' “[MASK] is a programmer”) and the likelihood of a gender word masked with an occupation word (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' “[MASK] is a [MASK]”) are calculated for male and female words, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' TBA then calculates the difference between them as the bias score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' SSS SSS (Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2021) uses stereotypi- cal and anti-stereotypical sentence pairs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' “She is a nurse” and “He is a nurse”) to evaluate bias in PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' Calculate the likelihood of masked modi- fied tokens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' She, He) given unmasked unmodi- fied tokens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' is, a, nurse) for each stereotypical and anti-stereotypical sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' The bias score is calculated by dividing the number of sentences for which the total likelihood is higher for stereotypical sentences compared to anti-stereotypical sentences by the total number of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' CPS CPS (Nangia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=', 2020) also uses stereo- typical and anti-stereotypical sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' On the other hand, calculate the likelihood of masked unmodified tokens given unmasked modified to- kens for each stereotypical and anti-stereotypical sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' The bias score is calculated by dividing the number of sentences for which the total like- lihood is higher for stereotypical sentences com- pared to anti-stereotypical sentences by the total number of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' As with SSS, the bias score is calculated using the sum of the likelihoods of the stereotyped and anti-stereotyped sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' AUL and AULA AUL and AULA (Kaneko and Bollegala, 2022) also uses stereotypical and anti- stereotypical sentence pairs, but they calculate the likelihood of unmasked unmodified tokens and modified tokens for each stereotypical and anti- stereotypical sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' As with SSS and CPS, the bias score is calculated using the sum of the likeli- hoods of the stereotyped and anti-stereotyped sen- tences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} +page_content=' AULA calculates the likelihood of the en- tire sentence by weighting and averaging with the attention weights to prioritize the likelihood of im- portant words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9FLT4oBgHgl3EQfaS_3/content/2301.12074v1.pdf'} diff --git a/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf b/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e5f9162882300f73d8a220353c42bbc83f0fcc4e --- /dev/null +++ b/hdE5T4oBgHgl3EQfFQ6l/content/2301.05421v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6fb6292df3159ec06f90d0026d4978d3008e01b746df7b87c867a027f2e6a2aa +size 7928408 diff --git a/kb_45/content/tmp_files/kb_45.pdf.txt b/kb_45/content/tmp_files/kb_45.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b87e89ff8a80c69bf43ccd0acac0181ff7fa7eb --- /dev/null +++ b/kb_45/content/tmp_files/kb_45.pdf.txt @@ -0,0 +1,2437 @@ +ARTICLE +Expansive microbial metabolic versatility and +biodiversity in dynamic Guaymas Basin +hydrothermal sediments +Nina Dombrowski +1, Andreas P. Teske2 & Brett J. Baker1 +Microbes in Guaymas Basin (Gulf of California) hydrothermal sediments thrive on hydro- +carbons and sulfur and experience steep, fluctuating temperature and chemical gradients. +The functional capacities of communities inhabiting this dynamic habitat are largely unknown. +Here, we reconstructed 551 genomes from hydrothermally influenced, and nearby cold +sediments belonging to 56 phyla (40 uncultured). These genomes comprise 22 unique +lineages, including five new candidate phyla. In contrast to findings from cold hydrocarbon +seeps, hydrothermal-associated communities are more diverse and archaea dominate over +bacteria. Genome-based metabolic inferences provide first insights into the ecological niches +of these uncultured microbes, including methane cycling in new Crenarchaeota and alkane +utilization in ANME-1. These communities are shaped by a high biodiversity, partitioning +among nitrogen and sulfur pathways and redundancy in core carbon-processing pathways. +The dynamic sediments select for distinctive microbial communities that stand out by +expansive biodiversity, and open up new physiological perspectives into hydrothermal eco- +system function. +DOI: 10.1038/s41467-018-07418-0 +OPEN +1 Department of Marine Science, Marine Science Institute, University of Texas Austin, Port Aransas, TX 78373, USA. 2 Department of Marine Sciences, +University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA. Correspondence and requests for materials should be addressed to +B.J.B. (email: acidophile@gmail.com) +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications +1 +1234567890():,; + +M +icrobial communities inhabit every environment and +are comprised of a multitude of different phyla, the +majority of which are uncultured1. Among these +environments, marine sediments contain abundant and phylo- +genetically diverse microbial communities2–4. High diversity has +been suggested to emerge as a strategy for survival of microbes +under fluctuating environmental conditions in nature5,6. While +single-gene surveys allow us to address the phylogenetic diversity +of microbial communities, metagenomic analyses provide a +connection between diversity and the functional potential enco- +ded within sedimentary communities. +Guaymas Basin (GB; Gulf of California, Mexico) is a young, +active seafloor-spreading center characterized by high water col- +umn productivity and fast sedimentation rates, leading to the +accumulation of massive layers of organic-rich sediments that +cover the hydrothermal spreading center and ridge flanks7–9. The +emplacement of hot basalt sills into organic-rich sediment +transforms buried organic matter into CO2, H2, low-molecular- +weight organic acids, ammonia, and hydrocarbons such as +methane, ethane and benzene8,10,11. These compounds migrate to +the sediment surface with rising vent fluids, where they fuel +hydrocarbon-degrading microbial communities11,12. Among all +hydrothermally generated hydrocarbons, methane has received +considerable interest as greenhouse gas shaping global climate13. +Porewater methane reaches millimolar concentrations while +ethane ranges from 40-100 µM. Also present in these sediments +are propane, n-butane and pentane, which accumulate at lower +concentrations compared to methane. Altogether, hydrocarbons +represent lucrative carbon sources for the resident microbial +community11,14–16. Additionally, hydrothermal circulation and +seawater in-mixing provide the upper sediments with electron +acceptors, among which sulfate is widely available in millimolar +porewater concentrations and rarely depleted within hydro- +thermal sediment cores11,14,17. In-situ microelectrode surveys +detect small oxygen peaks within hydrothermal sediments near +the mat-covered surface18,19. These results are consistent with +short-term dynamics of hydrothermal flow within minutes and +hours17. Additionally, short-term dynamics overlay with longer- +term hydrothermal activity changes over months and years18. +GB sediments have been shown to host diverse microbial +communities with distinct roles in carbon cycling11,17,20. In +particular, microbial consortia perform the anaerobic oxidation of +methane (AOM) in a syntrophic interaction consisting of anae- +robic methane-oxidizing archaea (ANME) and bacterial sulfate +reducers, typically Deltaproteobacteria, but including other +thermophilic bacterial lineages, such as Candidatus Desulfo- +fervidus auxilii21–23. Anaerobic hydrocarbon degraders include +Ca. Syntrophoarchaeum, which oxidizes butane in a syntrophic +interaction with Ca. Desulfofervidus auxilii, or the butane- and +propane oxidizing isolate BuS5, belonging to the Desulfosarcina- +Desulfococcus cluster15,16. Other common archaeal lineages +include Marine Benthic Group D and Bathyarchaeota, while +bacterial phyla include Proteobacteria (Delta-, Epsilon- and +Gammaproteobacteria), Bacteroidetes and Chloroflexi as well as +several candidate phyla11,17,24. Within the GB hydrothermal area +in the southern spreading center, a high degree of microbial +community connectivity exists among hydrothermal vent sites +and sediments within a few hundred meters25. A core micro- +biome is shared between microbial communities of GB hydro- +thermal sediments and cold seeps in the Sonora Margin, within a +few km distance; this microbiome is thought to be involved in +organic matter degradation as well as methane and carbon +cycling, suggesting microbial exchanges across neighboring sites +that share geochemical characteristics, such as abundant methane +concentrations26. Previously, we employed metagenomic recon- +structions of two GB sedimentary microbial communities, +showing the interconnectivity of carbon, sulfur and nitrogen +cycling among lineages20. However, despite these advances, we +still have a limited understanding of the spatial biodiversity and +full metabolic potential of microbes inhabiting the basin. +Here we characterize the biodiversity and physiological cap- +abilities of genomes from microbial communities inhabiting GB +sediments. The highly localized hydrothermal gradients in surfi- +cial GB sediments are ideal to compare adjacent sites with distinct +temperature and chemical regimes18,27. We selected samples from +methane- and sulfate-rich hydrothermal sediments covering a +wide thermal range, and contrasted them with cold, non- +hydrothermal sediments, as well as with hot, oil-rich sediments. +We hypothesize that microbial assemblages from hydrothermal +sediments are phylogenetically distinct from those in the sur- +rounding region and host a greater metabolic diversity. Therefore, +we sequenced a total of ~4 billion genomic reads from eleven +samples (two of which were from cool, background sediments) +from GB. Altogether, these data add 22 branches to the tree of life +and enabled to us determine the genetic repertoire and metabolic +versatility of these extreme hydrothermal communities. +Results +Phylogenetic +diversity +in Guaymas +Basin sediments. +To +examine the biodiversity of microbial communities inhabiting GB +sediments, we sampled and sequenced eleven sediments covering +different sampling locations, depths (0–24 cm), temperatures +(3–60 °C) and geochemical regimes (Fig. 1, Supplementary +Data 1, 2). Background sediments, represented by core 4567_28, +are not influenced by hydrothermal activity (temperature ~3 °C) +and occur interlaced with hydrothermal hot spots within the +spreading center18. All other samples are characterized by steep +thermal gradients, reflected by in-situ temperatures ranging from +4 °C to 60 °C. Dense mats of filamentous Gammaproteobacteria +(family Beggiatoaceae) covered hydrothermal sediments from +dive 4569, with an orange mat dominating core 4569_9 and a +white mat at the adjacent core 4569_2. Core 4569_4 was collected +from the periphery of this hydrothermal hotspot and did not +contain visible mats (Fig. 1). Porewater methane, sulfate, dis- +solved +inorganic +carbon +(DIC) +and +sulfide +co-occurred +throughout these cores (Supplementary Information), consistent +with hydrothermal circulation and inmixing of seawater-derived +electron acceptors. Cores 4571_4 and 4488_9 represent hot and +oily sediments with yellow-white sulfur precipitates on the surface +(Fig. 1). Among the hydrothermal cores, 4488_9 stands out by +steep thermal gradients (~150 °C at 30 cm depth), high sample +temperature (~60 °C), sulfate depletion at shallow depths, and +accumulation of non-methane hydrocarbons (Supplementary +Figure 1, Supplementary Data 1). After sequence assembly, we +reconstructed 551 draft genomes via tetranucleotide and coverage +binning. These metagenome-assembled genomes (MAGs), sim- +plified +as +‘genome’ +throughout +the +manuscript, +represent +medium-quality MAGs and were > 50% complete and < 10% +contaminated (301 genomes > 70% and 61 genomes > 90% com- +plete; Supplementary Data 2, 3)28. +Each genome was classified by constructing a phylogenetic tree +using 37 single-copy, protein-coding marker genes (Supplemen- +tary Data 4)29. Overall, the 551 genomes (247 archaea and 304 +bacteria) represented 16 cultured and 40 uncultured, candidate +phyla that comprise a substantial number of new microbial +lineages, many of which branch basal to those previously +described (Fig. 2, Supplementary Figure 2, Supplementary Data 5). +GB genomes form 22 new lineages on the tree of life based on a +phylogenetic distance analysis (collapsing branches at an average +branch length distance < 0.6). Among those lineages, we dis- +covered five new candidate phyla designated GB-AP1,2 and +ARTICLE +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +2 +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications + +GB-BP1-3 for archaeal and bacterial phyla, respectively. The +placement of these five phyla was confirmed by comparing the +average amino acid identity (AAI) of genomes within a phylum to +genomes of all other phyla (Supplementary Data 6). Within each +new phylum, GB-AP1,2 shared an AAI of ~44 and ~96% and GB- +BP1-3 of ~54, ~72 and ~60%, respectively, and were more similar +to themselves then to other genomes (~43% AAI summarized +across all genomes). While the genomes of GB-AP1 shared a low +AAI, we did not detect any lineage with a closer similarity. Two +16S rRNA gene sequences recovered from GB-BP1 clustered with +the uncultivated lineage MAT-CR-M4-B0730, which was pre- +viously detected in the Kazan mud volcano or Guerrero Negro +hypersaline mats (Supplementary Figure 3). In total, we defined +24 archaeal and 37 bacterial groups (or ‘clusters’) for closer +analysis (see Methods section, Supplementary Data 3 and Fig. 2). +Archaeal genomes were represented by Bathyarchaeota (n = 41), +Thermoproteales (n = 40) and Thermoplasmata (n = 36), and +bacteria belonged to Deltaproteobacteria (n = 56), Gammapro- +teobacteria (n = 39) and Bacteroidetes (n = 27; Supplementary +Data 3). Additionally, we detected several candidate lineages, +including Asgard archaea (n = 9), Verstraetearchaeota (n = 7) +and the bacterial CPR superphylum (n = 6). Overall, more +genomes were recovered from hydrothermal (average of ~60 +genomes per sample) than from background sediments (average +~9 genomes per sample; Supplementary Data 3). We detected +only one archaeal (Bathyarchaeota) and 7 bacterial lineages +(Chloroflexi, Deltaproteobacteria, Gammaproteobacteria) in the +background compared to 22 archaeal and 31 bacterial clusters in +the hydrothermal samples, suggesting a greater biodiversity in the +more extreme environment. +The +effect +of +environmental +parameters +on +community +assembly. To better understand the factors that drive community +assembly, we investigated the occurrence of major phylogenetic +clusters across sites. First, we confirmed that the genomes accu- +rately reflected the community as a whole based on the abun- +dance of ribosomal protein S3 across sites (Supplementary +Figure 4). Next, we used the genomes to estimate the occurrence +of different phylogenetic groups across all samples (Supplemen- +tary Figure 5, Supplementary Data 7). Several bacterial lineages, +such as Planctomycetes or Deltaproteobacteria, were more fre- +quently detected in background sediments than in hydrothermal +sediments. In contrast, archaea were increasingly detected within +the deeper, hotter hydrothermal samples, but not in cool surface +sediments on the periphery of hydrothermal hot spots. Dominant +lineages in the hot samples were Thaumarchaeota and Archae- +oglobales as well as Acetothermia, and Omnitrophica. Two gen- +otypes dominated hot sediments: B48_G6 (Methanosarcinales, +ANME-1) and B16_G6 (Thermodesulfobacteria, ~88% AAI to +Ca. Desulfofervidus auxilii) (Supplementary Data 3, Supplemen- +tary Data 7). While the hydrothermal sediments had an overall +similar distribution of taxa across depth profiles, the oily sedi- +ment from 4488_9 harbored only few abundant taxa, including +Thermoplasmata, Aerophobetes and Thermotoga (Supplemen- +tary Figures 4, 5). Core 4488_9 differs from other hydrothermal +samples in its high hydrocarbon content, quick downcore +depletion of sulfate, and steep thermal gradients (Supplementary +Data 1, Supplementary Methods). In combination these factors +appear to reduce the microbial diversity, especially of the archaeal +community. Altogether, the hydrothermal activity gives rise to a +unique community that shows a marked enrichment in archaea +that can represent up to 50% of recovered genomes (Supple- +mentary Data 7). This enrichment appears to be largely driven by +the rich substrate availability, by hydrothermal circulation and by +inmixing of the electron acceptor sulfate (Supplementary Meth- +ods). However, a greater sampling size would be needed to dis- +entangle the relative contribution of individual factors on +community assembly such as temperature, methane or hydro- +carbon availability. +Carbon cycling. Given that these genomes yielded such a large +number of unique microbial lineages, we inferred their potential +physiological capabilities by assigning metabolic functions to +proteins in each individual genome. First, we investigated the +ability of the community to degrade and metabolize complex +carbohydrates and peptides deposited in sediments by searching +genomes for the presence of carbohydrate-active enzymes +Vent 1: +4569_9 (center) +4569_2 (intermediate) +4569_4 (outside) +Background: +4567_28 +Vent 2: +4571_4 +Vent 3: +4488_9 +Fig. 1 Overview of sampling sites. In-situ photos of the three hydrothermal +sampling sites (Vent1, Vent2, Vent3) and the non-hydrothermal +background sediment, including Alvin dive and core number for the +sediment cores that were used for DNA extraction and metagenomic +analysis. White circle: spots where sediment cores were retrieved by push +coring. For Vent1, three sediment cores were taken inside the yellow mat +(4569_9), further outside in a white mat area (4569_2) and outside of the +mat area (4569_4); next to each core a thermal logging probe was inserted +into the sediment. At Vent2 and Vent3, one core each (4571_4 and +4488_9) was sampled. Metadata for all samples are summarized in +Supplementary Data 1 +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +ARTICLE +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications +3 + +(CAZYmes), peptidases and pathways for carbon metabolism. In +total, we detected ~30,000 and ~11,000 potential CAZYmes and +peptidases, respectively (Fig. 3, Supplementary Figure 6, Supple- +mentary Data 8, 9). Generally, bacteria encoded for a broader +repertoire of CAZYmes compared to archaea; for example GH13 +(α-amylase), GH23 (lytic transglycosylase) or GH74 (xylogluca- +nase) were more common in bacteria (Fig. 3, Supplementary +Data 8). Most CAZymes were assigned to Thermoproteales (n = +20) and Asgard archaea (n = 16) as well as Verrucomicrobia (n = +38) and Bacteroidetes (n = 30). Peptidases were more equally +distributed across both domains and abundant in Asgard archaea +(n = 34) and Thermococci (n = 24) as well as Aminicenantes +(n = 54) and Acidobacteria (n = 51; Supplementary Figure 6, +Supplementary Data 9). Approximately 2–3% of CAZYmes and +peptidases are potentially secreted, suggesting that complex sub- +strates are degraded outside of the cell and later taken up for +degradation. Potentially secreted enzymes include CE8 (pectin +methylesterase), and GH13 (α-amylase) as well as M28 (amino- +peptidases) and S08 (subtilisin-like peptidases). A subset of +CAZymes, such as GH23, may be involved in cell wall main- +tenance; however, the presence of sugar and peptide transporters +as well as downstream metabolic pathways in most genomes +suggest that other CAZymes might be involved in energy meta- +bolism (see below). +Common pathways for the degradation of substrates produced +by the activity of CAZymes and peptidases include glycolysis +(glucokinase (glk), phosphofructokinase (pfk), pyruvate kinase +(pyk)), gluconeogenesis (fructose-1,6-bisphosphatase (fbp), phos- +phoenolpyruvate carboxykinase (pckA)) and fermentation (Fig. 4 +and Supplementary Data 10). In several cases, archaeal genomes +encoded for more key genes of gluconeogenesis compared to +glycolysis, which could imply that some archaea prefer peptides +as an energy source; this finding is consistent with the occurrence +of a high number of peptidases in their genomes (Supplementary +Figure 6). Compared to archaea, bacteria contained a greater +metabolic +repertoire +and +might +use +both +glycolysis +and +gluconeogenesis. Most genomes encoded for the potential to +metabolize pyruvate produced during glycolysis to acetyl-CoA +and further into fermentation pathways, producing formate, +ethanol or acetate (Fig. 4). GB archaea were mainly capable of +acetate formation using the ADP-forming acetyl-CoA synthetase +(acdA), while bacteria encoded for phosphate acetyltransferase +(pta) and acetate kinase (ackA) for acetate production; formate +C-acetyltransferase (pflD) and formate dehydrogenase (fdoG) for +Tree scale: 0.1 +DPANN +Altiarchaeales +Archaeoglobus +Methanomicrobia +Thermoplasmata +Thermococci +Asgard +Korarchaeota +Bathyarchaeota +Thaumarchaeota +Desulfurococcales +Thermoproteales +CPR +Acetothermia +Thermotogae +WOR-3 +Hydrothermae +Omnitrophica +Verrucomicrobia +Planctomycetes +Aquificae +Actinobacteria +Chloroflexi +Spirochaeta +Cloaci +Hyd24-12 +GB-BP1 +Zixi +Marinimicrobia +KSB1 +Bacteroidetes +Acidobacteria +Aminicenantes +Epsilonproteobacteria +Gamma- +proteobacteria +Delta- +proteobacteria +Thermodesulfobacteria +Latesci +References +GB hydrothermal +GB background +GB-BP2 +GB- +BP3 +GB- +AP2 +GB- +AP1 +DeepCrenGroup1+2 +Verstraetearchaeota +Aerophobetes +Caldiserica +Fig. 2 Maximum likelihood phylogenetic tree of GB genomes based on 37 concatenated protein-coding genes. Grey: Reference Genomes. Blue: Genomes +assembled from cold background sediments. Red: Genomes recovered from hot, hydrothermal sediments. The full tree can be found in Supplementary +Figure 2 and the tree file is available in Supplementary Data 5 +ARTICLE +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +4 +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications + +formate production; and aldehyde dehydrogenase (aldh) and +alcohol dehydrogenase (adh) for ethanol production. +Not only is the GB microbiome able to process the deposited +organic carbon pool by fermentation, but we also detected +pathways for carbon fixation. The most common route of carbon +fixation was the Wood-Ljungdahl pathway in both archaea and +bacteria, while the Calvin-Benson-Bassham (CBB) and rTCA +cycles were restricted mostly to Proteobacteria (Fig. 4, Supple- +mentary Data 10). Although the Group III Ribulose-1,5-bispho- +sphate carboxylase-oxygenase (Rubisco, key marker gene of the +CBB cycle) was detected in most archaea, this subgroup is implied +in a nucleotide salvage pathway and not necessarily used for +carbon fixation (Supplementary Data 10)31. A Group I/II +Rubisco, feeding CO2 into the CBB cycle, was only detected in +some Gammaproteobacteria (orders Chromatiales and Thiotri- +chales). Additionally, marker genes for the rTCA cycle, including +ATP-citrate-lyase (aclAB), pyruvate ferredoxin oxidoreductase +(porABCD) and 2-oxoacid ferredoxin oxidoreductase (oorABCD), +were mainly detected in Epsilonproteobacteria (order Campylo- +bacterales; Supplementary Data 10). While several genes of the 3- +hydroxypropionate or related cycles were present in a subset of +genomes, a full pathway appeared to be absent (Supplementary +Data 10). Conversely, the Wood-Ljungdahl pathway was present +in several clusters, including Archaeoglobales and Methanosarci- +nales as well as Chloroflexi and Deltaproteobacteria (Fig. 4, +Supplementary Data 10). Interestingly, we also detected genes +from this pathway in candidate phyla, including Hydrother- +marchaeota and Latescibacteria, which might either oxidize +acetate or perform acetogenesis. +Alkyl-coenzyme M reductase linked hydrocarbon cycling. We +detected the methyl-Coenzyme M reductase (mcrA), a key +enzyme for methanogenesis and AOM, in Syntrophoarchaea, +Methanomicrobia, and a deep-branching Thermoproteales line- +age (designated DeepCrenGroup1; Fig. 4, Supplementary Data 3). +To our knowledge this is the first report of mcrABG genes in the +Crenarchaeota. The only bacteria able to utilize methane encoded +for the particulate methane monooxygenase (pmoA), which was +restricted to Gammaproteobacteria (orders Cellvibrionales and +Methylococcales; Supplementary Data 10). A closer phylogenetic +analysis of McrA might even suggest a broader substrate usage +potentially not restricted to methane (Fig. 5, Supplementary +Data 11). McrA from most ANME-1, ANME-2c and Deep- +CrenGroup1 clustered with known methane oxidizers, while the +McrA from one Syntrophoarchaeum (B49_G1) clustered with +butane-oxidizers (Fig. 5). McrA from GoM-Arc1 branched +between those two clusters, which is consistent with earlier work +that suggested that GoM-Arc1 might utilize a different alkane, +perhaps ethane, which can reach relatively high concentrations of +40-100 µM in GB11,20. However, further experimental evidence, +preferably from enrichment cultures, is needed to confirm the +substrate usage of these McrA proteins. +Surprisingly, ANME-1 bin B39_G2 contains two McrA +proteins (on two different contigs, both mate-paired to other +contigs from that bin) that are phylogenetically related to those +from Ca. Syntrophoarchaeum spp. (Fig. 5). Similarly to Ca. +Syntrophoarchaeum spp. B39_G2 contains genes with homology +to those that encode for the butyryl-CoA oxidation pathway, such +as acyl-CoA dehydrogenase and enoyl-CoA dehydratase (Supple- +mentary Data 10, Supplementary Figure 7). This pathway appears +to be involved in butane oxidation in Ca. Syntrophoarchaeum +butanivorans16, making this the first example of an ANME-1 +archaeon potentially able to use short-chain alkanes. The +detection of these unique methyl coenzyme-M reductase genes +and pathways suggests that ANME-1 archaea are not limited to +methane utilization and potentially able to oxidize alkanes +anaerobically. +Lipid and hydrocarbon utilization. Pathways for lipid degra- +dation were widespread in bacteria and less common in archaea, +where they were mainly detected in Archaeoglobales, Bath- +yarchaeota +and Geothermarchaeota +(Fig. +4, +Supplementary +Carbohydrate +esterase +Glycoside hydrolase +Polysaccharide +lyase + +CE1 +CE11 +CE12 +CE13 +CE14 +CE15 +CE16 +CE2 +CE3 +CE4 +CE6 +CE7 +CE8 +CE9 +GH1 +GH10 +GH100 +GH102 +GH103 +GH105 +GH106 +GH108 +GH109 +GH110 +GH113 +GH114 +GH116 +GH117 +GH12 +GH120 +GH122 +GH123 +GH127 +GH128 +GH129 +GH13 +GH130 +GH133 +GH135 +GH136 +GH137 +GH139 +GH140 +GH141 +GH142 +GH144 +GH145 +GH15 +GH16 +GH17 +GH18 +GH2 +GH20 +GH23 +GH24 +GH25 +GH26 +GH27 +GH28 +GH29 +GH3 +GH30 +GH31 +GH32 +GH33 +GH35 +GH36 +GH37 +GH38 +GH39 +GH4 +GH42 +GH43 +GH46 +GH47 +GH5 +GH50 +GH51 +GH53 +GH57 +GH62 +GH63 +GH65 +GH66 +GH73 +GH74 +GH76 +GH77 +GH78 +GH79 +GH8 +GH84 +GH87 +GH88 +GH89 +GH9 +GH91 +GH92 +GH93 +GH94 +GH95 +GH97 +GH99 +PL1 +PL10 +PL11 +PL12 +PL14 +PL15 +PL17 +PL21 +PL22 +PL25 +PL26 +PL6 +PL8 +PL9 +Zixibacteria (6) +Verrucomicrobia (3) +Thermotoga (11) +Thermodesulfobacteria (5) +Spirochaetes (11) +Planctomycetes (11) +Omnitrophica (WOR–2) (13) +Marinimicrobia (7) +Latescibacteria (8) +KSB1 (8) +Hydrothermae (5) +GB–BP3 (4) +GB–BP2 (5) +GB–BP1 (6) +Gammaproteobacteria (39) +Epsilonproteobacteria (8) +Deltaproteobacteria (56) +Coatesbacteria (4) +Cloacimonetes (5) +Chloroflexi (20) +Caldiserica (4) +Bacteroidetes (27) +Aquificae (3) +Aminicenantes (3) +Aerophobetes (4) +Acidobacteria (4) +Acetothermia (6) +Woesearchaeota (7) +Verstraetearchaeota (14) +Thermoproteales (33) +Thermoplasmata (36) +Thermococci (15) +Thaumarchaeota (9) +Nanoarchaeota (3) +Methanomicrobia (11) +Korarchaeota (10) +Geothermarchaeota (3) +GB–AP2 (3) +Desulfurococcales (12) +Bathyarchaeota (41) +Asgard (9) +Archaeoglobales (6) +Altiarchaeales (8) +Aenigmarchaeota (14) +Genomes +(%) +0 +25 +50 +75 +100 +* +* +* +* +* +* +* +* +* +* +Fig. 3 Number of carbohydrate-active enzymes (CAZymes) encoded in GB genomes. Percentage of carbohydrate esterases (CE), glycoside hydrolases +(GH) and polysaccharide lyases (PL) encoded in GB genomes summarized for each phylogenetic cluster. Brackets: Total number of genomes encoded in +each phylogenetic cluster. Asterisk: CAZyme with potential secretion signal (see also Supplementary Data 8) +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +ARTICLE +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications +5 + +Data 10). The acyl-CoA dehydrogenase (acd) represents a key +gene catalyzing the first step in beta-oxidation and accommodates +a broad substrate range32,33. GB ACDs fell alongside described +glutaryl-CoA dehydrogenases, small/medium- and long-chain +acyl-CoA dehydrogenases, potential butyryl-CoA dehydrogenases +and isovaleryl-CoA dehydrogenases (Supplementary Figure 7, +Supplementary Data 12). Only ~50% of archaeal lineages encoded +for acd, which was found scattered across taxa, for example only +~30% of Verstraetearchaeota encoded for acd. This gene was +common in Archaeoglobales, Asgard archaea and Geother- +marchaeota, all of which encoded for other beta-oxidation genes, +such as enoyl-CoA hydratase (EC 4.2.1.17) or 3-hydroxyacyl-CoA +dehydrogenase (EC 1.1.1.35; Fig. 4, Supplementary Data 10). In +contrast, 33 out of 37 bacterial lineages encoded for acd. How- +ever, only a subset of those lineages - including Aquificae, +Chloroflexi or Deltaproteobacteria - encoded for further beta- +oxidation genes. In these cases, enzymes, such as the glutaryl-CoA +dehydrogenase, might be involved in amino acid catabolism or in +benzoyl-CoA degradation32,34. +Hydrocarbons are another abundant source for energy and +biomass generation in GB. While we did not detect genes for +aerobic hydrocarbon degradation, we found indications that GB +genomes might anaerobically degrade hydrocarbons using glycyl +radical enzymes (GREs, Supplementary Figure 8, Supplementary +Data 13). GREs use a radical-based chemistry to carry out +challenging metabolic reactions under anaerobic conditions and +are involved in a multitude of pathways, such as fermentation, +DNA synthesis or hydrocarbon degradation35,36. Compared to +ACDs, GREs had a sparser distribution and were found in only 6 +out of 24 archaeal and 21 out of 37 bacterial lineages. GREs were +common in Deltaproteobacteria (n = 32), Bacteroidetes (n = 23) +or Asgard archaea (n = 15). Several GREs encoded for enzymes +involved in anaerobic hydrocarbon degradation, such as benzyl- +succinate +synthase +(bssA) +in +Deltaproteobacteria +(B38_G6, +B7_G9), alkylsuccinate synthase (assA) in Deltaproteobacteria +(B2_G1, B111_G9) or hydroxyphenylacetate decarboxylase in +Bathyarchaeota (B26_G17) and Chloroflexi (B43_G15). Some +GREs grouped neither with the previously mentioned enzymes +Tree scale: 0.1 +Aenigmarchaeota (14) +Nanohaloarchaeota (2) +Nanoarchaeota (3) +Woesearchaeota (7) +Diapherotrites (2) +Micrarchaeota (2) +Altiarchaeales (8) +Thermoplasmata (36) +Methanomicrobia (11) +Archaeoglobales (6) +Hadesarchaea (1) +Hydrothermarchaeota (2) +Thermococci (15) +Korarchaeota (10) +Desulfurococcales (12) +Thermoproteales (40) +GB-AP2 (3) +Bathyarchaeota (41) +GB-AP1 (2) +Asgard (9) +Geothermarc (3) +Thaumar (9) +Tenericutes (1) +CPR (6) +Aerophobetes (4) +Caldiserica (4) +Acetothermia (6) +Thermotoga (11) +TA06 (2) +GB-BP2 (5) +Hydrothermae (4) +Stahlbacteria (1) +Omnitrophica (13) +Planctomycetes (11) +Chlamydia (1) +Verrucomicrobia (3) +Aquificae (3) +Poribacteria (1) +Chloroflexi (20) +Coatesbacteria (4) +Actinobacteria (2) +Aminicenantes (3) +Acidobacteria (4) +Deltaprot (61)* +Epsilonpr (8) +Gammapr (39) +Spirochaetes (11) +Cloacimonetes (5) +GB-BP3 (4) +Hyd24-12 (2) +Latescibacteria (8) +GB-BP1 (6) +Zixibacteria (6) +Marinimicrobia (7) +KSB1 (7) +Calditrichaeota (1) +Ignavibac (2) +Bacteroidet (27) +Pacearchaeota (2) +Verstraetearchaeota (7) +C1 +mcrABG +fdoG +cooS +acsB +Central metabolism +glk +pfk +pyk +pckA +fbp +korAB +porA +pflD1 +ldh +aldh +adh +acdA +pta +ackA +fadD +acd 2 +4.2.1.17 +1.1.1.35 +fadA +pccB +epi +mcmA +Hydrocarbons +Gly +Glu +Fermentation +Lipids +Prop +HCs +cdhA +cdhB +acsE +H2 +S +FeFe +NiFe G1 +NiFe G2 +NiFe G3 +NiFe G4 +dsrAB +cysIJ +soxABC +phsA +sqr +N +napA +narG +nirB +nrfA +NirS +norB +nosZ +nifH +hao +hcp +octR +sir +AbcA +assAD1 +dhaA +asrA +O2 +coxABC +ccoNOP +cydAB +Present in >50% genomes +Present in 30–50% genomes +Other +ArsRed +SeRed +merA +Fig. 4 Core metabolic genes detected across phylogenetic clusters inhabiting GB sediments. Presence of core metabolic genes involved in carbon +metabolism, hydrocarbon (HC) degradation and respiration. Shaded colors: Gene present in 30–50% of genomes/phylogenetic cluster. Solid colors: Gene +present in 50–100% of genomes/cluster. C1 C1- compound metabolism, H2 hydrogen metabolism, N nitrogen metabolism, S sulfur metabolism, O2 oxygen +metabolism, ArsRed arsenate reductase, SeRed selenate reductase, Gly glycolysis, Glu gluconeogenesis, Prop propane. Number in brackets: number of +genomes belonging to individual phylogenetic clusters. Grey circle: Bootstrap support > 70%. Asterisk: Deltaproteobacteria includes genomes from both +Deltaproteobacteria and Thermodesulfobacteria. 1pflD and assA are often difficult to discriminate from other glycyl radical enzymes, therefore, an additional +phylogenetic analysis can be found in Supplementary Figure 8. 2Phylogenetical analyses of substrate specificity of acd genes can be found in +Supplementary Figure 7. A complete list of metabolic genes can be found in Supplementary Data 10 +ARTICLE +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +6 +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications + +nor with the pyruvate formate lyase or other characterized +GREs35, suggesting that those might utilize different substrates, +such as carbohydrates or peptides. +Respiratory processes. Next, we investigated the GB microbial +communities for their involvement in respiratory processes. +Overall, more bacterial and archaeal genomes contained genes +that encode anaerobic rather than aerobic respiratory pathways, +consistent with rapidly depleted oxygen levels within the first few +millimeters of the sediment (Fig. 4, Supplementary Data 10)18. +Cytochrome c oxidases occurred in ~10% of genomes, but were +mainly limited to Bacteroidetes, Epsilon-/and Gammaproteo- +bacteria, and Verrucomicrobia. Conversely, genes for hydrogen, +nitrogen, sulfur and potentially arsenate and selenate cycling were +more widespread. We detected [FeFe]-hydrogenases in ~10% of +genomes and these mostly belonged to Group A, which can be +involved in fermentative hydrogen evolution37. Approximately +70% of genomes encoded for [NiFe]-hydrogenases belonging to +Group 1 (a–e and h; membrane-bound hydrogen-uptake hydro- +genases involved in hydrogenotrophic respiration), Group 3 (a–d; +cytosolic bidirectional hydrogenases) and Group 4 (b,d,e and g; +membrane-bound, +hydrogen-evolving +hydrogenases; +Supple- +mentary Data 10)37. The most common [NiFe]-hydrogenase was +found in ~25% of genomes, and belongs to Group 3b that is +involved in NADPH oxidation coupled to hydrogen evolution. +Genes involved in the nitrogen and sulfur cycle were mostly +restricted to bacteria, whereas archaeal nitrogen cycling genes +were limited to nifH in Methanomicrobia (Fig. 4, Supplementary +Data 10). Genes for dissimilatory nitrate reduction to ammonium +(DNRA) (narGH/napAB and nirBD/nrfAH) were present in few +Bacteroidetes +(i.e. +B27_G6, +B58_G6), +Epsilonproteobacteria +(B6_G4, B37_G6) and several Gammaproteobacteria (Methylo- +coccaceae, Thiotrichales). More commonly, we detected DNRA +GB-EvMd-Methermicoc-1 AIX10984 +B22_G9 (ANME-1) +Methanocorpusculum labreanum ABN07725 +B64_G16 (ANME-1) +Methanohalobium evestigatum YP_003726594 +GB-EvMd-Mhalo-mcrA AIX10982 +GB-EvMd-ANME-1 AIX11003 +Bathyarchaeota BA2 KPV61791 +Methanosarcina spp. +Methanococcoides burtonii YP_567018 +GB-EvMd-ANME-2 AIX10993 +Methanomassiliicoccus intestinalis YP_008072226 +otorris spp. +Methan +B49_G1 (Syntrophoarchaeum) +B39_G2 (ANME-1) +Syntrophoarchaeum caldarius OFV68281 +GZfos13E1 AAU82276 +Methanofollis liminatans WP_004037742 +Methanocaldococcus spp. +Methanothermobacter spp. +Methanosphaerula palustris YP_002467317 +Methanothermococcus spp. +Uncultured KT387810 +Syntrophoarchaeum butanivorans OFV65760 +Methanobacterium spp +. +p +p +s +alu +g +e +r +o +n +a +h +t +e +M +Methanoculleus bourgensis YP_006545160 +ex4484_138 +Methanosaeta harundinacea YP_005919503 +nocella spp. +Metha +Methanolacinia petrolearia YP_003895599 +GB-EvMd-GrpE AIX10996 +B49_G1 (Syntrophoarchaeum) +Syntrophoarchaeum caldarius OFV67100 +GB-EvMd-DBrGrpIV AIX11002 +Methanobrevibacter spp. +Methanotorris spp. +GB-EvMd-Methermicoc-2 AIX10985 +B49_G1 (Syntrophoarchaeum) +Methanopyrus spp. +Syntrophoarchaeum caldarius OFV67773 +Syntrophoarchaeum caldarius OFV68676 +Bathyarchaeota BA1 KT387805 +GB-EvMd-Mmseep-1 AIX10987 +Methanolobus psychrophilus YP_006922405 +B75_G16 (DeepCrenGroup1) +B9_G1 (ANME-2) +Methanosalsum zhilinae YP_004615938 +Syntrophoarchaeum butanivorans OFV65745 +Methanocaldococcus spp. +Syntrophoarchaeum butanivorans OFV67021 +Methanolacinia petrolearia YP_003895179 +GB-EvMd-DBrGrpIII AIX10991 +B25_G9 (GomArc1) +B39_G2 (ANME-1) +B48_G6 (ANME-1) +Methanomethylophilus alvus YP_007713068 +Methanothermus sp. +Methanothermobacter spp. +Methanococcus spp. +Methanobacterium spp. +Syntrophoarchaeum butanivorans OFV66176 +GB-EvMd-Mplanus AIX10988 +GB-EvMd-Mcoc AIX10989 +GB-EvMd-Mmseep AIX10986 +Methanosphaera stadtmanae CAE48306 +B65_G16 (GomArc1) +Methanospirillum hungatei ABD41854 +Methanobrevibacter spp. +GB-EvMd-DBrGrpII AIX11000 +Tree scale: 0.1 +Butane +X-Alkane +Methane +Fig. 5 Maximum likelihood phylogenetic tree of the methyl-Coenzyme M reductase (McrA) protein detected in GB genomes. Bold labels: McrA detected in +GB genomes (see also Supplementary Data 10). Black circle: Bootstrap support ≥ 70 (number of bootstraps determined using the extended majority-rule +consensus tree criterion). RaxML was run as raxmlHPC-PTHREADS-AVX -f a -m PROTGAMMAAUTO -N autoMRE. The tree file is available in +Supplementary Data 11 +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +ARTICLE +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications +7 + +genes distributed separately over several genomes. A complete +denitrification pathway (napA/narGH, nirK/nirS, norBC, nosZ) +was present in a few genomes, including one Bacteroidetes +(B2_G4), some Epsilonproteobacteria (i.e. B135_G9) and several +Gammaproteobacteria genomes (Halieaceae, Thiotrichales); indi- +vidual denitrification genes were found scattered across different +taxonomic lineages. Genes involved in anaerobic ammonium +oxidation (anammox) were not found, consistent with low nitrate +and nitrite concentrations in GB sediments38. Genes for the +dissimilatory reduction of sulfate to sulfide (sat, aprAB and +dsrAB) were found in few archaea (i.e. Archaeoglobales) and +several bacteria including Deltaproteobacteria, Gammaproteo- +bacteria and Zixibacteria. The sulfur-oxidation (SOX) system +(soxAX, soxYZ, soxB, soxCD) showed a restricted phylogenetic +distribution and was only located in Epsilonproteobacteria and +Gammaproteobacteria. While on average ~10% of all genomes +contained genes for sulfur and nitrogen cycling, complete +pathways for these processes were present in only few genomes. +Redundancy and interconnectivity among GB microbes. To +assess whether hydrothermal sediments not only host a greater +phylogenetic but also metabolic diversity than background sam- +ples (Fig. 2), we next investigated the spatial distribution of core +metabolic genes across all sites and taxa. Regardless of their +origin, most genomes encoded genes for general carbon cycling +(CAZymes, peptidases, gluconeogenesis, glycolysis), fermentation +and lipid oxidation (Fig. 6 and Supplementary Data 10). +Respiratory genes were restricted to cooler, shallower samples but +present in both background and hydrothermal sediment cores. +For example, denitrification genes, SOX genes or the cytochrome +c oxidase were found only in the shallower, colder sediments +(temperature ~5 °C) and were present in ~20-30% of genomes. In +contrast, these genes were represented in only ~0-4% of genomes +in deeper, hotter samples (temperature range of 10 °C-60 °C). +Exceptions were genes for sulfate/sulfite reduction, such as dsrAB, +that were still found in ~8% of genomes in deeper, hotter sedi- +ments. Compared to background samples, genes involved in C1- +metabolism and hydrogenases were more frequently found in +hydrothermal sediments. In background sediments only one +Bathyarchaeotal genome contained carbon fixation-related genes +(cdhAB), while genes for methane cycling (mcrA) were unde- +tectable. Hydrogenases belonging to Group 4 g, which represent +membrane-bound hydrogenases that generate a proton-motive +force for energy generation, were absent from the background but +present in ~25-30% of genomes across all hydrothermal samples +(Fig. 6 and Supplementary Data 10). These findings suggest that +methane and hydrogen might be important drivers of metabolic +processes in GB hydrothermal sediments. +With few exceptions most metabolic genes were encoded in +several taxonomically distinct lineages. For example, C1-related +genes (with the exception of mcrA) and genes related to beta- +oxidation, hydrogen, nitrogen, sulfur and oxygen cycling were +found in ~10 different phylogenetic lineages; fermentation genes +were present in most phylogenetic clusters of both the archaeal +and bacterial community. While the studied genomic dataset +from the cold and hydrothermal samples were not represented by +an equal number of genomes (average of ~9 and ~60 genomes per +habitat type, respectively), we still find that those genomes +represent the community well in terms of phylogenetic diversity +(Supplementary Figure 4). Additionally, when searching for a +subset of these core metabolic genes in binned and unbinned +contigs from the complete assembly (only considering contigs +> 2,000 bp), we observed a similar trend (Supplementary Data 14). +For example, fermentation genes were abundant across all sites, +denitrification genes were more common in cold and shallow +samples and mcrA was completely absent from the background +samples. Overall, these findings suggest that the GB genomes are +representative of the community as a whole, and that they reflect +key metabolic differences between the microbial communities +present in hydrothermal and background samples. +Discussion +In this study, we employed the largest genomic sampling of GB +sediments to date to investigate the interplay of community +composition and functional diversity. Compared to earlier work +on Guaymas Basin sediments20, the higher sampling number and +inclusion of background samples allowed to better describe the +enhanced diversity present in these sediments and shed light on +the drivers of community assembly. In contrast to previous stu- +dies showing that sulfidic- and methane-rich seep sediments host +a lower microbial diversity compared to non-seep marine +sediments39,40, we demonstrate that GB hydrothermal sediments +contain a diverse community that is enriched in archaea com- +pared to a less diverse, bacterial-dominated community found in +nearby cold sediments. Therefore, the more extreme conditions in +hydrothermal sediments, which include steep thermal and geo- +chemical gradients17,27, appear not to inhibit microbial diversity. +Due to difficulties in isolating sufficient amounts of DNA from +deeper, hotter samples, we cannot exclude that diversity may +decline in those sediments. Earlier work reported a decrease in +cell numbers with increasing depth that did not necessarily cor- +relate with a decrease in OTU numbers25, potentially explaining +our difficulties in isolating sufficient amounts of DNA but sup- +porting our assumption that steep temperature gradients do not +necessarily inhibit microbial diversity. Especially samples from +core 4569_9 experience a highly variable, fluctuating thermal +regime over time, where even surficial layers can vary from 20 °C +to 70 °C, as determined by multi-day continuous thermal logging +(Supplementary Figure 1)17. In response to such conditions, +microbes must either adapt, have a wide thermal optimum, as +shown for some ANME-1 archaea23, or be able to recolonize the +sediment after a temperature sweep from a surficial reservoir41. +Here, we propose that the diverse communities inhabiting +hydrothermal sediments could serve as a flexible seed bank for +the deeper, hotter sediments as well as for highly fluctuating +environmental gradients in shallow sediments5,25,42. +The differences we observed in community composition across +sites were not always translated into obvious changes in func- +tional capacities of those communities. For example, we detected +abundant genes for carbon cycling and fermentation across all +sites, while other metabolic processes such as respiration, were +limited to shallow sediments but present in both background and +hydrothermal sediments. Respiratory processes were often parti- +tioned among the community and only few genomes were +encoding for full pathways. Metabolic handoffs have been +observed in other microbial communities and could allow a +flexible +interchange +of +metabolites +between +changing +populations43,44. Another metabolic feature that could allow for +greater ecosystem stability could be metabolic plasticity, i.e. +switching metabolic processes in response to changes in envir- +onmental conditions. We found indications for such plasticity in +several bacterial genomes, especially within the Delta- and +Gammaproteobacteria that might couple the reduction of sulfur +with the oxidation of carbon, lipids or hydrocarbons. While we +cannot determine which processes are active, enhanced genotypic +diversity might provide an additional adaptation strategy to +variable environmental conditions. +The only functional categories that were consistently enriched +across all hydrothermal sites and almost absent in background +sediments were group 4g hydrogenases and pathways for +ARTICLE +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +8 +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications + +6 +3 +3 +3 +5 +3 +5 +0 +2 +0 +1 +0 +0 +1 +2 +0 +2 +0 +2 +0 +2 +1 +1 +0 +3 +0 +0 +0 +0 +1 +1 +0 +0 +1 +0 +1 +0 +0 +1 +0 +1 +0 +1 +0 +4 +1 +3 +0 +6 +3 +5 +3 +0 +0 +3 +2 +6 +3 +1 +0 +2 +1 +0 +0 +0 +0 +1 +0 +6 +0 +3 +0 +5 +0 +0 +0 +0 +0 +1 +0 +2 +0 +0 +0 +2 +1 +0 +1 +1 +0 +0 +0 +5 +0 +1 +0 +0 +1 +0 +0 +15 +10 +10 +9 +10 +9 +0 +3 +6 +3 +3 +2 +1 +2 +2 +3 +3 +0 +0 +0 +1 +1 +1 +5 +7 +12 +6 +2 +9 +10 +6 +8 +3 +6 +6 +8 +4 +6 +5 +5 +5 +5 +5 +4 +7 +4 +5 +6 +4 +2 +2 +2 +2 +2 +2 +2 +3 +3 +3 +1 +7 +4 +0 +3 +4 +6 +1 +6 +3 +5 +6 +2 +1 +5 +2 +3 +2 +0 +1 +7 +2 +4 +2 +8 +9 +1 +4 +1 +6 +0 +1 +1 +1 +8 +1 +1 +9 +8 +0 +1 +0 +1 +2 +1 +1 +9 +6 +6 +4 +2 +5 +2 +1 +1 +8 +4 +3 +3 +3 +4 +0 +0 +1 +0 +2 +0 +2 +2 +1 +1 +1 +1 +7 +8 +8 +8 +4 +7 +7 +5 +4 +7 +5 +9 +0 +0 +2 +0 +0 +2 +1 +0 +2 +9 +5 +7 +7 +7 +7 +6 +9 +1 +5 +1 +0 +0 +1 +8 +8 +1 +1 +1 +1 +0 +0 +5 +3 +0 +0 +2 +0 +0 +2 +0 +0 +4 +0 +0 +5 +1 +7 +5 +4 +9 +9 +0 +1 +1 +3 +2 +2 +8 +2 +6 +6 +9 +1 +3 +1 +3 +7 +1 +2 +1 +4 +4 +4 +4 +5 +5 +5 +4 +1 +0 +1 +0 +0 +4 +13 +4 +4 +5 +3 +3 +2 +2 +2 +1 +3 +2 +1 +8 +4 +5 +5 +2 +28 +20 +27 +22 +13 +14 +11 +4 +11 +12 +14 +8 +11 +10 +4 +3 +0 +1 +4 +5 +7 +5 +0 +0 +2 +1 +14 +12 +6 +4 +4 +4 +6 +6 +6 +6 +1 +1 +1 +1 +0 +0 +1 +1 +2 +4 +6 +5 +19 +14 +12 +10 +1 +0 +7 +7 +10 +9 +2 +3 +4 +6 +0 +0 +1 +1 +2 +2 +3 +2 +2 +1 +1 +2 +3 +2 +2 +2 +6 +3 +8 +10 +14 +9 +17 +13 +4 +1 +14 +11 +2 +2 +4 +2 +9 +7 +5 +3 +8 +3 +4 +1 +6 +3 +9 +1 +8 +2 +9 +5 +1 +2 +8 +1 +4 +6 +2 +6 +4 +1 +1 +1 +0 +1 +0 +6 +1 +3 +0 +1 +0 +0 +3 +1 +5 +3 +8 +3 +7 +8 +5 +7 +4 +0 +2 +0 +1 +0 +3 +1 +3 +8 +2 +1 +4 +0 +2 +5 +1 +1 +8 +1 +6 +0 +7 +2 +1 +5 +0 +0 +0 +6 +1 +2 +4 +1 +0 +5 +1 +9 +1 +2 +1 +2 +3 +1 +6 +1 +0 +1 +1 +8 +0 +2 +9 +0 +3 +0 +4 +9 +1 +0 +7 +9 +4 +0 +6 +4 +4 +1 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +1 +1 +1 +1 +4 +1 +0 +3 +3 +1 +1 +0 +1 +1 +1 +1 +1 +7 +5 +5 +1 +1 +1 +1 +1 +0 +0 +2 +0 +0 +2 +1 +0 +4 +1 +0 +Background +Vent1 +(out) +Vent1 +(intermediate) +Vent1 +(center) +Vent2 +Vent3 +cydA +ccoN +coxA +sqr +phsA +soxB +soxX +soxA +sir +cysJ +asrA +dsrB +dsrA +octR +hpc +hao +nifH +nosZ +norB +nirS +nirK +nrfH +nirB +narG +napA +NiFe G4 +NiFe G3d +NiFe G3c +NiFe G3b +NiFe G3a +NiFe G1 +FeFe +AbcA +epi +pccB +acd +fadD +cdhB +cdhA +acsE +acsB +cooS +mcrA +acs +ackA +adh +ald +fdoG +porA +korA +21–24 cm +3°C +0–3 cm +3°C +0–3 cm +3°C +0–3 cm +6°C +12–15 cm +28°C +21–24 cm +41°C +4–6 cm +~60°C +0–3 cm +10°C +12–15 cm +34°C +0–3 cm +21°C +9–12 cm +48°C +Fermentation +HCs +C1 +Hydrogenases +Nitrogen +Sulfur +O2 +4567_28 +4488_9 +4569_4 +4569_2 +4569_9 +4571_4 +0 +25 +50 +75 +100 +Size of circle: +No. of genes/ +Total no. of genomes +No. of phylogenetic +clusters that encode +for a metabolic gene: +x +Fig. 6 Metabolic profile across different GB sediment sites, depth profiles and temperature regimes. Shown is the number of core metabolic genes relative +to the total number of genomes (in %) per site, depth and temperature regime. Temperatures are averages for the 2 or 3 cm thick sediment layers from +which DNA was isolated. Background samples: Cold GB samples without hydrothermal activity. Vent1–3: Hydrothermal sediment sampling locations, see +also Fig. 1. ID at the bottom: number codes designating every Alvin dive and sediment core (see also Supplementary Data 1 for further explanation). A +complete list of metabolic genes can be found in Supplementary Data 10. Number in circles: Number of phylogenetic clusters that encode for individual +core metabolic genes at each site +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +ARTICLE +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications +9 + +methanogenesis and methane oxidation. Group 4g hydrogenases +are not well characterized but are generally described to be +membrane-bound hydrogenases that allow for energy-generation +by establishing ion gradients over the membrane45. These com- +plexes are often found in thermophiles, such as Pyrococcus fur- +iosus45, and could potentially provide a selective advantage in +hydrothermal sediments over other energy-generating systems. +While trace concentrations of biogenic methane are present in +background sediments (Supplementary Data 1, Supplementary +Methods), the inability to detect mcrA in these samples could be +because of sequencing depth; in contrast detecting mcrA in +hydrothermal sediments appears to be linked to microbial +methane oxidation produced by pyrolysis of organic matter17. +Within the phylogenetically and functionally diverse commu- +nity inhabiting GB, the metabolic repertoire shows a high degree +of functional redundancy across different phyla, i.e. different taxa +encode the same metabolic function and thus might substitute for +one another. Therefore, even if community composition varies, +metabolic function is predicted to be relatively stable. Like phy- +logenetic diversity, functional redundancy could benefit the +community when dealing with perturbations in environmental +conditions and has been observed in other environments +including the global marine or humane microbiome46,47. While +any stressor, such as temperature, might result in the removal of a +given taxon, functional redundancy across different lineages that +are each tolerant to some degree of environmental fluctuations, +and together cover a wide window of environmental conditions, +ensures the stability of community function. This is consistent +with the ‘it’s the song not the singer’ (ITSNTS) theory, which +assumes that surviving taxa replace perturbed taxa (‘the singers’) +and thereby allow nutrient cycles (‘the song’) to persist in the +environment48. This theory is consistent with our findings, in +which we not only observe phylogenetically diverse but also +functionally redundant communities. Altogether, the phyloge- +netic diversity, metabolic partitioning as well as functional +redundancy that we observe appear to be characteristics of +microbial communities in these dynamic hydrothermal vent +sediments. +One +question +that +arises +when +observing +functional +redundancy within a microbial community is whether this +redundancy enhances species competition and de-stabilizes the +community49,50. While it is not in the scope of this study to +discern niche patterns, we would assume that the high redun- +dancy in our dataset might still allow microbes to inhabit dif- +ferent niches. Two mechanisms that could allow co-existence of +supposedly redundant microbes could be metabolic auxotrophies +or heterogeneity in limiting resources and/or environmental +conditions50–52. Amino acid auxotrophies can create community +interdependencies, which could balance competition and thereby +stabilize microbial communities53. We do see indications for such +interdependencies in our dataset, where auxotrophies are com- +mon in small genomes belonging to CPR bacteria (Supplementary +Data 10). Additionally, we assume that the diverse GB-inhabiting +communities are stabilized by the high abundance of substrates +present in hydrothermal sediments, which might reduce com- +petition and allow taxa to coexist. Finally, while genes for core +metabolic processes showed a high redundancy across our data- +set, we hypothesize that enzymes involved in substrate degrada- +tion are undergoing substantial diversification with respect to +their substrate spectra. The diversity of genes involved in car- +bohydrate (mcrA, CAZYmes), lipid (acyl-CoA dehydrogenase) +and peptide degradation and the expanding substrate range and +diversity of hydrocarbon-degrading genes, such as mcrA, supports +this notion16,20,54. A limitation of the current study that com- +plicates a definite description of the diversity patterns and func- +tional redundancy present in Guaymas sediments is the low +sample number and limited number of bins recovered from a +subset of samples (i.e. 4567_28 and 4488_9); given the limitations +of deep-sea sampling, different habitat and sediment types are +represented unevenly. Activity-based analyses of large sample +numbers, i.e., metatranscriptomics, would more rigorously link +genetic patterns to their environmental determinants. +Guaymas Basin is a hotspot for microbial biodiversity and an +ideal study site to investigate the functional diversity of hydro- +thermally influenced seafloor sediments. Here we establish that +these hydrothermal sediments contain a large number of archaeal +and bacterial lineages, including several uncultivated phylum- +level lineages that have not been described from other habitats. +Intriguingly, hydrothermal GB sediments hosted a greater +diversity compared to surrounding non-hydrothermal sediments +contrasting previous work on methane seep communities39,40. +These differences are likely linked to the unique environment in +GB sediments characterized by by convective mixing of fluids +resulting in variable thermal regimes, and admixture of hydro- +thermal carbon and energy sources. Most functional properties +were shared widely among different phylogenetic lineages across +different sampling sites with a greater functional redundancy of +metabolic processes found in hydrothermal sediments. One +unique functional trait of hydrothermal compared to background +sediments was the presence of methane cycling genes among +novel lineages, including a new deep-branching Crenarchaeota +group. We propose that the combination of dynamic seep and +hydrothermal conditions in Guaymas Basin enhances microbial +diversity, and sustains a distinctive microbial community, whose +functional complexity and redundancy reflects the intricate and +dynamic geochemical and thermal landscape of this habitat. +Methods +Sampling. Guaymas Basin sediment samples were collected from the Gulf of +California (27°N0.388, 111°W24.560) at a depth of approximately 2000 m below +the water surface. Sediment cores were collected during four Alvin dives (4488, +4569, 4567, and 4571) in 2008 and 2009 (Supplementary Data 1). Sample site +photos were compiled from the Alvin frame grabber site (http://4dgeo.whoi.edu/ +alvin). Intact sediments were collected during Alvin dives using polycarbonate +cores (45-60 cm in length, 6.25 cm interior diameter), subsampled into cm layers +under N2 gas in the ship’s laboratory and immediately frozen at −80 °C. Eleven +sediment subsamples for DNA isolation from different depth profiles yielded +sufficient genomic DNA for metagenomic sequencing (Supplementary Data 1). +Higher temperature samples were tested as well but did not yield sufficient DNA +for metagenomic sequencing. Metadata for all dives, including details on the +geochemistry (i.e. methane concentrations and dissolved organic carbon con- +centrations and δ13C values, sulfate and sulfide concentrations) and thermal pro- +files of the sampling sites, are available to compare microbial community +composition across sediment cores (Supplementary Data 1, Supplementary +Methods)17. Additional images and descriptions of the sampling locations are +published in a survey of different Guaymas Basin habitats18. +Metagenomic sequencing and assembly. Total DNA from ≥ 10 g of sediment +from each of the eleven samples (see above) was extracted using the MoBio +PowerMax soil kit using the manufacturer’s instructions. DNA concentrations were +measured using a Qubit™ 3.0 Fluorometer and a final concentration of 10 ng/µl of +each sample (using a total amount of 100 ng) was used to prepare libraries for +paired-end Illumina (HiSeq–2500 1TB) sequencing. Illumina library preparation +and sequencing was performed at the Joint Genome Institute (JGI). Sequencing was +performed on an Illumina HiSeq 2500 machine using the paired end 2 × 125 bp +run-type mode. All runs combined provided a total of ~280 gigabases of sequen- +cing data (Supplementary Data 2) Quality control and sequence assembly was +performed by JGI. Briefly, sequences were trimmed and screened for low quality +sequences using bbtools (https://jgi.doe.gov/data-and-tools/bbtools/) and assem- +bled using megahit v1.0.6 using the following options: --k-list +23,43,63,83,103,12355. Summary statistics for the number of generated reads and +the quality of the metagenomic assembly is provided in Supplementary Data 2. For +further binning, only scaffolds ≥ 2000 bp were included. +Metagenomic binning. Metagenomic binning was performed on individual +assemblies using the binning tools ESOM, Anvi’o and Metabat. ESOM binning was +performed by calculating tetranucleotide frequencies of scaffolds with a minimum +length of 2000 bp using the K-batch algorithm for training after running the perl +ARTICLE +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +10 +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications + +script esomWrapper.pl56. The resulting Emerging Self-Organizing Maps (ESOM) +were manually sorted and curated. Bins were extracted using getClassFasta.pl +(using -loyal 51). The binning process was enhanced by incorporating reference +genomes as genetic signatures for the assembled contigs into ESOM. For Anvi’o +(v2.2.2) the metagenomic workflow pipeline that incorporates CONCOCT was +used for binning57. Briefly, coverage information was obtained by generating eleven +mapping files for each assembly file by mapping all high-quality reads of each of +the eleven samples against the assembly of one sample using the BWA-MEM +algorithm in paired-end mode (bwa-0.7.12-r1034; using default settings)58. The +resulting sam file was sorted and converted to bam using samtools (version 0.1.19)59. +The bam file was prepared for Anvi’o using the script anvi-init-bam and a contigs +database generated using anvi-gen-contigs-database. These two files were further +used as input for anvi-profile. Generated profiles for the eleven different assemblies +were combined using anvi-merge and the resulting bins summarized using anvi- +summarize (-C CONCOT). If not mentioned otherwise, the scripts were used with +default settings. Finally, binning was performed using metabat (v1)60. As described +for Anvi’o the used input files consisted of the scaffold files (≥2000 bp) and the +mapping files to recover bins both by sequence composition and abundance across +samples. First, each of the mapping files were summarized using jgi_summar- +ize_bam_contig_depths and then metabat was run using the following settings: +--minProb 75 --minContig 2000 --minContigByCorr 2000. Results from the three +different binning tools were combined using DAS Tool (version 1.0)61. Therefore, +for each of the binning tools a scaffold-to-bin list was prepared and DAS Tool run +on each of the eleven scaffold files as follows: DAS_Tool.sh -i Anvio_contig_list.tsv, +Metabat_contig_list.tsv,ESOM_contig_list.tsv -l Anvio,Metabat,ESOM -c scaffolds. +fasta --write_bins 1. +The accuracy of the binning approach was evaluated by calculating the +percentage of completeness and contamination using CheckM lineage_wf (v1.0.5; +Supplementary Data 3)62. Genomes were only analyzed further if they were more +than 50% complete and showed a contamination below 10%. Contaminants that +were identified based on their phylogenetic placement (wrong taxonomic +assignment compared to the average taxonomic assignment of the genes assigned +to each bin), GC content (>25% difference compared to the mean of all scaffolds +assigned to each bin) or abundance (>25% differences compared to the mean +abundance of all scaffolds assigned to each bin) were manually removed from +individual genomes. This yielded a total of 247 archaea and 304 bacterial genomes. +Relative abundance. To determine the relative abundance of each genome across +the elven sequenced sediment samples, we mapped the contigs from all binned +genomes (i.e., using the “whole MAG”) against the high-quality reads of each +individual metagenome (generating eleven sam files). The sam output was sorted +and converted to bam as described above and we then used the metabat output, +which describes the read counts recruited by each contig, for further analyses. All +analyses were performed in R (version 3.3.3). On average, ~47% of the high-quality +metagenomic sequences could be binned, with the notable exception of the sample +from 4567_28, from which the recovered MAGs only recruited ~18% of reads for +an undetermined reason. +To determine the average abundance of major taxonomic groups (referred to as +cluster, which were determined by the phylogenetic analysis described below), +contigs were first assigned to their phylogenetic cluster (see description for the +phylogenetic analysis below) and were then summarized using the ddply function +from the plyr package63. These clusters do not represent a specific taxonomic rank +but were chosen to account for both phylogenetic diversity (i.e., Crenarchaeota are +usually represented at order rank or lower if possible) as well as available genomes +(the different phyla of the CPR superphylum were ranked together because they +were represented by only few genomes). The counts recruited by each taxonomic +group were normalized by the total length of contigs belonging to each cluster, the +library size of the individual metagenomes and multiplied by 1000 for better +readability. The normalized relative abundance was plotted using the heatmap.2 +function in the gplots package. The summary statistics are provided in +Supplementary Data 7 (only includes clusters with ≥3 lineages). +To determine the relative abundance of the ribosomal protein S3 (RPS3) across +samples, RPS3 was extracted from all eleven assemblies (only considering contigs +>2000 bp) using phylosift (v1.0.1 using options: phylosift all --keep_search +--custom marker_list.txt). In total, we identified 1227 RPS3 sequences in the +dataset, 486 of which belonged to binned contigs (~40%) and, therefore, RPS3 +could be successfully recovered from ~82% of bins; (Supplementary Table S4). The +unaligned nucleotide sequences were concatenated and used as an input to run bwa +against all eleven metagenomes to determine their relative abundance across +samples. Read counts were extracted using samtools, normalized by gene length +and library size and plotted using the ggplot2 package in R. +Phylogenetic analyses. Phylosift was used to extract marker genes for the phy- +logenetic placement of the assembled metagenomic bins64. A set of 37 single-copy, +protein-coding housekeeping genes was chosen for a further phylogenetic analysis +(Supplementary Data 4). To generate a reference dataset, archaeal (all available +genomes) and bacterial genomes (selected genomes that include at least three +members from each genus and a preference for type strains whenever possible) +were downloaded from NCBI on March 2017. Next, all reference genomes and GB +genomes (fasta files) were used as an input for phylosift (v1.0.1) using the ‘phylosift +search’ followed by the ‘phylosift align’ mode. The concatenated protein alignments +of 37 elite marker genes (concat.updated.1.fasta) were combined for all genomes of +interest and trimmed using TrimAL (version 1.2) using the automated1 setting65. A +phylogenetic tree was generated using a maximum likelihood-based approach +using RAxML (version 8.2.10, called as: raxmlHPC-PTHREADS-AVX -f a -m +PROTGAMMAAUTO -N autoMRE)66. The tree was visualized using the Inter- +active Tree Of Life (iTOL) webtool67. For better visualization, the initial tree was +reduced to only include references that were branching close to GB genomes and +included a 224 genomes from cultured representatives and 330 genomes from +uncultured genomes (including metagenome-assembled genomes, enrichment +cultures, co-cultures and single-cell assembled genomes). All of these genomes were +used to calculate an average amino acid identity across all genomes using com- +parem (v0.0.23, function aai_wf; https://github.com/dparks1134/CompareM). The +AAI was used as a main measure to distinguish the new phylum-level genomes that +were discovered with the phylogenetic approach. Therefore, the average AAI of +each phylum was calculated and compared to all remaining phyla, especially those +branching close to the phyla of interest (see also Supplementary Data 6). +The 16S rRNA gene sequences were extracted using phylosift (settings are +described above) and barrnap (https://github.com/tseemann/barrnap, v0.7, +settings: --kingdom arc/bac --lencutoff 0.2 --reject 0.3 --evalue 1e-05) and aligned +to the SILVA SSURef_NR99 database (release 13.12.2017) using the SILVA +webaligner68. The alignment was manually curated in ARB. A phylogenetic tree +was generated using a maximum likelihood-based approach using RAxML +(settings: raxmlHPC-PTHREADS-AVX -T 10 -f a -m GTRGAMMA -N autoMRE +-p 12345 -x 12345). 16S rRNA gene sequences were manually checked for +contamination in cases with an inconsistent phylogenetic assignment between 16S +rRNA gene sequences or the 37 protein-coding marker genes. The whole contig +was discarded, when all assigned proteins on the contig with the 16S rRNA gene +showed a different taxonomic assignment (using blastp) compared to the +remaining scaffolds of the respective genome. +A similar phylogenetic approach was taken to phylogenetically characterize +other key genes of interest (i.e. hydrogenases, mcrA, glycyl radical enzymes, acyl- +CoA dehydrogenases (acd)). Genes of interest were identified in GB genomes using +KAAS, HMMER, blastp or the HydDB webserver (for details see below). Published +reference genes were extracted using the NCBI and Uniprot webservers (McrA, +glycyl radical enzymes, ACD) as well as the HydDB webserver (hydrogenases). For +the glycyl radical enzymes, proteins identified as PflA, AssA, BssA, HbsA, MasD, +NmsA in KEGG or a custom blast search were combined in a single analysis. For +the ACD phylogeny, the KAAS IDs K00248, K00249, K06445, K00255, K06446 and +K09479 were included to build a phylogenetic tree. Protein sequences from GB and +reference genomes were combined and aligned using muscle (v3.8.31, default +settings), trimmed using TrimAL and a phylogenetic tree generated using RAxML +as described above. Protein-coding genes falling on long branches were manually +checked using blastp on the NCBI webserver and discarded if the annotation was +not hydrogenase, acyl-CoA dehydrogenase or glycyl radical enzyme. +Annotations and metabolic analyses. Gene prediction for individual genomes +was performed using prodigal (V2.6.2, default settings)69. The genomes contained +on average 1,665 predicted proteins for archaea (min = 500 and max = 4,685) and +2,491 for bacteria (min = 636 and max = 6,964) (Supplementary Data 3). Meta- +bolic reconstructions were done for each individual genome, but in several cases +the results were summarized for major taxonomic lineages, or clusters. These +clusters do not represent a specific taxonomic rank but were chosen to account for +both phylogenetic diversity (Crenarchaeota are usually represented at order rank) +as well as available genomes (the different phyla of the CPR superphylum were +ranked together because they were represented by only few genomes). +Predicted genes of individual genomes were further characterized using KAAS +(KEGG Automatic Annotation Server; Supplementary Data 10)70. Therefore, +protein sequences of each of the individual genomes were uploaded to the KAAS +webserver using the ‘Complete or Draft Genome’ setting (used parameters: +GHOSTX, custom genome dataset, BBH assignment method). For a detailed +pathway analysis the KO numbers were downloaded, concatenated and merged +with a KO-to-pathway metadata file in R (Supplementary Data 10). +Additionally, we searched for key metabolic genes using custom blastp and +hmmer databases43. A Curated Database of Anaerobic Hydrocarbon Degradation +Genes (AnHyDeg) and the MEROPS database were used to identify hydrocarbon +degradation genes as well as peptidases in the concatenated proteins sequences +of all GB genomes using blastp (e-value threshold of 1e-20; Supplementary +Data 10)71–73. Hits were discarded if they were related to core metabolic processes +(i.e., pyrimidine synthesis) or included heat-shock resistance proteins, precursor +proteins and signal peptides. Additionally, we utilized a custom hmmer as well as +the Pfam and TIGRFAM databases to search for key metabolic marker genes using +hmmsearch and custom bit-score cutoffs43,74,75. Hydrogenases were extracted from +the genomes using hmmsearch (e-value cut-off of 1e-20) and confirmed using a +web-based search using the hydrogenase classifier HydDB76. Finally, genes +encoding for carbohydrate degradation enzymes described in the Carbohydrate- +Active enZYmes (CAZYmes) database were identified using the dbcan webtool and +applying an e-value threshold of 1e-577. Protein localization was determined for +CAZYmes and peptidases using the command-line version of Psort (V3.0) using +the option --archaea for archaeal genomes. The results for the MEROPS and +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +ARTICLE +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications +11 + +CAZymes database searches are summarized in Supplementary Data 8 and 9. In +the case of protein-coding genes hitting to multiple genes in the before-mentioned +databases, the best hit was chosen based on their e-value and bit-score using +blast_best.pl (http://alrlab.research.pdx.edu/aquificales/scripts/). +Genes assigned to core metabolic pathways are summarized in Supplementary +Data 10. Hits for key metabolic marker genes found in major taxonomic clusters +(Fig. 3) were verified across different databases (KAAS, PFAM and TIGRPFAMs) +and cross-checked with results from close reference genomes that fell within the +same phylogenetic group as the genome of interest to reduce the chance of +contamination. Genes not found in close reference genomes were further validated +with blastp using the NCBI webserver tool. If a hit could not be confirmed or if the +top phylogenetic hit for whole contig was not consistent with the phylogenetic +assignment of the genome, it was removed from the genome. +Data availability +All sequence data and sample information are available at NCBI under BioProject +ID PRJNA362212. Accession numbers for individual genomes can be found in +Supplementary Data 3. Additionally, the raw data is provided in IMG/MER and the +IMG Genome IDs for the individual metagenomes are provided in Supplementary +Data 1. +Received: 13 June 2018 Accepted: 31 October 2018 +References +1. +Hug, L. A. et al. A new view of the tree of life. Nat. Microbiol 1, 16048 (2016). +2. +Parkes, R. J. et al. A review of prokaryotic populations and processes in sub- +seafloor sediments, including biosphere:geosphere interactions. Mar. Geol. +352, 409–425 (2014). +3. +Lozupone, C. A. & Knight, R. 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The work conducted +by the U.S. Department of Energy Joint Genome Institute, a DOE Office of Science User +Facility, is supported by the Office of Science of the U.S. Department of Energy under +Contract No. DE-AC02-05CH11231 provided to ND. This work was funded by a Sloan +Foundation Ocean Sciences fellowship (FG-2016-6301) and National Science Foundation +DEB: Systematics and Biodiversity Sciences (grant number 1753661) provided to B.J.B. +A.P.T. and Guaymas Basin fieldwork was supported by U.S. National Science Foundation +grants OCE-0647633 and OCE-1357238. +Author contributions +B.J.B., A.P.T. and N.D. conceived, designed the study, and were involved in writing the +manuscript. N.D. processed the data, reconstructed the genomes and performed the +analyses. +Additional information +Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- +018-07418-0. +Competing interests: The authors declare no competing interests. +Reprints and permission information is available online at http://npg.nature.com/ +reprintsandpermissions/ +Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in +published maps and institutional affiliations. +Open Access This article is licensed under a Creative Commons +Attribution 4.0 International License, which permits use, sharing, +adaptation, distribution and reproduction in any medium or format, as long as 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 images or other third party +material in this article are included in the article’s Creative Commons license, unless +indicated otherwise in a credit line to the material. If material is not included in the +article’s Creative Commons license and your intended use is not permitted by statutory +regulation or exceeds the permitted use, you will need to obtain permission directly from +the copyright holder. To view a copy of this license, visit http://creativecommons.org/ +licenses/by/4.0/. +© The Author(s) 2018 +NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-07418-0 +ARTICLE +NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.1038/s41467-018-07418-0 | www.nature.com/naturecommunications +13 + diff --git a/kb_45/content/tmp_files/load_file.txt b/kb_45/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e5998dce5346281b298f6a80368e868a13dc2fa --- /dev/null +++ b/kb_45/content/tmp_files/load_file.txt @@ -0,0 +1,2191 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf,len=2190 +page_content='ARTICLE Expansive microbial metabolic versatility and biodiversity in dynamic Guaymas Basin hydrothermal sediments Nina Dombrowski 1, Andreas P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Teske2 & Brett J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Baker1 Microbes in Guaymas Basin (Gulf of California) hydrothermal sediments thrive on hydro- carbons and sulfur and experience steep, fluctuating temperature and chemical gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The functional capacities of communities inhabiting this dynamic habitat are largely unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Here, we reconstructed 551 genomes from hydrothermally influenced, and nearby cold sediments belonging to 56 phyla (40 uncultured).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These genomes comprise 22 unique lineages, including five new candidate phyla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In contrast to findings from cold hydrocarbon seeps, hydrothermal-associated communities are more diverse and archaea dominate over bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genome-based metabolic inferences provide first insights into the ecological niches of these uncultured microbes, including methane cycling in new Crenarchaeota and alkane utilization in ANME-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These communities are shaped by a high biodiversity, partitioning among nitrogen and sulfur pathways and redundancy in core carbon-processing pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The dynamic sediments select for distinctive microbial communities that stand out by expansive biodiversity, and open up new physiological perspectives into hydrothermal eco- system function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 OPEN 1 Department of Marine Science, Marine Science Institute, University of Texas Austin, Port Aransas, TX 78373, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 2 Department of Marine Sciences, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Correspondence and requests for materials should be addressed to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' (email: acidophile@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com) NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications 1 1234567890():,; M icrobial communities inhabit every environment and are comprised of a multitude of different phyla, the majority of which are uncultured1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Among these environments, marine sediments contain abundant and phylo- genetically diverse microbial communities2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' High diversity has been suggested to emerge as a strategy for survival of microbes under fluctuating environmental conditions in nature5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While single-gene surveys allow us to address the phylogenetic diversity of microbial communities, metagenomic analyses provide a connection between diversity and the functional potential enco- ded within sedimentary communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Guaymas Basin (GB; Gulf of California, Mexico) is a young, active seafloor-spreading center characterized by high water col- umn productivity and fast sedimentation rates, leading to the accumulation of massive layers of organic-rich sediments that cover the hydrothermal spreading center and ridge flanks7–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The emplacement of hot basalt sills into organic-rich sediment transforms buried organic matter into CO2, H2, low-molecular- weight organic acids, ammonia, and hydrocarbons such as methane, ethane and benzene8,10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These compounds migrate to the sediment surface with rising vent fluids, where they fuel hydrocarbon-degrading microbial communities11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Among all hydrothermally generated hydrocarbons, methane has received considerable interest as greenhouse gas shaping global climate13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Porewater methane reaches millimolar concentrations while ethane ranges from 40-100 µM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Also present in these sediments are propane, n-butane and pentane, which accumulate at lower concentrations compared to methane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Altogether, hydrocarbons represent lucrative carbon sources for the resident microbial community11,14–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, hydrothermal circulation and seawater in-mixing provide the upper sediments with electron acceptors, among which sulfate is widely available in millimolar porewater concentrations and rarely depleted within hydro- thermal sediment cores11,14,17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In-situ microelectrode surveys detect small oxygen peaks within hydrothermal sediments near the mat-covered surface18,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These results are consistent with short-term dynamics of hydrothermal flow within minutes and hours17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, short-term dynamics overlay with longer- term hydrothermal activity changes over months and years18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' GB sediments have been shown to host diverse microbial communities with distinct roles in carbon cycling11,17,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In particular, microbial consortia perform the anaerobic oxidation of methane (AOM) in a syntrophic interaction consisting of anae- robic methane-oxidizing archaea (ANME) and bacterial sulfate reducers, typically Deltaproteobacteria, but including other thermophilic bacterial lineages, such as Candidatus Desulfo- fervidus auxilii21–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Anaerobic hydrocarbon degraders include Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Syntrophoarchaeum, which oxidizes butane in a syntrophic interaction with Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Desulfofervidus auxilii, or the butane- and propane oxidizing isolate BuS5, belonging to the Desulfosarcina- Desulfococcus cluster15,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Other common archaeal lineages include Marine Benthic Group D and Bathyarchaeota, while bacterial phyla include Proteobacteria (Delta-, Epsilon- and Gammaproteobacteria), Bacteroidetes and Chloroflexi as well as several candidate phyla11,17,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Within the GB hydrothermal area in the southern spreading center, a high degree of microbial community connectivity exists among hydrothermal vent sites and sediments within a few hundred meters25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A core micro- biome is shared between microbial communities of GB hydro- thermal sediments and cold seeps in the Sonora Margin, within a few km distance; this microbiome is thought to be involved in organic matter degradation as well as methane and carbon cycling, suggesting microbial exchanges across neighboring sites that share geochemical characteristics, such as abundant methane concentrations26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Previously, we employed metagenomic recon- structions of two GB sedimentary microbial communities, showing the interconnectivity of carbon, sulfur and nitrogen cycling among lineages20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' However, despite these advances, we still have a limited understanding of the spatial biodiversity and full metabolic potential of microbes inhabiting the basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Here we characterize the biodiversity and physiological cap- abilities of genomes from microbial communities inhabiting GB sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The highly localized hydrothermal gradients in surfi- cial GB sediments are ideal to compare adjacent sites with distinct temperature and chemical regimes18,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' We selected samples from methane- and sulfate-rich hydrothermal sediments covering a wide thermal range, and contrasted them with cold, non- hydrothermal sediments, as well as with hot, oil-rich sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' We hypothesize that microbial assemblages from hydrothermal sediments are phylogenetically distinct from those in the sur- rounding region and host a greater metabolic diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Therefore, we sequenced a total of ~4 billion genomic reads from eleven samples (two of which were from cool, background sediments) from GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Altogether, these data add 22 branches to the tree of life and enabled to us determine the genetic repertoire and metabolic versatility of these extreme hydrothermal communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Results Phylogenetic diversity in Guaymas Basin sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' To examine the biodiversity of microbial communities inhabiting GB sediments, we sampled and sequenced eleven sediments covering different sampling locations, depths (0–24 cm), temperatures (3–60 °C) and geochemical regimes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 1, Supplementary Data 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Background sediments, represented by core 4567_28, are not influenced by hydrothermal activity (temperature ~3 °C) and occur interlaced with hydrothermal hot spots within the spreading center18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' All other samples are characterized by steep thermal gradients, reflected by in-situ temperatures ranging from 4 °C to 60 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Dense mats of filamentous Gammaproteobacteria (family Beggiatoaceae) covered hydrothermal sediments from dive 4569, with an orange mat dominating core 4569_9 and a white mat at the adjacent core 4569_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Core 4569_4 was collected from the periphery of this hydrothermal hotspot and did not contain visible mats (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Porewater methane, sulfate, dis- solved inorganic carbon (DIC) and sulfide co-occurred throughout these cores (Supplementary Information), consistent with hydrothermal circulation and inmixing of seawater-derived electron acceptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Cores 4571_4 and 4488_9 represent hot and oily sediments with yellow-white sulfur precipitates on the surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Among the hydrothermal cores, 4488_9 stands out by steep thermal gradients (~150 °C at 30 cm depth), high sample temperature (~60 °C), sulfate depletion at shallow depths, and accumulation of non-methane hydrocarbons (Supplementary Figure 1, Supplementary Data 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' After sequence assembly, we reconstructed 551 draft genomes via tetranucleotide and coverage binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These metagenome-assembled genomes (MAGs), sim- plified as ‘genome’ throughout the manuscript, represent medium-quality MAGs and were > 50% complete and < 10% contaminated (301 genomes > 70% and 61 genomes > 90% com- plete; Supplementary Data 2, 3)28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Each genome was classified by constructing a phylogenetic tree using 37 single-copy, protein-coding marker genes (Supplemen- tary Data 4)29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Overall, the 551 genomes (247 archaea and 304 bacteria) represented 16 cultured and 40 uncultured, candidate phyla that comprise a substantial number of new microbial lineages, many of which branch basal to those previously described (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 2, Supplementary Figure 2, Supplementary Data 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' GB genomes form 22 new lineages on the tree of life based on a phylogenetic distance analysis (collapsing branches at an average branch length distance < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Among those lineages, we dis- covered five new candidate phyla designated GB-AP1,2 and ARTICLE NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 2 NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications GB-BP1-3 for archaeal and bacterial phyla, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The placement of these five phyla was confirmed by comparing the average amino acid identity (AAI) of genomes within a phylum to genomes of all other phyla (Supplementary Data 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Within each new phylum, GB-AP1,2 shared an AAI of ~44 and ~96% and GB- BP1-3 of ~54, ~72 and ~60%, respectively, and were more similar to themselves then to other genomes (~43% AAI summarized across all genomes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While the genomes of GB-AP1 shared a low AAI, we did not detect any lineage with a closer similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Two 16S rRNA gene sequences recovered from GB-BP1 clustered with the uncultivated lineage MAT-CR-M4-B0730, which was pre- viously detected in the Kazan mud volcano or Guerrero Negro hypersaline mats (Supplementary Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In total, we defined 24 archaeal and 37 bacterial groups (or ‘clusters’) for closer analysis (see Methods section, Supplementary Data 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Archaeal genomes were represented by Bathyarchaeota (n = 41), Thermoproteales (n = 40) and Thermoplasmata (n = 36), and bacteria belonged to Deltaproteobacteria (n = 56), Gammapro- teobacteria (n = 39) and Bacteroidetes (n = 27; Supplementary Data 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, we detected several candidate lineages, including Asgard archaea (n = 9), Verstraetearchaeota (n = 7) and the bacterial CPR superphylum (n = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Overall, more genomes were recovered from hydrothermal (average of ~60 genomes per sample) than from background sediments (average ~9 genomes per sample; Supplementary Data 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' We detected only one archaeal (Bathyarchaeota) and 7 bacterial lineages (Chloroflexi, Deltaproteobacteria, Gammaproteobacteria) in the background compared to 22 archaeal and 31 bacterial clusters in the hydrothermal samples, suggesting a greater biodiversity in the more extreme environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The effect of environmental parameters on community assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' To better understand the factors that drive community assembly, we investigated the occurrence of major phylogenetic clusters across sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' First, we confirmed that the genomes accu- rately reflected the community as a whole based on the abun- dance of ribosomal protein S3 across sites (Supplementary Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Next, we used the genomes to estimate the occurrence of different phylogenetic groups across all samples (Supplemen- tary Figure 5, Supplementary Data 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Several bacterial lineages, such as Planctomycetes or Deltaproteobacteria, were more fre- quently detected in background sediments than in hydrothermal sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In contrast, archaea were increasingly detected within the deeper, hotter hydrothermal samples, but not in cool surface sediments on the periphery of hydrothermal hot spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Dominant lineages in the hot samples were Thaumarchaeota and Archae- oglobales as well as Acetothermia, and Omnitrophica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Two gen- otypes dominated hot sediments: B48_G6 (Methanosarcinales, ANME-1) and B16_G6 (Thermodesulfobacteria, ~88% AAI to Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Desulfofervidus auxilii) (Supplementary Data 3, Supplemen- tary Data 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While the hydrothermal sediments had an overall similar distribution of taxa across depth profiles, the oily sedi- ment from 4488_9 harbored only few abundant taxa, including Thermoplasmata, Aerophobetes and Thermotoga (Supplemen- tary Figures 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Core 4488_9 differs from other hydrothermal samples in its high hydrocarbon content, quick downcore depletion of sulfate, and steep thermal gradients (Supplementary Data 1, Supplementary Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In combination these factors appear to reduce the microbial diversity, especially of the archaeal community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Altogether, the hydrothermal activity gives rise to a unique community that shows a marked enrichment in archaea that can represent up to 50% of recovered genomes (Supple- mentary Data 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' This enrichment appears to be largely driven by the rich substrate availability, by hydrothermal circulation and by inmixing of the electron acceptor sulfate (Supplementary Meth- ods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' However, a greater sampling size would be needed to dis- entangle the relative contribution of individual factors on community assembly such as temperature, methane or hydro- carbon availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Carbon cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Given that these genomes yielded such a large number of unique microbial lineages, we inferred their potential physiological capabilities by assigning metabolic functions to proteins in each individual genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' First, we investigated the ability of the community to degrade and metabolize complex carbohydrates and peptides deposited in sediments by searching genomes for the presence of carbohydrate-active enzymes Vent 1: 4569_9 (center) 4569_2 (intermediate) 4569_4 (outside) Background: 4567_28 Vent 2: 4571_4 Vent 3: 4488_9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 1 Overview of sampling sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In-situ photos of the three hydrothermal sampling sites (Vent1, Vent2, Vent3) and the non-hydrothermal background sediment, including Alvin dive and core number for the sediment cores that were used for DNA extraction and metagenomic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' White circle: spots where sediment cores were retrieved by push coring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For Vent1, three sediment cores were taken inside the yellow mat (4569_9), further outside in a white mat area (4569_2) and outside of the mat area (4569_4); next to each core a thermal logging probe was inserted into the sediment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' At Vent2 and Vent3, one core each (4571_4 and 4488_9) was sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Metadata for all samples are summarized in Supplementary Data 1 NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 ARTICLE NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications 3 (CAZYmes), peptidases and pathways for carbon metabolism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In total, we detected ~30,000 and ~11,000 potential CAZYmes and peptidases, respectively (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 3, Supplementary Figure 6, Supple- mentary Data 8, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Generally, bacteria encoded for a broader repertoire of CAZYmes compared to archaea; for example GH13 (α-amylase), GH23 (lytic transglycosylase) or GH74 (xylogluca- nase) were more common in bacteria (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 3, Supplementary Data 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Most CAZymes were assigned to Thermoproteales (n = 20) and Asgard archaea (n = 16) as well as Verrucomicrobia (n = 38) and Bacteroidetes (n = 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Peptidases were more equally distributed across both domains and abundant in Asgard archaea (n = 34) and Thermococci (n = 24) as well as Aminicenantes (n = 54) and Acidobacteria (n = 51; Supplementary Figure 6, Supplementary Data 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Approximately 2–3% of CAZYmes and peptidases are potentially secreted, suggesting that complex sub- strates are degraded outside of the cell and later taken up for degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Potentially secreted enzymes include CE8 (pectin methylesterase), and GH13 (α-amylase) as well as M28 (amino- peptidases) and S08 (subtilisin-like peptidases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A subset of CAZymes, such as GH23, may be involved in cell wall main- tenance; however, the presence of sugar and peptide transporters as well as downstream metabolic pathways in most genomes suggest that other CAZymes might be involved in energy meta- bolism (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Common pathways for the degradation of substrates produced by the activity of CAZymes and peptidases include glycolysis (glucokinase (glk), phosphofructokinase (pfk), pyruvate kinase (pyk)), gluconeogenesis (fructose-1,6-bisphosphatase (fbp), phos- phoenolpyruvate carboxykinase (pckA)) and fermentation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4 and Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In several cases, archaeal genomes encoded for more key genes of gluconeogenesis compared to glycolysis, which could imply that some archaea prefer peptides as an energy source; this finding is consistent with the occurrence of a high number of peptidases in their genomes (Supplementary Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Compared to archaea, bacteria contained a greater metabolic repertoire and might use both glycolysis and gluconeogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Most genomes encoded for the potential to metabolize pyruvate produced during glycolysis to acetyl-CoA and further into fermentation pathways, producing formate, ethanol or acetate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' GB archaea were mainly capable of acetate formation using the ADP-forming acetyl-CoA synthetase (acdA), while bacteria encoded for phosphate acetyltransferase (pta) and acetate kinase (ackA) for acetate production; formate C-acetyltransferase (pflD) and formate dehydrogenase (fdoG) for Tree scale: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='DPANN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Altiarchaeales ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Archaeoglobus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Methanomicrobia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermoplasmata ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermococci ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Asgard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Korarchaeota ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Bathyarchaeota ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thaumarchaeota ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Desulfurococcales ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermoproteales ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CPR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Acetothermia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermotogae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='WOR-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Hydrothermae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Omnitrophica ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Verrucomicrobia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Planctomycetes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aquificae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Actinobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Chloroflexi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Spirochaeta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Cloaci ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Hyd24-12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB-BP1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Zixi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Marinimicrobia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='KSB1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Bacteroidetes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Acidobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aminicenantes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Epsilonproteobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Gamma- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='proteobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Delta- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='proteobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermodesulfobacteria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Latesci ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='References ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB hydrothermal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB background ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB-BP2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='BP3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='AP2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='AP1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='DeepCrenGroup1+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Verstraetearchaeota ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aerophobetes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Caldiserica ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 2 Maximum likelihood phylogenetic tree of GB genomes based on 37 concatenated protein-coding genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Grey: Reference Genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Blue: Genomes assembled from cold background sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Red: Genomes recovered from hot, hydrothermal sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The full tree can be found in Supplementary Figure 2 and the tree file is available in Supplementary Data 5 ARTICLE NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 4 NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications formate production; and aldehyde dehydrogenase (aldh) and alcohol dehydrogenase (adh) for ethanol production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Not only is the GB microbiome able to process the deposited organic carbon pool by fermentation, but we also detected pathways for carbon fixation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The most common route of carbon fixation was the Wood-Ljungdahl pathway in both archaea and bacteria, while the Calvin-Benson-Bassham (CBB) and rTCA cycles were restricted mostly to Proteobacteria (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4, Supple- mentary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Although the Group III Ribulose-1,5-bispho- sphate carboxylase-oxygenase (Rubisco, key marker gene of the CBB cycle) was detected in most archaea, this subgroup is implied in a nucleotide salvage pathway and not necessarily used for carbon fixation (Supplementary Data 10)31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A Group I/II Rubisco, feeding CO2 into the CBB cycle, was only detected in some Gammaproteobacteria (orders Chromatiales and Thiotri- chales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, marker genes for the rTCA cycle, including ATP-citrate-lyase (aclAB), pyruvate ferredoxin oxidoreductase (porABCD) and 2-oxoacid ferredoxin oxidoreductase (oorABCD), were mainly detected in Epsilonproteobacteria (order Campylo- bacterales; Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While several genes of the 3- hydroxypropionate or related cycles were present in a subset of genomes, a full pathway appeared to be absent (Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Conversely, the Wood-Ljungdahl pathway was present in several clusters, including Archaeoglobales and Methanosarci- nales as well as Chloroflexi and Deltaproteobacteria (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4, Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Interestingly, we also detected genes from this pathway in candidate phyla, including Hydrother- marchaeota and Latescibacteria, which might either oxidize acetate or perform acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Alkyl-coenzyme M reductase linked hydrocarbon cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' We detected the methyl-Coenzyme M reductase (mcrA), a key enzyme for methanogenesis and AOM, in Syntrophoarchaea, Methanomicrobia, and a deep-branching Thermoproteales line- age (designated DeepCrenGroup1; Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4, Supplementary Data 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' To our knowledge this is the first report of mcrABG genes in the Crenarchaeota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The only bacteria able to utilize methane encoded for the particulate methane monooxygenase (pmoA), which was restricted to Gammaproteobacteria (orders Cellvibrionales and Methylococcales; Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A closer phylogenetic analysis of McrA might even suggest a broader substrate usage potentially not restricted to methane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 5, Supplementary Data 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' McrA from most ANME-1, ANME-2c and Deep- CrenGroup1 clustered with known methane oxidizers, while the McrA from one Syntrophoarchaeum (B49_G1) clustered with butane-oxidizers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' McrA from GoM-Arc1 branched between those two clusters, which is consistent with earlier work that suggested that GoM-Arc1 might utilize a different alkane, perhaps ethane, which can reach relatively high concentrations of 40-100 µM in GB11,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' However, further experimental evidence, preferably from enrichment cultures, is needed to confirm the substrate usage of these McrA proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Surprisingly, ANME-1 bin B39_G2 contains two McrA proteins (on two different contigs, both mate-paired to other contigs from that bin) that are phylogenetically related to those from Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Syntrophoarchaeum spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Similarly to Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Syntrophoarchaeum spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' B39_G2 contains genes with homology to those that encode for the butyryl-CoA oxidation pathway, such as acyl-CoA dehydrogenase and enoyl-CoA dehydratase (Supple- mentary Data 10, Supplementary Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' This pathway appears to be involved in butane oxidation in Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Syntrophoarchaeum butanivorans16, making this the first example of an ANME-1 archaeon potentially able to use short-chain alkanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The detection of these unique methyl coenzyme-M reductase genes and pathways suggests that ANME-1 archaea are not limited to methane utilization and potentially able to oxidize alkanes anaerobically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Lipid and hydrocarbon utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Pathways for lipid degra- dation were widespread in bacteria and less common in archaea, where they were mainly detected in Archaeoglobales, Bath- yarchaeota and Geothermarchaeota (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4, ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Supplementary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Carbohydrate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='esterase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Glycoside hydrolase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Polysaccharide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='lyase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CE3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='PL9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Zixibacteria (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Verrucomicrobia (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermotoga (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermodesulfobacteria (5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Spirochaetes (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Planctomycetes (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Omnitrophica (WOR–2) (13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Marinimicrobia (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Latescibacteria (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='KSB1 (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Hydrothermae (5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB–BP3 (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB–BP2 (5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB–BP1 (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Gammaproteobacteria (39) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Epsilonproteobacteria (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Deltaproteobacteria (56) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Coatesbacteria (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Cloacimonetes (5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Chloroflexi (20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Caldiserica (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Bacteroidetes (27) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aquificae (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aminicenantes (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aerophobetes (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Acidobacteria (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Acetothermia (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Woesearchaeota (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Verstraetearchaeota (14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermoproteales (33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermoplasmata (36) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermococci (15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thaumarchaeota (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Nanoarchaeota (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Methanomicrobia (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Korarchaeota (10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Geothermarchaeota (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB–AP2 (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Desulfurococcales (12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Bathyarchaeota (41) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Asgard (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Archaeoglobales (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Altiarchaeales (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aenigmarchaeota (14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Genomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 3 Number of carbohydrate-active enzymes (CAZymes) encoded in GB genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Percentage of carbohydrate esterases (CE), glycoside hydrolases (GH) and polysaccharide lyases (PL) encoded in GB genomes summarized for each phylogenetic cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Brackets: Total number of genomes encoded in each phylogenetic cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Asterisk: CAZyme with potential secretion signal (see also Supplementary Data 8) NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 ARTICLE NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications 5 Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The acyl-CoA dehydrogenase (acd) represents a key gene catalyzing the first step in beta-oxidation and accommodates a broad substrate range32,33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' GB ACDs fell alongside described glutaryl-CoA dehydrogenases, small/medium- and long-chain acyl-CoA dehydrogenases, potential butyryl-CoA dehydrogenases and isovaleryl-CoA dehydrogenases (Supplementary Figure 7, Supplementary Data 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Only ~50% of archaeal lineages encoded for acd, which was found scattered across taxa, for example only ~30% of Verstraetearchaeota encoded for acd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' This gene was common in Archaeoglobales, Asgard archaea and Geother- marchaeota, all of which encoded for other beta-oxidation genes, such as enoyl-CoA hydratase (EC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='17) or 3-hydroxyacyl-CoA dehydrogenase (EC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='35; Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4, Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In contrast, 33 out of 37 bacterial lineages encoded for acd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' How- ever, only a subset of those lineages - including Aquificae, Chloroflexi or Deltaproteobacteria - encoded for further beta- oxidation genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In these cases, enzymes, such as the glutaryl-CoA dehydrogenase, might be involved in amino acid catabolism or in benzoyl-CoA degradation32,34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Hydrocarbons are another abundant source for energy and biomass generation in GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While we did not detect genes for aerobic hydrocarbon degradation, we found indications that GB genomes might anaerobically degrade hydrocarbons using glycyl radical enzymes (GREs, Supplementary Figure 8, Supplementary Data 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' GREs use a radical-based chemistry to carry out challenging metabolic reactions under anaerobic conditions and are involved in a multitude of pathways, such as fermentation, DNA synthesis or hydrocarbon degradation35,36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Compared to ACDs, GREs had a sparser distribution and were found in only 6 out of 24 archaeal and 21 out of 37 bacterial lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' GREs were common in Deltaproteobacteria (n = 32), Bacteroidetes (n = 23) or Asgard archaea (n = 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Several GREs encoded for enzymes involved in anaerobic hydrocarbon degradation, such as benzyl- succinate synthase (bssA) in Deltaproteobacteria (B38_G6, B7_G9), alkylsuccinate synthase (assA) in Deltaproteobacteria (B2_G1, B111_G9) or hydroxyphenylacetate decarboxylase in Bathyarchaeota (B26_G17) and Chloroflexi (B43_G15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Some GREs grouped neither with the previously mentioned enzymes Tree scale: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aenigmarchaeota (14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Nanohaloarchaeota (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Nanoarchaeota (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Woesearchaeota (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Diapherotrites (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Micrarchaeota (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Altiarchaeales (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermoplasmata (36) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Methanomicrobia (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Archaeoglobales (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Hadesarchaea (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Hydrothermarchaeota (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermococci (15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Korarchaeota (10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Desulfurococcales (12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermoproteales (40) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB-AP2 (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Bathyarchaeota (41) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB-AP1 (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Asgard (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Geothermarc (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thaumar (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Tenericutes (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='CPR (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aerophobetes (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Caldiserica (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Acetothermia (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Thermotoga (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='TA06 (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB-BP2 (5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Hydrothermae (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Stahlbacteria (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Omnitrophica (13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Planctomycetes (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Chlamydia (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Verrucomicrobia (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aquificae (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Poribacteria (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Chloroflexi (20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Coatesbacteria (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Actinobacteria (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Aminicenantes (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Acidobacteria (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Deltaprot (61)* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Epsilonpr (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Gammapr (39) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Spirochaetes (11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Cloacimonetes (5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB-BP3 (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Hyd24-12 (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Latescibacteria (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='GB-BP1 (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Zixibacteria (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Marinimicrobia (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='KSB1 (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Calditrichaeota (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Ignavibac (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Bacteroidet (27) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Pacearchaeota (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Verstraetearchaeota (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='mcrABG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='fdoG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='cooS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='acsB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Central metabolism ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='glk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pfk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pyk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pckA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='fbp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='korAB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='porA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pflD1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='ldh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='aldh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='adh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='acdA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='ackA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='fadD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='acd 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='35 fadA pccB epi mcmA Hydrocarbons Gly Glu Fermentation Lipids Prop HCs cdhA cdhB acsE H2 S FeFe NiFe G1 NiFe G2 NiFe G3 NiFe G4 dsrAB cysIJ soxABC phsA sqr N napA narG nirB nrfA NirS norB nosZ nifH hao hcp octR sir AbcA assAD1 dhaA asrA O2 coxABC ccoNOP cydAB Present in >50% genomes Present in 30–50% genomes Other ArsRed SeRed merA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4 Core metabolic genes detected across phylogenetic clusters inhabiting GB sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Presence of core metabolic genes involved in carbon metabolism, hydrocarbon (HC) degradation and respiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Shaded colors: Gene present in 30–50% of genomes/phylogenetic cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Solid colors: Gene present in 50–100% of genomes/cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' C1 C1- compound metabolism, H2 hydrogen metabolism, N nitrogen metabolism, S sulfur metabolism, O2 oxygen metabolism, ArsRed arsenate reductase, SeRed selenate reductase, Gly glycolysis, Glu gluconeogenesis, Prop propane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Number in brackets: number of genomes belonging to individual phylogenetic clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Grey circle: Bootstrap support > 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Asterisk: Deltaproteobacteria includes genomes from both Deltaproteobacteria and Thermodesulfobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 1pflD and assA are often difficult to discriminate from other glycyl radical enzymes, therefore, an additional phylogenetic analysis can be found in Supplementary Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 2Phylogenetical analyses of substrate specificity of acd genes can be found in Supplementary Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A complete list of metabolic genes can be found in Supplementary Data 10 ARTICLE NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 6 NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications nor with the pyruvate formate lyase or other characterized GREs35, suggesting that those might utilize different substrates, such as carbohydrates or peptides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Respiratory processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Next, we investigated the GB microbial communities for their involvement in respiratory processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Overall, more bacterial and archaeal genomes contained genes that encode anaerobic rather than aerobic respiratory pathways, consistent with rapidly depleted oxygen levels within the first few millimeters of the sediment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4, Supplementary Data 10)18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Cytochrome c oxidases occurred in ~10% of genomes, but were mainly limited to Bacteroidetes, Epsilon-/and Gammaproteo- bacteria, and Verrucomicrobia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Conversely, genes for hydrogen, nitrogen, sulfur and potentially arsenate and selenate cycling were more widespread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' We detected [FeFe]-hydrogenases in ~10% of genomes and these mostly belonged to Group A, which can be involved in fermentative hydrogen evolution37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Approximately 70% of genomes encoded for [NiFe]-hydrogenases belonging to Group 1 (a–e and h; membrane-bound hydrogen-uptake hydro- genases involved in hydrogenotrophic respiration), Group 3 (a–d; cytosolic bidirectional hydrogenases) and Group 4 (b,d,e and g; membrane-bound, hydrogen-evolving hydrogenases; Supple- mentary Data 10)37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The most common [NiFe]-hydrogenase was found in ~25% of genomes, and belongs to Group 3b that is involved in NADPH oxidation coupled to hydrogen evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genes involved in the nitrogen and sulfur cycle were mostly restricted to bacteria, whereas archaeal nitrogen cycling genes were limited to nifH in Methanomicrobia (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4, Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genes for dissimilatory nitrate reduction to ammonium (DNRA) (narGH/napAB and nirBD/nrfAH) were present in few Bacteroidetes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' B27_G6, B58_G6), Epsilonproteobacteria (B6_G4, B37_G6) and several Gammaproteobacteria (Methylo- coccaceae, Thiotrichales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' More commonly, we detected DNRA GB-EvMd-Methermicoc-1 AIX10984 B22_G9 (ANME-1) Methanocorpusculum labreanum ABN07725 B64_G16 (ANME-1) Methanohalobium evestigatum YP_003726594 GB-EvMd-Mhalo-mcrA AIX10982 GB-EvMd-ANME-1 AIX11003 Bathyarchaeota BA2 KPV61791 Methanosarcina spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methanococcoides burtonii YP_567018 GB-EvMd-ANME-2 AIX10993 Methanomassiliicoccus intestinalis YP_008072226 otorris spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methan B49_G1 (Syntrophoarchaeum) B39_G2 (ANME-1) Syntrophoarchaeum caldarius OFV68281 GZfos13E1 AAU82276 Methanofollis liminatans WP_004037742 Methanocaldococcus spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methanothermobacter spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methanosphaerula palustris YP_002467317 Methanothermococcus spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Uncultured KT387810 Syntrophoarchaeum butanivorans OFV65760 Methanobacterium spp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' p p s alu g e r o n a h t e M Methanoculleus bourgensis YP_006545160 ex4484_138 Methanosaeta harundinacea YP_005919503 nocella spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Metha Methanolacinia petrolearia YP_003895599 GB-EvMd-GrpE AIX10996 B49_G1 (Syntrophoarchaeum) Syntrophoarchaeum caldarius OFV67100 GB-EvMd-DBrGrpIV AIX11002 Methanobrevibacter spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methanotorris spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' GB-EvMd-Methermicoc-2 AIX10985 B49_G1 (Syntrophoarchaeum) Methanopyrus spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Syntrophoarchaeum caldarius OFV67773 Syntrophoarchaeum caldarius OFV68676 Bathyarchaeota BA1 KT387805 GB-EvMd-Mmseep-1 AIX10987 Methanolobus psychrophilus YP_006922405 B75_G16 (DeepCrenGroup1) B9_G1 (ANME-2) Methanosalsum zhilinae YP_004615938 Syntrophoarchaeum butanivorans OFV65745 Methanocaldococcus spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Syntrophoarchaeum butanivorans OFV67021 Methanolacinia petrolearia YP_003895179 GB-EvMd-DBrGrpIII AIX10991 B25_G9 (GomArc1) B39_G2 (ANME-1) B48_G6 (ANME-1) Methanomethylophilus alvus YP_007713068 Methanothermus sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methanothermobacter spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methanococcus spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methanobacterium spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Syntrophoarchaeum butanivorans OFV66176 GB-EvMd-Mplanus AIX10988 GB-EvMd-Mcoc AIX10989 GB-EvMd-Mmseep AIX10986 Methanosphaera stadtmanae CAE48306 B65_G16 (GomArc1) Methanospirillum hungatei ABD41854 Methanobrevibacter spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' GB-EvMd-DBrGrpII AIX11000 Tree scale: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1 Butane X-Alkane Methane Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 5 Maximum likelihood phylogenetic tree of the methyl-Coenzyme M reductase (McrA) protein detected in GB genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Bold labels: McrA detected in GB genomes (see also Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Black circle: Bootstrap support ≥ 70 (number of bootstraps determined using the extended majority-rule consensus tree criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' RaxML was run as raxmlHPC-PTHREADS-AVX -f a -m PROTGAMMAAUTO -N autoMRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The tree file is available in Supplementary Data 11 NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 ARTICLE NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications 7 genes distributed separately over several genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A complete denitrification pathway (napA/narGH, nirK/nirS, norBC, nosZ) was present in a few genomes, including one Bacteroidetes (B2_G4), some Epsilonproteobacteria (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' B135_G9) and several Gammaproteobacteria genomes (Halieaceae, Thiotrichales); indi- vidual denitrification genes were found scattered across different taxonomic lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genes involved in anaerobic ammonium oxidation (anammox) were not found, consistent with low nitrate and nitrite concentrations in GB sediments38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genes for the dissimilatory reduction of sulfate to sulfide (sat, aprAB and dsrAB) were found in few archaea (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Archaeoglobales) and several bacteria including Deltaproteobacteria, Gammaproteo- bacteria and Zixibacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The sulfur-oxidation (SOX) system (soxAX, soxYZ, soxB, soxCD) showed a restricted phylogenetic distribution and was only located in Epsilonproteobacteria and Gammaproteobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While on average ~10% of all genomes contained genes for sulfur and nitrogen cycling, complete pathways for these processes were present in only few genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Redundancy and interconnectivity among GB microbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' To assess whether hydrothermal sediments not only host a greater phylogenetic but also metabolic diversity than background sam- ples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 2), we next investigated the spatial distribution of core metabolic genes across all sites and taxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Regardless of their origin, most genomes encoded genes for general carbon cycling (CAZymes, peptidases, gluconeogenesis, glycolysis), fermentation and lipid oxidation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 6 and Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Respiratory genes were restricted to cooler, shallower samples but present in both background and hydrothermal sediment cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For example, denitrification genes, SOX genes or the cytochrome c oxidase were found only in the shallower, colder sediments (temperature ~5 °C) and were present in ~20-30% of genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In contrast, these genes were represented in only ~0-4% of genomes in deeper, hotter samples (temperature range of 10 °C-60 °C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Exceptions were genes for sulfate/sulfite reduction, such as dsrAB, that were still found in ~8% of genomes in deeper, hotter sedi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Compared to background samples, genes involved in C1- metabolism and hydrogenases were more frequently found in hydrothermal sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In background sediments only one Bathyarchaeotal genome contained carbon fixation-related genes (cdhAB), while genes for methane cycling (mcrA) were unde- tectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Hydrogenases belonging to Group 4 g, which represent membrane-bound hydrogenases that generate a proton-motive force for energy generation, were absent from the background but present in ~25-30% of genomes across all hydrothermal samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 6 and Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These findings suggest that methane and hydrogen might be important drivers of metabolic processes in GB hydrothermal sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' With few exceptions most metabolic genes were encoded in several taxonomically distinct lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For example, C1-related genes (with the exception of mcrA) and genes related to beta- oxidation, hydrogen, nitrogen, sulfur and oxygen cycling were found in ~10 different phylogenetic lineages; fermentation genes were present in most phylogenetic clusters of both the archaeal and bacterial community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While the studied genomic dataset from the cold and hydrothermal samples were not represented by an equal number of genomes (average of ~9 and ~60 genomes per habitat type, respectively), we still find that those genomes represent the community well in terms of phylogenetic diversity (Supplementary Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, when searching for a subset of these core metabolic genes in binned and unbinned contigs from the complete assembly (only considering contigs > 2,000 bp), we observed a similar trend (Supplementary Data 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For example, fermentation genes were abundant across all sites, denitrification genes were more common in cold and shallow samples and mcrA was completely absent from the background samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Overall, these findings suggest that the GB genomes are representative of the community as a whole, and that they reflect key metabolic differences between the microbial communities present in hydrothermal and background samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Discussion In this study, we employed the largest genomic sampling of GB sediments to date to investigate the interplay of community composition and functional diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Compared to earlier work on Guaymas Basin sediments20, the higher sampling number and inclusion of background samples allowed to better describe the enhanced diversity present in these sediments and shed light on the drivers of community assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In contrast to previous stu- dies showing that sulfidic- and methane-rich seep sediments host a lower microbial diversity compared to non-seep marine sediments39,40, we demonstrate that GB hydrothermal sediments contain a diverse community that is enriched in archaea com- pared to a less diverse, bacterial-dominated community found in nearby cold sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Therefore, the more extreme conditions in hydrothermal sediments, which include steep thermal and geo- chemical gradients17,27, appear not to inhibit microbial diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Due to difficulties in isolating sufficient amounts of DNA from deeper, hotter samples, we cannot exclude that diversity may decline in those sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Earlier work reported a decrease in cell numbers with increasing depth that did not necessarily cor- relate with a decrease in OTU numbers25, potentially explaining our difficulties in isolating sufficient amounts of DNA but sup- porting our assumption that steep temperature gradients do not necessarily inhibit microbial diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Especially samples from core 4569_9 experience a highly variable, fluctuating thermal regime over time, where even surficial layers can vary from 20 °C to 70 °C, as determined by multi-day continuous thermal logging (Supplementary Figure 1)17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In response to such conditions, microbes must either adapt, have a wide thermal optimum, as shown for some ANME-1 archaea23, or be able to recolonize the sediment after a temperature sweep from a surficial reservoir41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Here, we propose that the diverse communities inhabiting hydrothermal sediments could serve as a flexible seed bank for the deeper, hotter sediments as well as for highly fluctuating environmental gradients in shallow sediments5,25,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The differences we observed in community composition across sites were not always translated into obvious changes in func- tional capacities of those communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For example, we detected abundant genes for carbon cycling and fermentation across all sites, while other metabolic processes such as respiration, were limited to shallow sediments but present in both background and hydrothermal sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Respiratory processes were often parti- tioned among the community and only few genomes were encoding for full pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Metabolic handoffs have been observed in other microbial communities and could allow a flexible interchange of metabolites between changing populations43,44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Another metabolic feature that could allow for greater ecosystem stability could be metabolic plasticity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' switching metabolic processes in response to changes in envir- onmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' We found indications for such plasticity in several bacterial genomes, especially within the Delta- and Gammaproteobacteria that might couple the reduction of sulfur with the oxidation of carbon, lipids or hydrocarbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While we cannot determine which processes are active, enhanced genotypic diversity might provide an additional adaptation strategy to variable environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The only functional categories that were consistently enriched across all hydrothermal sites and almost absent in background sediments were group 4g hydrogenases and pathways for ARTICLE NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 8 NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Nitrogen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Sulfur ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='O2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='4567_28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='4488_9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='4569_4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='4569_2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='4569_9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='4571_4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='Size of circle: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' of genes/ Total no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' of genomes No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' of phylogenetic clusters that encode for a metabolic gene: x Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 6 Metabolic profile across different GB sediment sites, depth profiles and temperature regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Shown is the number of core metabolic genes relative to the total number of genomes (in %) per site, depth and temperature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Temperatures are averages for the 2 or 3 cm thick sediment layers from which DNA was isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Background samples: Cold GB samples without hydrothermal activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Vent1–3: Hydrothermal sediment sampling locations, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' ID at the bottom: number codes designating every Alvin dive and sediment core (see also Supplementary Data 1 for further explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A complete list of metabolic genes can be found in Supplementary Data 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Number in circles: Number of phylogenetic clusters that encode for individual core metabolic genes at each site NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 ARTICLE NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications 9 methanogenesis and methane oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Group 4g hydrogenases are not well characterized but are generally described to be membrane-bound hydrogenases that allow for energy-generation by establishing ion gradients over the membrane45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These com- plexes are often found in thermophiles, such as Pyrococcus fur- iosus45, and could potentially provide a selective advantage in hydrothermal sediments over other energy-generating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While trace concentrations of biogenic methane are present in background sediments (Supplementary Data 1, Supplementary Methods), the inability to detect mcrA in these samples could be because of sequencing depth; in contrast detecting mcrA in hydrothermal sediments appears to be linked to microbial methane oxidation produced by pyrolysis of organic matter17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Within the phylogenetically and functionally diverse commu- nity inhabiting GB, the metabolic repertoire shows a high degree of functional redundancy across different phyla, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' different taxa encode the same metabolic function and thus might substitute for one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Therefore, even if community composition varies, metabolic function is predicted to be relatively stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Like phy- logenetic diversity, functional redundancy could benefit the community when dealing with perturbations in environmental conditions and has been observed in other environments including the global marine or humane microbiome46,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While any stressor, such as temperature, might result in the removal of a given taxon, functional redundancy across different lineages that are each tolerant to some degree of environmental fluctuations, and together cover a wide window of environmental conditions, ensures the stability of community function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' This is consistent with the ‘it’s the song not the singer’ (ITSNTS) theory, which assumes that surviving taxa replace perturbed taxa (‘the singers’) and thereby allow nutrient cycles (‘the song’) to persist in the environment48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' This theory is consistent with our findings, in which we not only observe phylogenetically diverse but also functionally redundant communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Altogether, the phyloge- netic diversity, metabolic partitioning as well as functional redundancy that we observe appear to be characteristics of microbial communities in these dynamic hydrothermal vent sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' One question that arises when observing functional redundancy within a microbial community is whether this redundancy enhances species competition and de-stabilizes the community49,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' While it is not in the scope of this study to discern niche patterns, we would assume that the high redun- dancy in our dataset might still allow microbes to inhabit dif- ferent niches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Two mechanisms that could allow co-existence of supposedly redundant microbes could be metabolic auxotrophies or heterogeneity in limiting resources and/or environmental conditions50–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Amino acid auxotrophies can create community interdependencies, which could balance competition and thereby stabilize microbial communities53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' We do see indications for such interdependencies in our dataset, where auxotrophies are com- mon in small genomes belonging to CPR bacteria (Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, we assume that the diverse GB-inhabiting communities are stabilized by the high abundance of substrates present in hydrothermal sediments, which might reduce com- petition and allow taxa to coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Finally, while genes for core metabolic processes showed a high redundancy across our data- set, we hypothesize that enzymes involved in substrate degrada- tion are undergoing substantial diversification with respect to their substrate spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The diversity of genes involved in car- bohydrate (mcrA, CAZYmes), lipid (acyl-CoA dehydrogenase) and peptide degradation and the expanding substrate range and diversity of hydrocarbon-degrading genes, such as mcrA, supports this notion16,20,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A limitation of the current study that com- plicates a definite description of the diversity patterns and func- tional redundancy present in Guaymas sediments is the low sample number and limited number of bins recovered from a subset of samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 4567_28 and 4488_9); given the limitations of deep-sea sampling, different habitat and sediment types are represented unevenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Activity-based analyses of large sample numbers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e., metatranscriptomics, would more rigorously link genetic patterns to their environmental determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Guaymas Basin is a hotspot for microbial biodiversity and an ideal study site to investigate the functional diversity of hydro- thermally influenced seafloor sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Here we establish that these hydrothermal sediments contain a large number of archaeal and bacterial lineages, including several uncultivated phylum- level lineages that have not been described from other habitats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Intriguingly, hydrothermal GB sediments hosted a greater diversity compared to surrounding non-hydrothermal sediments contrasting previous work on methane seep communities39,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These differences are likely linked to the unique environment in GB sediments characterized by by convective mixing of fluids resulting in variable thermal regimes, and admixture of hydro- thermal carbon and energy sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Most functional properties were shared widely among different phylogenetic lineages across different sampling sites with a greater functional redundancy of metabolic processes found in hydrothermal sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' One unique functional trait of hydrothermal compared to background sediments was the presence of methane cycling genes among novel lineages, including a new deep-branching Crenarchaeota group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' We propose that the combination of dynamic seep and hydrothermal conditions in Guaymas Basin enhances microbial diversity, and sustains a distinctive microbial community, whose functional complexity and redundancy reflects the intricate and dynamic geochemical and thermal landscape of this habitat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Methods Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Guaymas Basin sediment samples were collected from the Gulf of California (27°N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='388, 111°W24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='560) at a depth of approximately 2000 m below the water surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Sediment cores were collected during four Alvin dives (4488, 4569, 4567, and 4571) in 2008 and 2009 (Supplementary Data 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Sample site photos were compiled from the Alvin frame grabber site (http://4dgeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='whoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='edu/ alvin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Intact sediments were collected during Alvin dives using polycarbonate cores (45-60 cm in length, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='25 cm interior diameter), subsampled into cm layers under N2 gas in the ship’s laboratory and immediately frozen at −80 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Eleven sediment subsamples for DNA isolation from different depth profiles yielded sufficient genomic DNA for metagenomic sequencing (Supplementary Data 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Higher temperature samples were tested as well but did not yield sufficient DNA for metagenomic sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Metadata for all dives, including details on the geochemistry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' methane concentrations and dissolved organic carbon con- centrations and δ13C values, sulfate and sulfide concentrations) and thermal pro- files of the sampling sites, are available to compare microbial community composition across sediment cores (Supplementary Data 1, Supplementary Methods)17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additional images and descriptions of the sampling locations are published in a survey of different Guaymas Basin habitats18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Metagenomic sequencing and assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Total DNA from ≥ 10 g of sediment from each of the eleven samples (see above) was extracted using the MoBio PowerMax soil kit using the manufacturer’s instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' DNA concentrations were measured using a Qubit™ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0 Fluorometer and a final concentration of 10 ng/µl of each sample (using a total amount of 100 ng) was used to prepare libraries for paired-end Illumina (HiSeq–2500 1TB) sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Illumina library preparation and sequencing was performed at the Joint Genome Institute (JGI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Sequencing was performed on an Illumina HiSeq 2500 machine using the paired end 2 × 125 bp run-type mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' All runs combined provided a total of ~280 gigabases of sequen- cing data (Supplementary Data 2) Quality control and sequence assembly was performed by JGI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Briefly, sequences were trimmed and screened for low quality sequences using bbtools (https://jgi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='doe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='gov/data-and-tools/bbtools/) and assem- bled using megahit v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='6 using the following options: --k-list 23,43,63,83,103,12355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Summary statistics for the number of generated reads and the quality of the metagenomic assembly is provided in Supplementary Data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For further binning, only scaffolds ≥ 2000 bp were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Metagenomic binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Metagenomic binning was performed on individual assemblies using the binning tools ESOM, Anvi’o and Metabat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' ESOM binning was performed by calculating tetranucleotide frequencies of scaffolds with a minimum length of 2000 bp using the K-batch algorithm for training after running the perl ARTICLE NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 10 NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications script esomWrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pl56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The resulting Emerging Self-Organizing Maps (ESOM) were manually sorted and curated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Bins were extracted using getClassFasta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pl (using -loyal 51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The binning process was enhanced by incorporating reference genomes as genetic signatures for the assembled contigs into ESOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For Anvi’o (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2) the metagenomic workflow pipeline that incorporates CONCOCT was used for binning57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Briefly, coverage information was obtained by generating eleven mapping files for each assembly file by mapping all high-quality reads of each of the eleven samples against the assembly of one sample using the BWA-MEM algorithm in paired-end mode (bwa-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='12-r1034; using default settings)58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The resulting sam file was sorted and converted to bam using samtools (version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='19)59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The bam file was prepared for Anvi’o using the script anvi-init-bam and a contigs database generated using anvi-gen-contigs-database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These two files were further used as input for anvi-profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Generated profiles for the eleven different assemblies were combined using anvi-merge and the resulting bins summarized using anvi- summarize (-C CONCOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' If not mentioned otherwise, the scripts were used with default settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Finally, binning was performed using metabat (v1)60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' As described for Anvi’o the used input files consisted of the scaffold files (≥2000 bp) and the mapping files to recover bins both by sequence composition and abundance across samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' First, each of the mapping files were summarized using jgi_summar- ize_bam_contig_depths and then metabat was run using the following settings: --minProb 75 --minContig 2000 --minContigByCorr 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Results from the three different binning tools were combined using DAS Tool (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0)61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Therefore, for each of the binning tools a scaffold-to-bin list was prepared and DAS Tool run on each of the eleven scaffold files as follows: DAS_Tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='sh -i Anvio_contig_list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='tsv, Metabat_contig_list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='tsv,ESOM_contig_list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='tsv -l Anvio,Metabat,ESOM -c scaffolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' fasta --write_bins 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The accuracy of the binning approach was evaluated by calculating the percentage of completeness and contamination using CheckM lineage_wf (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='5; Supplementary Data 3)62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genomes were only analyzed further if they were more than 50% complete and showed a contamination below 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Contaminants that were identified based on their phylogenetic placement (wrong taxonomic assignment compared to the average taxonomic assignment of the genes assigned to each bin), GC content (>25% difference compared to the mean of all scaffolds assigned to each bin) or abundance (>25% differences compared to the mean abundance of all scaffolds assigned to each bin) were manually removed from individual genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' This yielded a total of 247 archaea and 304 bacterial genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Relative abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' To determine the relative abundance of each genome across the elven sequenced sediment samples, we mapped the contigs from all binned genomes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e., using the “whole MAG”) against the high-quality reads of each individual metagenome (generating eleven sam files).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The sam output was sorted and converted to bam as described above and we then used the metabat output, which describes the read counts recruited by each contig, for further analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' All analyses were performed in R (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' On average, ~47% of the high-quality metagenomic sequences could be binned, with the notable exception of the sample from 4567_28, from which the recovered MAGs only recruited ~18% of reads for an undetermined reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' To determine the average abundance of major taxonomic groups (referred to as cluster, which were determined by the phylogenetic analysis described below), contigs were first assigned to their phylogenetic cluster (see description for the phylogenetic analysis below) and were then summarized using the ddply function from the plyr package63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These clusters do not represent a specific taxonomic rank but were chosen to account for both phylogenetic diversity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e., Crenarchaeota are usually represented at order rank or lower if possible) as well as available genomes (the different phyla of the CPR superphylum were ranked together because they were represented by only few genomes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The counts recruited by each taxonomic group were normalized by the total length of contigs belonging to each cluster, the library size of the individual metagenomes and multiplied by 1000 for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The normalized relative abundance was plotted using the heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2 function in the gplots package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The summary statistics are provided in Supplementary Data 7 (only includes clusters with ≥3 lineages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' To determine the relative abundance of the ribosomal protein S3 (RPS3) across samples, RPS3 was extracted from all eleven assemblies (only considering contigs >2000 bp) using phylosift (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1 using options: phylosift all --keep_search --custom marker_list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='txt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In total, we identified 1227 RPS3 sequences in the dataset, 486 of which belonged to binned contigs (~40%) and, therefore, RPS3 could be successfully recovered from ~82% of bins; (Supplementary Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The unaligned nucleotide sequences were concatenated and used as an input to run bwa against all eleven metagenomes to determine their relative abundance across samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Read counts were extracted using samtools, normalized by gene length and library size and plotted using the ggplot2 package in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Phylogenetic analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Phylosift was used to extract marker genes for the phy- logenetic placement of the assembled metagenomic bins64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A set of 37 single-copy, protein-coding housekeeping genes was chosen for a further phylogenetic analysis (Supplementary Data 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' To generate a reference dataset, archaeal (all available genomes) and bacterial genomes (selected genomes that include at least three members from each genus and a preference for type strains whenever possible) were downloaded from NCBI on March 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Next, all reference genomes and GB genomes (fasta files) were used as an input for phylosift (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1) using the ‘phylosift search’ followed by the ‘phylosift align’ mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The concatenated protein alignments of 37 elite marker genes (concat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='fasta) were combined for all genomes of interest and trimmed using TrimAL (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2) using the automated1 setting65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A phylogenetic tree was generated using a maximum likelihood-based approach using RAxML (version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='10, called as: raxmlHPC-PTHREADS-AVX -f a -m PROTGAMMAAUTO -N autoMRE)66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The tree was visualized using the Inter- active Tree Of Life (iTOL) webtool67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For better visualization, the initial tree was reduced to only include references that were branching close to GB genomes and included a 224 genomes from cultured representatives and 330 genomes from uncultured genomes (including metagenome-assembled genomes, enrichment cultures, co-cultures and single-cell assembled genomes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' All of these genomes were used to calculate an average amino acid identity across all genomes using com- parem (v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='23, function aai_wf; https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/dparks1134/CompareM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The AAI was used as a main measure to distinguish the new phylum-level genomes that were discovered with the phylogenetic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Therefore, the average AAI of each phylum was calculated and compared to all remaining phyla, especially those branching close to the phyla of interest (see also Supplementary Data 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The 16S rRNA gene sequences were extracted using phylosift (settings are described above) and barrnap (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/tseemann/barrnap, v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='7, settings: --kingdom arc/bac --lencutoff 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2 --reject 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='3 --evalue 1e-05) and aligned to the SILVA SSURef_NR99 database (release 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2017) using the SILVA webaligner68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The alignment was manually curated in ARB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A phylogenetic tree was generated using a maximum likelihood-based approach using RAxML (settings: raxmlHPC-PTHREADS-AVX -T 10 -f a -m GTRGAMMA -N autoMRE p 12345 -x 12345).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 16S rRNA gene sequences were manually checked for contamination in cases with an inconsistent phylogenetic assignment between 16S rRNA gene sequences or the 37 protein-coding marker genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The whole contig was discarded, when all assigned proteins on the contig with the 16S rRNA gene showed a different taxonomic assignment (using blastp) compared to the remaining scaffolds of the respective genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A similar phylogenetic approach was taken to phylogenetically characterize other key genes of interest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' hydrogenases, mcrA, glycyl radical enzymes, acyl- CoA dehydrogenases (acd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genes of interest were identified in GB genomes using KAAS, HMMER, blastp or the HydDB webserver (for details see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Published reference genes were extracted using the NCBI and Uniprot webservers (McrA, glycyl radical enzymes, ACD) as well as the HydDB webserver (hydrogenases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For the glycyl radical enzymes, proteins identified as PflA, AssA, BssA, HbsA, MasD, NmsA in KEGG or a custom blast search were combined in a single analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For the ACD phylogeny, the KAAS IDs K00248, K00249, K06445, K00255, K06446 and K09479 were included to build a phylogenetic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Protein sequences from GB and reference genomes were combined and aligned using muscle (v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='31, default settings), trimmed using TrimAL and a phylogenetic tree generated using RAxML as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Protein-coding genes falling on long branches were manually checked using blastp on the NCBI webserver and discarded if the annotation was not hydrogenase, acyl-CoA dehydrogenase or glycyl radical enzyme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Annotations and metabolic analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Gene prediction for individual genomes was performed using prodigal (V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='2, default settings)69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The genomes contained on average 1,665 predicted proteins for archaea (min = 500 and max = 4,685) and 2,491 for bacteria (min = 636 and max = 6,964) (Supplementary Data 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Meta- bolic reconstructions were done for each individual genome, but in several cases the results were summarized for major taxonomic lineages, or clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' These clusters do not represent a specific taxonomic rank but were chosen to account for both phylogenetic diversity (Crenarchaeota are usually represented at order rank) as well as available genomes (the different phyla of the CPR superphylum were ranked together because they were represented by only few genomes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Predicted genes of individual genomes were further characterized using KAAS (KEGG Automatic Annotation Server; Supplementary Data 10)70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Therefore, protein sequences of each of the individual genomes were uploaded to the KAAS webserver using the ‘Complete or Draft Genome’ setting (used parameters: GHOSTX, custom genome dataset, BBH assignment method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' For a detailed pathway analysis the KO numbers were downloaded, concatenated and merged with a KO-to-pathway metadata file in R (Supplementary Data 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, we searched for key metabolic genes using custom blastp and hmmer databases43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A Curated Database of Anaerobic Hydrocarbon Degradation Genes (AnHyDeg) and the MEROPS database were used to identify hydrocarbon degradation genes as well as peptidases in the concatenated proteins sequences of all GB genomes using blastp (e-value threshold of 1e-20; Supplementary Data 10)71–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Hits were discarded if they were related to core metabolic processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='e., pyrimidine synthesis) or included heat-shock resistance proteins, precursor proteins and signal peptides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, we utilized a custom hmmer as well as the Pfam and TIGRFAM databases to search for key metabolic marker genes using hmmsearch and custom bit-score cutoffs43,74,75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Hydrogenases were extracted from the genomes using hmmsearch (e-value cut-off of 1e-20) and confirmed using a web-based search using the hydrogenase classifier HydDB76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Finally, genes encoding for carbohydrate degradation enzymes described in the Carbohydrate- Active enZYmes (CAZYmes) database were identified using the dbcan webtool and applying an e-value threshold of 1e-577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Protein localization was determined for CAZYmes and peptidases using the command-line version of Psort (V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='0) using the option --archaea for archaeal genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The results for the MEROPS and NATURE COMMUNICATIONS | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 ARTICLE NATURE COMMUNICATIONS | (2018) 9:4999 | DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467-018-07418-0 | www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='com/naturecommunications 11 CAZymes database searches are summarized in Supplementary Data 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' In the case of protein-coding genes hitting to multiple genes in the before-mentioned databases, the best hit was chosen based on their e-value and bit-score using blast_best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pl (http://alrlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='pdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='edu/aquificales/scripts/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genes assigned to core metabolic pathways are summarized in Supplementary Data 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Hits for key metabolic marker genes found in major taxonomic clusters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 3) were verified across different databases (KAAS, PFAM and TIGRPFAMs) and cross-checked with results from close reference genomes that fell within the same phylogenetic group as the genome of interest to reduce the chance of contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Genes not found in close reference genomes were further validated with blastp using the NCBI webserver tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' If a hit could not be confirmed or if the top phylogenetic hit for whole contig was not consistent with the phylogenetic assignment of the genome, it was removed from the genome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Data availability All sequence data and sample information are available at NCBI under BioProject ID PRJNA362212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Accession numbers for individual genomes can be found in Supplementary Data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additionally, the raw data is provided in IMG/MER and the IMG Genome IDs for the individual metagenomes are provided in Supplementary Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Received: 13 June 2018 Accepted: 31 October 2018 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' 40, W445–W451 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Acknowledgements We thank Kiley W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Seitz for detailed comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' The work conducted by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Department of Energy Joint Genome Institute, a DOE Office of Science User Facility, is supported by the Office of Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Department of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' DE-AC02-05CH11231 provided to ND.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' This work was funded by a Sloan Foundation Ocean Sciences fellowship (FG-2016-6301) and National Science Foundation DEB: Systematics and Biodiversity Sciences (grant number 1753661) provided to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' and Guaymas Basin fieldwork was supported by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' National Science Foundation grants OCE-0647633 and OCE-1357238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Author contributions B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='B., A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' conceived, designed the study, and were involved in writing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' processed the data, reconstructed the genomes and performed the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Additional information Supplementary Information accompanies this paper at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content='1038/s41467- 018-07418-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} +page_content=' Competing interests: The authors declare no competing interests.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_45/content/kb_45.pdf'} diff --git a/l9E3T4oBgHgl3EQf6Qta/content/tmp_files/2301.04788v1.pdf.txt b/l9E3T4oBgHgl3EQf6Qta/content/tmp_files/2301.04788v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..606161a537d3eefcc74572c4785c1e8e7eb731b0 --- /dev/null +++ b/l9E3T4oBgHgl3EQf6Qta/content/tmp_files/2301.04788v1.pdf.txt @@ -0,0 +1,1692 @@ +arXiv:2301.04788v1 [cs.CL] 12 Jan 2023 +Language Cognition and Language Computation Human and +Machine Language Understanding∗ +Shaonan Wang1,2,∗, Nai Ding3,4,∗, Nan Lin5,6, Jiajun Zhang1,2, Chengqing Zong1,2 +1National Laboratory of Pattern Recognition, Institute of Automation, CAS, Beijing, China +2School of Artificial Intelligence, University of Chinese Academy of Sciences, Beijing, China +3Key Laboratory for Biomedical Engineering of Ministry of Education, College of Biomedical Engineering +and Instrument Sciences, Zhejiang University, Hangzhou, China +4Zhejiang Lab, Zhejiang University, Hangzhou, China +5CAS Key Laboratory of Behavioural Sciences, Institute of Psychology, Beijing, China +6Department of Psychology, University of Chinese Academy of Sciences, Beijing, China +Corresponding authors. Email: shaonan.wang@nlpr.ia.ac.cn, ding nai@zju.edu.cn +Abstract +Language understanding is a key scientific issue in the fields of cognitive and computer science. +However, the two disciplines differ substantially in the specific research questions. Cognitive sci- +ence focuses on analyzing the specific mechanism of the brain and investigating the brain’s response +to language; few studies have examined the brain’s language system as a whole. By contrast, com- +puter scientists focus on the efficiency of practical applications when choosing research questions +but may ignore the most essential laws of language. Given these differences, can a combination of +the disciplines offer new insights for building intelligent language models and studying language +cognitive mechanisms? In the following text, we first review the research questions, history, and +methods of language understanding in cognitive and computer science, focusing on the current +progress and challenges. We then compare and contrast the research of language understanding +in cognitive and computer sciences. Finally, we review existing work that combines insights from +language cognition and language computation and offer prospects for future development trends. +1 +Introduction +Language is a multilevel symbolic system that includes multiple levels: phonetics, morphology, syntax, +semantics, and pragmatics. The most basic language symbols can be combined to form more complex +and endless symbol sequences to allow flexible expression of meaning. +As such, language is also +considered the carrier of human thought and the most natural tool through which humans exchange +ideas and express emotions. +Because of the diverse and flexible characteristics of language, it is difficult to study the mech- +anism of human language understanding and to build a computation model that can understand +language. In the early days of computer science, language research pioneers attempted to conduct +cross-disciplinary research in computer science, linguistics, and cognitive science. They aimed to es- +tablish connections between human language-understanding mechanisms and language-computation +models [1, 2, 3, 4, 5, 6]. However, owing to the complexity of the problem, interdisciplinary research +has gradually become separated over the decades, forming subfields such as natural language under- +standing in computer science, psycholinguistics in cognitive psychology, and neurobiology of language +research in cognitive neuroscience. In this paper, ”cognitive science” mainly refers to the two fields +of cognitive psychology and cognitive neuroscience, particularly the branches of psycholinguistics and +the cognitive neuroscience of language [7]. +Figure 1 shows the relationship between cognitive and computer science in the direction of language +understanding. There are substantial differences in the research questions and methods adopted in +the two fields. Computer scientists primarily adopt rationalist (represented by rule-based methods) +∗This paper is originally written in Chinese and published in SCIENTIA SINICA Informationis. Here we translate +it into English with an extension of recent work. +1 + +Language cognition +♦Human language understanding +� +Build a computational +framework to analyze +natural languages +Analyzing the working +mechanisms of the brain +language understanding +Research methods +Cognitive Science, +Psychology, Neuroscience +Language behavior and neural +response analysis (Behavior +experiment, fMRI, MEG, EEG, +et al.) +Research problems +Foundations +Computational modeling +(Using computational methods to +model the mechanisms of brain +language understanding) +Rule-based approach +(Rationalist model) +Data-based approach +(Empiricism model) +Language computation +♦Machine language understanding +� +Linguistics +Computer Science, Statistics +Figure 1: Connections between cognitive sciences and computer sciences on the language understanding +problem +and empirical methods (data-driven modeling methods, represented by statistical machine learning +and neural network methods) and pay more attention to applied research–that is, how to build an +intelligent system to understand natural language to complete various practical applications (such +as machine translation, dialogue system, and automatic summarization). The ”understanding” here +refers, more precisely, to application-oriented ”processing”; thus, in many cases, ”natural language +processing” is commonly used to collectively refer to this research direction. +On the other hand, cognitive scientists adopt neuroimaging and behavioral analysis methods and +pay more attention to the psychological and neural basis of human language understanding, such +as the functions of each brain region in language understanding and how neural activities encode +different levels of language information. The commonality between these two fields is that both use +linguistics as the subject basis and computational modeling as tools for analysis in terms of language +understanding. In general, language cognition and language computation have achieved fruitful results +in their respective directions, and new theories and methods have been continuously proposed and +successfully applied. +In recent years, improvements in computing resources and deep learning algorithms have led to the +rapid development of artificial intelligence and computer science. Computers have defeated professional +human players in tasks such as chess, quizzes, Go, and video games. In the field of natural language +understanding, automatic dialogue systems and question answering systems, such as Siri and Watson, +have also emerged as well as more practical machine translation systems [8, 9, 10]. +However, the +current artificial intelligence system relies on largescale training data that lack basic common-sense +knowledge; thus, a large gap remains between human and artificial intelligence in terms of learning and +generalization abilities [11, 12, 13]. Therefore, the human brain, as the only example of the realization +of intelligence, has once again attracted the attention of computer scientists, and the development of +brain-like intelligence by drawing on the neural and cognitive behavior mechanisms of the human brain +has become a hotspot at the forefront of research worldwide. +Simultaneously, cognitive science, including cognitive neuroscience and cognitive psychology, has +developed rapidly. +Noninvasive and real-time monitoring of language processing in the brain has +become possible. Through various sophisticated experimental designs, researchers in the field of cog- +nitive science have made valuable discoveries regarding the neurological basis of language processing +[14, 15, 16, 17, 18]. Traditional cognitive science experiments rely on strict experimental controls, lead- +ing to significant deficiencies in terms of ecology and globality. However, in recent years, an increasing +2 + +number of studies have begun to adopt high ecological paradigms and use advanced data-analysis +methods to analyze the information-processing mechanism of the human brain under high ecological +validity tasks. In terms of language research, researchers have gradually begun to use computational +modeling methods to study the language understanding process of the human brain under experimental +stimulation conditions of natural texts [19, 20, 21, 22, 23, 24, 25, 26]. +With the rapid accumulation of interdisciplinary research in the fields of cognitive science and +computer science in recent years, researchers have begun to summarize the changes that computer +science methods, especially deep learning models, can contribute to research in the field of cognitive +science [27, 28, 29, 30] and how the conclusions of cognitive science discoveries can help in building +artificial intelligence models [31, 32, 33, 34]. The above work explored the possibility of combining +the two fields at the macro level but did not discuss how to combine the two fields to carry out +work on subdivided issues. To provide a reference for cognitive and computer scientists to conduct +interdisciplinary research in the direction of language understanding, this paper summarizes and looks +forward to the existing interdisciplinary research work on language understanding. +In summary, in terms of language comprehension, computational models can help cognitive sci- +entists to quantitatively study and model the brain while understanding brain mechanisms can help +computer scientists build smarter language and computation models. Therefore, to promote a new +round of development in human and machine language understanding research, it is imperative to +conduct interdisciplinary research that combines cognitive and computer science. +The following sections first introduce the definitions, main research issues, research status, and +research methods of human language cognition (Section 2) and machine language computation (Section +3) as well as the limitations of existing research. We then compare the main ideas and concepts of +language cognition and language computation (Section 4) and analyze the similarities and differences +between the two at different levels. Section 5 summarizes the existing work on combining language +cognition and computation. On this basis, the limitations of the existing combination methods are +analyzed, and feasible future research directions are proposed (Section 6). Finally, we present the main +conclusions of this study (Section 7). +2 +Research on language cognition +2.1 +Definition of language cognition +The language cognition mentioned in this article refers to the human brain’s understanding of language. +Specifically, it refers to the process of extracting abstract symbolic information from auditory, visual, +and other sensory information when an individual receives information, such as speech and text. +Language cognition is a complex process with varying structures and mechanisms of different levels. +Moreover, the brain networks on which they depend are also very complex. For example, in the process +of speech comprehension, the auditory system must encode the basic acoustic features of speech and +then follow multiple steps, such as vocabulary recognition, syntax construction, and semantic analysis, +before finally realizing language comprehension. +2.2 +Main research questions +Language is a complex sequence, and the human brain is a complex system. +This makes it very +challenging to study the language-processing mechanism of the human brain. On one hand, a language +contains units of different sizes; which language unit should be used as the starting point? Does the +brain use a certain unit as the most important unit in language processing? What about the core +processing unit? This is the first research question presented below. +On the other hand, information processing in the brain is very complicated. Different types of +information are processed by different brain areas in a certain order. Research on the related brain +regions and time courses of language processing is summarized below in the second and third research +questions, respectively. +Ultimately, both the observation of the brain area and the processing timing are only phenomeno- +logical descriptions of language processing in the brain. What cognitive and computational mechanisms +underlie these phenomena? This is the fourth research question introduced below. +1. Units and dimensions of language cognition +3 + +Linguists have defined many language units of different sizes and types such as phonemes, sylla- +bles, morphemes, words, phrases, and sentences. A key concern in the study of language cognition +is whether these language units are merely concepts proposed by linguists for the convenience of +research or truly processing units on which the brain relies for language understanding. In the +normal process of language understanding, what type of language unit does the brain analyze? +Does the brain construct different neural representations for different types of language informa- +tion (such as phonetics, grammar, and semantics)? These are the concerns of language cognition +research. For example, Liberman et al. [35] believe that the phoneme is the basic unit of speech +processing and that phoneme recognition is the function of the brain motor system. However, +Greenberg [36] and Hickok et al. [37] hold that the syllable is the more central processing unit +and that phoneme processing is possible only for certain tasks. For another example, Townsend +et al. [38] believe that large phrases or even sentences are the basic units of semantic under- +standing, but many connectionists think that words are the basic processing units of the brain +and that phrase and sentence structures have little effect on the brain’s language processing [39]. +2. Brain networks that localize different types of language information +Language is a function of the brain; but which parts of the brain are crucial for this function? +Neuroscience research has found that the brain can be divided structurally and functionally, and +the earliest evidence for functional division of the brain comes from studies of aphasia, which +will be introduced in the next section. Aphasia research and modern neuroimaging research have +found that language is not a single function but includes many functional modules [40, 41, 42]. +Therefore, current language cognition research pays more attention to the brain network involved +in locating specific functional modules. +3. Time course and control of language information processing +What is the processing order of different modules for language understanding in the brain? For +example, will the brain parse the grammatical structure first and then process the meaning [43]? +How long does it take to process each step? Can different features in a vocabulary be identified +for a long time [44]? Are the steps and sequences of brain processing language automatic and +invariable, or do they require the influence and regulation of cognitive functions such as attention +and working memory [45]? These are also the focus of research on language cognition. +4. Neural coding and computational mechanism of language information +Studies of brain regions and time courses have focused on describing language processing phe- +nomenologically. How do these phenomena arise? From a computing point of view, what are +the ”data structures” for computing in the brain, and what algorithms are used to operate these +data structures? Research on processing mechanisms will inevitably involve mathematical mod- +els, which also presents difficulty. At present, one type of model seeks to directly explain the +neural response of the brain [46, 47, 48], and the other type simulates language behavior, such +as by simulating the language-acquisition process [49]. +2.3 +Development of language cognition +Early research focused primarily on aphasia. These studies emerged around the middle of the nine- +teenth century and mainly analyzed the relationship between brain damage and language behav- +ior in patients. In the past 50 years, the maturity of technologies, including electroencephalogram +(EEG), magnetoencephalography (MEG), positron emission computed tomography (PET), and func- +tional magnetic resonance imaging (fMRI), has provided powerful tools for studying the language +function of the normal brain. Studies based on behavioral or neuroscience experiments have achieved +many results. Combined with the research questions in the previous section, we introduce four studies +as examples. +The first study focused on units of language comprehension. An extreme view is that sentences (or +large phrases) form the basis for processing. In this unit, the process of listening or reading a sentence +is simply a process of acquiring information, and the information is processed and integrated when +it reaches the boundary of the sentence. Another extreme perspective is that language processing is +carried out in real time; that is, the brain will process the current information at every moment. Thus, +information obtained at all times can be fully processed, and there is no need for centralized processing +4 + +at some important language boundaries. Evidence for the first view is that language understanding is +heavily dependent on context; George Miller found that, in noisy environments, the same word can be +better recognized if it is placed in a sentence [50]. +As another example, in the process of listening to an article, if the article is suddenly interrupted +and the listener is asked to recall what he heard before, the listener can only accurately recall the +vocabulary in the current sentence [51]. The second view is supported by considerable evidence. For +example, if you play a voice and ask people to read along, some people can follow at a speed of +approximately 300 ms. This means that the voice spoken to the reader is only approximately one +word slower than the voice heard. In this case, if you heard the word ”tomorrane,” but, according to +the contextual information, the word should mean ”tomorrow,” you would have a higher probability +of saying ”tomorrow” instead of ”tomorrane.” However, when contextual information is lacking, the +reader will not perform this type of correction [52]. +In this case, the reader integrated the above +information in real time rather than waiting to process it until the end of the sentence. +Another study found that, if a person saw four objects on a screen (such as a horse, an apple, a +table, and a newspaper), when they heard ”this kid is riding,” the person’s gaze would often fall to the +”horse object. This indicates that people’s language processing is predictive and that people instantly +generate expectations based on the above. Combining these two perspectives, one can argue that the +brain performs both immediate predictive processing and additional integration at sentence or phrase +boundaries. +The second study concerned the modules included in language processing and the corresponding +regions or networks in the brain. Early research on aphasia found that, when certain brain regions are +damaged by trauma or disease, language function is impaired. More importantly, language is not a +single function but a complex system of functions in different brain regions. For example, patients with +Broca’s aphasia cannot produce language but can understand it, and patients with Wernicke’s aphasia +can speak language but cannot understand it. These two aphasias suggest that language production +and comprehension are in separate brain areas and, therefore, can be selectively impaired. +More detailed studies have found that some patients have impaired recognition of nouns but pre- +served recognition of verbs while others have the opposite, suggesting that verbs and nouns are pro- +cessed differently in the brain [53, 54]. Similarly, some patients have an impaired ability to distinguish +phonemes (such as being unable to distinguish /ba/ and /da/) but normal auditory word compre- +hension while others have the opposite, which shows that phoneme discrimination and auditory word +recognition involve different brain regions [37]. +All these phenomena indicate that language com- +prehension involves many modules; thus, damaging specific brain areas affects only part of language +function. +Studies based on fMRI methods, which observe the activation of different brain regions in processing +different information or performing different tasks in people with typical brain functioning, have also +revealed this functional division. Moreover, they have found that different brain areas are activated by +grammatical and semantic processing [55, 56, 57], and different brain regions are activated by different +word categories (such as tools versus seeing animals) [58, 59]. In general, aphasia studies have found +that damage to some key brain areas can affect a certain function; however, MRI studies have generally +found that this function actually involves a more widespread brain network. For example, traditional +aphasia studies generally hold that temporal lobe damage is more likely to cause noun comprehension +problems, and frontal lobe damage is more likely to cause verb comprehension problems. However, +recent MRI studies have shown that processing verbs and nouns involves very complex brain networks +in which internal connection properties are related to the processing of two types of words [60, 61]. +The third study examined vocabulary recognition and processing. Numerous studies have shown +that word recognition is a parallel process. For example, when hearing the English syllable ”/kp/,” +it is generally believed that the brain will activate all the words at the beginning of this syllable in +parallel, such as cap, captain, and caption. Among these words, those with a higher word frequency +or that are more consistent with the context have stronger activation. How do psychologists conclude +that many words are activated simultaneously? Classic experiments used the cross-modal priming +effect. These experiments found that, after hearing ”/kp/,” people recognize visual presentation of +words including ”captains” and ”captions” faster than after hearing other syllables (such as ”/da/”). +Moreover, after hearing a word, vocabulary related to the semantics of the word is activated. For +example, after hearing ”captains,” people recognize words like ”ships” more quickly [62]. +From the perspective of neuroscience, vocabulary induces an EEG response N400 with a latency +5 + +of approximately 400 ms, and the amplitude of N400 is closely related to word frequency and the +previous context. For example, the amplitude of N400 induced by ”ship” depends on the preceding +word. If the preceding word is ”captain,” the amplitude of N400 induced by ”ship” is relatively small; +if the preceding word is an irrelevant word (such as ”apple), the amplitude of N400 induced by ship is +relatively large [63]. +Vocabulary recognition is generally believed to be a parallel process. After hearing part of the +vocabulary information, a large amount of vocabulary is activated; however, as the information in- +creases, the vocabulary that does not match the new information is suppressed until the brain finally +determines a possible vocabulary [62]. The brain processes possible words in parallel because language +is full of ambiguity, and information is presented very quickly. If you wait until all ambiguity is resolved +before starting processing, not only may the reaction speed be too slow, but the information that has +already exceeded the brains working memory capacity may be forgotten. Similar problems are more +common in sentence processing, such as the sentence ”The horse raced past the fence fell. Before seeing +the last word, we will think that ”raced” is the predicate verb of the sentence, but after seeing the +last word, we will find that the previous understanding was wrong; ”fell” is the predicate verb of the +sentence. Sentences that contain ambiguity so that the analysis of sentence structure changes during +the comprehension process are called ”garden path” sentences. At present, some theories suggest that +the human brain will also construct a variety of possible sentence structures at the same time and +then continue to screen, but others suggest that the brain will first construct the most likely sentence +structure and reanalyze if the structure is found to be wrong. +The fourth study focuses on speech understanding in complex environments. In the 1950s, British +scientist Colin Cherry discovered that attention plays a crucial role in speech understanding in complex +environments [64]. Cherry’s and subsequent studies found that, if two different speeches (speeches from +different speakers or different spatial orientations) were played simultaneously in the experiment and +the listener was asked to focus on one of the speeches, they could understand the speech that they +paid attention to very well but could not recall the content of the other speech they did not pay +attention to afterwards. Psychologists also quantitatively analyze the impact of various factors on +speech recognition, such as measuring the speech recognition rate under different noise intensities, +and then draw the psychological curve of the speech recognition rate changing with noise intensity. +American scientist George Miller found that speech recognition rate is related not only to noise intensity +and listener attention but also to the listener’s prior language knowledge [50]. In noisy environments, +humans can recognize grammatical sentences better than random word strings. +Cognitive neuroscience experiments have further shown that both attention and prior knowledge +can directly regulate the processing of the acoustic features of speech in the auditory cortex. Under +this mediation, the neural activity of the auditory cortex mainly encodes the attended speech. These +studies demonstrate the important roles of attention and prior knowledge in language comprehension. +The research on the above four aspects shows that predictive processing, language structure pro- +cessing, parallel processing, attention, and prior knowledge are all important characteristics of human +language cognition. +2.4 +Research methods +Studies in cognitive and life sciences can be divided into hypothesis- and data-driven research. Ac- +cordingly, linguistic studies can also be roughly divided into these two types of research. +1. Hypothesis-driven research +Most language cognition research is hypothesis-driven; that is, researchers will clarify the hypoth- +esis to be verified by the experiment (generally referred to as H1) and its opposite hypothesis +(H0) before the experiment. They will also clarify how the experimental results are consistent +with the expectations of H1 or H0 unanimously. If the final experimental result is consistent +with the expectation of H1 and inconsistent with the expectation of H0, then H0 is falsified (that +is, the findings support H1). In contrast, if the final experimental result is consistent with the +expectation of H0 and inconsistent with the expectation of H1, then H1 is proven false. +For example, suppose researchers wanted to test the hypothesis that the auditory cortex encodes +not only the acoustic features of speech but also phonemic information. Based on this hypothesis, +the researchers designed experiments to distinguish acoustic features from phonemic features and +6 + +then analyzed whether the latter affected the responses of the auditory cortex. Two specific +examples below illustrate hypothesis-driven research. +As mentioned above, the unit of brain processing language is a controversial issue. The hypothesis +proposed by one study is (H1) that the brain can encode multiple levels of language units in +parallel and that the neural activity encoding a language unit should be synchronized in time +with the unit; that is, when a language unit appears, the neural activity also occurs, and when +a language unit ends, so does the corresponding neural activity. The counter hypothesis (H0) is +that the brain processes only according to a single level (such as words) or that different levels of +language units do not show neural responses synchronized with language units. If this assumption +holds, then the update rates of neural activity encoding language units of different sizes, such as +syllables, words, phrases, and sentences, will be different. For instance, if there are four syllables +in speech per second, the response to the syllables will also change four times per second. If every +two syllables are combined into a phrase and every two phrases form a sentence, then the neural +response of words and phrases should be updated two times per second, and the neural response +of sentences should be updated one time per second. Therefore, the researchers designed the +experiment according to the above ideas and found that the MEG/EEG response of the human +brain in the process of listening to speech does contain 4 Hz, 2 Hz, and 1 Hz components, +corresponding to the neural responses of hypothetical syllables, phrases, and sentences [65]. This +experimental result supports H1. +Another example is the study of word meanings. The semantics of a word can be learned in +two ways. One way is to directly establish the relationship between the word and the objective +entity it refers to, such as seeing the fruit ”guava” and being told it is a ”guava.” This learning +method directly establishes the relationship between the language symbol ”guava” and the sen- +sory characteristics (visual, taste, etc.) of the object it refers to. Another way is to describe the +meaning of the acquired vocabulary through words, such as reading in the dictionary, ”Guava +is a plant of the myrtle family, and the fruit is edible.” The hypothesis (H1) here is that the +semantics acquired through the above two methods are encoded in different regions of the brain. +The opposite hypothesis (H0) is that the representation of a word in the brain is independent +of the acquisition pathway. To distinguish between these two modes of acquisition, the study +compared the processing of colors by sighted people and congenitally blind people; both groups +can acquire color concepts through language, but only sighted people can directly establish color +words and visual correspondence between color information. The study found that some brain +regions encode color in the same way in both groups, but others encode color in a way that is +only present in sighted people. This result also supports H1 hypothesis [66]. +2. Data-driven research +Hypothesis-driven research is often highly targeted research–that is, experiments specifically +designed to test a particular hypothesis. The opposite of hypothesis-driven research is data- +driven research. Data-driven research is exploratory; it does not put forward a hypothesis first +but explores possible results by collecting experimental data. The purpose of hypothesis-driven +research experiments is clear, so it is easier to obtain stable results. +For example, to verify +the hypothesis that neural activity and hierarchical language structure are synchronized in the +first experiment above, a constant rate was used to play speech to simplify data analysis (the +researchers only needed to analyze the response of a specific frequency in the frequency spectrum). +Moreover, to distinguish the two methods of acquiring word meaning in the second experiment, +two groups of people were selected for comparison. +However, it is often difficult to strictly distinguish between hypothesis- and data-driven research. +Without exploratory research, it is difficult to formulate a hypothesis; without a hypothesis, it is +difficult to determine which aspect of the data to analyze. For example, the above two hypothesis- +driven studies also contain data-driven components. The first study did not determine which +MEG/EEG channels can obtain responses, and the second did not assume in advance the brain +region in which the experimental phenomenon would be found. +7 + +2.5 +Limitations of existing research +Language cognition research initially revealed some patterns of human language understanding, but +much more is needed to truly analyze the mechanism of language understanding in the human brain. +Currently, the main problem is that, in theory, the existing research focuses on the qualitative expla- +nation of local problems and relies on small samples and strict experimental control, which leads to a +lack of ecological and global research conclusions. The overview is as follows. +1. Lack of discussion on the quantitative mechanism +Most cognitive science research is described at the phenomenon level, and even the discussion of +the mechanism is often qualitative and subjective. For example, as mentioned before, when the +brain processes a word, it produces an EEG response of N400. Many of studies have investigated +this response, clarifying how the previous contextual information of various properties affects +the magnitude of the N400. +However, these studies only show that the previous contextual +information can affect the N400 response and do not answer what computational mechanism +this effect reflects. Cognitive science literature discusses multiple mechanisms for the generation +of N400. One hypothesis is that N400 represents the current word that can be predicted by +the brain; that is, words that can be predicted produce smaller N400. +Another hypothesis +suggests that N400 represents the ease of integration of a current word with previous context; +that is, N400 is smaller if it is easy to integrate. +However, these hypotheses are qualitative +language descriptions, and it is not clear how the brain’s prediction and integration reflect the +computational mechanism. +2. Targeting specific linguistic phenomena +Cognitive science experiments often use strictly controlled experimental designs to study specific, +even very detailed language phenomena. Due to the strict control of experimental variables, +the corpus in the experiment tends to be consistent, so the experimental conclusions are likely +to be applicable only to the highly consistent corpus involved in the experiment. Poeppel and +Embick [67] identified a mismatch of research scales between linguistics and neuroscience research. +Linguistics is often concerned with very fine-grained issues (such as the usage of a word and how +the syntactic structure of a sentence should be divided) while neuroscience is concerned with +relatively macro issues (such as which part of the brain processes grammar). However, even +neurolinguistics studies often only use a relatively consistent and typical corpus for research, so +the universality of the conclusions is not strong. +3. Research conclusions are difficult to integrate +Closely related to the previous study, tightly controlled experiments lead to fragmentation of +research with one study only concerned with one particular linguistic phenomenon. If each study +focuses on one linguistic phenomenon and language contains a limited number of linguistic phe- +nomena, then an overall conclusion can be drawn by integrating different local studies. However, +because the language is too complex to be uniformly divided into several basic phenomena and +the experimental methods are too diverse, it is very difficult to integrate various research conclu- +sions. For instance, cognitive experimental studies have found that different types of language +materials can activate different brain regions and induce different types of EEG responses. How- +ever, if these studies are combined, can they tell us how the brain understands even a simple +sentence step-by-step? In fact, they cannot. +3 +Research on language computation +3.1 +Definition of language computation +The language computation mentioned in this article refers to the process of machine understanding of +language. Taking Chinese as an example, the process of language computation includes the recogni- +tion and representation of characters, the structure and semantic analysis of texts (including words, +phrases, sentences, and discourses), and the analysis of the association between text symbols and +the external world and finally achieves the goal of enabling machines to understand language. The +8 + +language computation we refer to below can be compared with the basic research problems in nat- +ural language processing (also called natural language analysis), such as lexical, syntactic, semantic +and discourse analysis, knowledge representation, and computing, and does not involve application +technology research. +3.2 +Main research questions +To make a machine understand natural language, we must first encode the information in a language +into a form that can be processed by the computer, which is called the text representation task. +To further analyze the information in a text, it is necessary to analyze its structure and semantic +information–that is, to perform structural analysis and semantic analysis tasks. Thus far, the machine +has known the relationship between different language symbols. It is necessary to further associate +language symbols with the external world and knowledge to understand natural languages like humans. +The main research questions are as follows: +1. Text representation method +Language is composed of small elements hierarchically and recursively, which in turn form words, +phrases, sentences, and discourses. In language communication, word is the most basic semantic +unit, and the combination of words needs to be based on specific rules. These limited rules can +combine different concepts to construct endless text units. How to represent lexical semantics, +such as by using symbols [68], functions [69], vectors [70], and tensors [71]? +How to build +efficient lexical semantics learning methods[72, 73, 74]? +How to combine lexical meaning to +form the meaning of larger-grained text units [75, 76, 77, 78]? These are key issues in linguistic +computing. +2. Structural Analysis Methods +Structural analysis is generally divided into syntactic structure analysis and discourse structure +analysis tasks. Among them, syntactic structure analysis studies the combination and depen- +dence relationship between words in a sentence, and discourse structure analysis studies the +combination and dependence relationship between sentences in a paragraph or discourse. These +two types of analysis can resolve the ambiguity of the structure in the input text, analyze the in- +ternal structure of the input text, and provide structural information for the semantic analysis of +the text, which is considered to be an important part of language understanding [79, 80, 81]. The +main research issues in this direction include how to design or select formal rules for grammars +and how to design automatic analysis algorithms. Representative of this kind of work are the +rule-based phrase structure analysis method proposed by Klein et al. [82] and the dependency +structure analysis method based on neural networks proposed by Chen et al. [83]. +3. Semantic analysis method +For different language units, the task of semantic analysis is different. +For word, semantic +analysis focuses on how to disambiguate the meaning of words and how to identify the semantic +relationship between words (including antonyms, synonyms, part-whole and event relations, etc.); +for sentences, semantic analysis includes semantic role labeling, semantic parsing, calculation +of semantic similarity between texts, and identification of implication relations; for discourses, +semantic analysis includes how to resolve references and identify inter sentence relations in texts. +The identification and calculation of the above-mentioned semantic information and semantic +relationship is the basis for understanding the meaning of a text. It is also a difficult problem in +language computation to build an efficient semantic analysis method [84, 85]. +4. Knowledge representation and symbol association method +Knowledge in this paper refers to world knowledge, historical knowledge, commonsense knowl- +edge, and professional knowledge of various disciplines. Knowledge representation is a description +of knowledge. The current model represents knowledge in the form of symbols [86] or distributed +vectors [87] and realizes the association between language symbols and knowledge through re- +trieval or mapping to a unified representation space. Among them, how to design the encoding +form of knowledge and automatically learn the representation of knowledge is the key to this type +of research [88, 89]. In addition, no matter it is a human or a machine, to understand the meaning +9 + +encoded in language symbols, it is necessary to associate it with world knowledge. Otherwise, as +the ”Chinese room” described, the people in the room do not know Chinese and cannot truly un- +derstand the received Chinese information, but he can make Chinese native speakers think that +he can speak Chinese fluently, creating an intelligent impression [90]. Therefore, how to associate +language symbols and knowledge is also the core issue of language computation research. +3.3 +Developments of language cognition +Language is a serialized and structured symbolic expression. How to represent the meaning of text and +automatically analyze its semantics and structure is a crucial step in research on language computation. +Moreover, this has always been a major challenge in machine language understanding. Almost all +natural language processing tasks, such as machine translation, question answering, and dialogue +systems, rely on semantic representation and computation of input language sequences. +Amidst the decades of the development of natural language processing, text-representation methods +have undergone a systematic transformation from discrete symbol representation to continuous vector +representation. With discrete symbol representation, words are regarded as discrete symbols, and each +word can be expressed as a one-hot vector whose dimensions are equal to the size of the vocabulary, +where one dimension is 1 and the other dimensions are 0. In this representation system, sentences and +discourses are usually represented by a bag-of-words model. +In 1954, Harris proposed the concept of a bag of words in the article ”Distributional Structure.” In +the following decades, the bag of words model has been the mainstream model of text representation +[91]. This text representation method, based on discrete symbols, can only use string matching to +extract features and calculate the similarity between language units, which easily leads to data sparsity +problems and cannot capture the semantic similarity between words. +On the other hand, distributed continuous vector representation is convenient for semantic cal- +culation and measurement and can theoretically solve the problem of semantic gaps between words, +sentences, and discourses. Harris and Firth proposed and clarified the distributed hypothesis of words +in 1954 and 1957, respectively, in which the semantics of a word are determined by its context; that is, +words with similar contexts have similar semantics [91, 92]. Matrix decomposition and neural networks +are the two main models for learning the distributed vector representations of words. Among these, +neural networks have been the mainstream model for learning distributed vector representations in +recent years. +In 2003, Yoshua Bengio et al. [93] proposed a neural network language model that uses a low- +dimensional continuous real number vector to represent each word and learns an n-gram grammar +model based on this, marking the beginning of distributed text representation. Tomas Mikolov et +al. [70] proposed the Word2Vec method, including two models of CBOW and Skip-gram in 2013, +which greatly simplifies the distributed vector learning method of words so that it can make full use +of massive unlabeled text data to learn words efficiently. +In 2017, the transformer model proposed by Google [94] combined the semantics of vocabulary +more efficiently through pairwise calculations between words to obtain a semantic representation of +text. Since then, largescale pre-training models based on transformer architectures, such as BERT [95], +TransformerXL [96], GPT3 [97], PaLM [98], and ChatGPT 1, have been developed for various language +processing applications, further establishing the dominance of distributed vector representation. +The distributed text representation model greatly facilitates the representation and calculation +of natural language, thus becoming the cornerstone of deep learning applied to natural language +processing tasks and further promoting the breakthrough development of applications such as text +understanding and machine translation. However, existing methods lack the modeling of fine-grained +text semantics and structural information and cannot effectively deal with linguistic phenomena such +as lexical ambiguity, antonyms, extended meanings, and structural ambiguities of sentences and texts +[99, 100, 101]. On the other hand, semantic, syntactic, and discourse analysis methods study how +to represent the meaning, combination, and dependency of language units and provide structural +information for text representation, which may be helpful for existing text representation models. +Semantic parsing is the most representative task in semantic analysis, which studies how to parse +natural language into a complete semantic representation (such as logical expressions) that can be +recognized or calculated by machines. Specifically, it includes how to obtain the semantics of words in +1https://openai.com/blog/chatgpt/ +10 + +a sentence, the semantic relationship between words, and how to combine word representations into +sentence semantic representations [102]. Similar to most natural language processing tasks, semantic +analysis has undergone a shift from rule-based, statistical, and neural-network approaches. Although +the performance of the neural-network-based semantic analysis model has made great progress, this +method still strongly relies on a large amount of manually labeled data and cannot truly realize +semantic understanding by learning statistical laws for prediction. +In addition to the abovementioned semantic analysis, understanding the meaning of a text is +inseparable from the analysis of its structure. At the sentence level, structural analysis refers to the +analysis of the input word sequence in the syntactic structure of a grammatical sentence, which is +mainly divided into phrase and dependency structure analysis. Phrase structure analysis analyzes +sentences in a hierarchical phrase structure tree based on phrase structure grammar, which is part of +the theory of language transformation and generation created by Chomsky [103] in 1957. Dependency +structure grammar is mainly used to describe the semantic dependencies between words and was first +proposed by Tesniere in 1959 [104]. +At the discourse level, structural analysis studies the combination and dependency relationship +between sentences in paragraphs or discourses with the goal of analyzing discourse structure as a +whole to understand discourse meaning. Computational linguists generally believe that syntax and +discourse analyses are necessary to realize language understanding. Only by analyzing the structure of +the text correctly can we understand its meaning. However, the results of various language processing +tasks in recent years have shown that explicitly modeling text structures does not significantly improve +model performance [105]. This leads researchers to rethink the role of text structure analysis, whether +existing language models can implicitly learn certain syntactic rules, and whether artificially defined +syntactic structures are necessary for machine-language understanding. +In addition to expressing and analyzing shallow meaning units such as words, phrases, and sen- +tences, language understanding also requires higher-level expressive abilities, such as the connection +with ”knowledge,” the ability to perceive context, and reasoning. Traditional knowledge representa- +tion methods use a computational framework based on logical symbols, such as a logical reasoning +language based on first-order predicate knowledge representation represented by Prolog and a prob- +abilistic graphical model represented by a Markov logic network. These methods are good at precise +reasoning and computation but lack generalization and approximate semantic computation capabilities +under incomplete knowledge. In recent years, with the development of deep learning, knowledge repre- +sentation has gradually adopted neural-network-based methods to learn the representation of entities +and relations. This method based on representation learning has better generalization performance +and is more conducive to the widespread uncertainty calculation problems in information processing, +but it cannot perform precise semantic calculations and reasoning [106]. Therefore, the combination +of precise semantic computing based on symbolic logic and approximate semantic computing based on +representation learning is the future research trend of knowledge representation methods. +3.4 +Research methods +Consistent with most research directions in the field of computer science, natural language methods +are mainly divided into two camps: rationalist and empiricist (or a data-driven approach). The rule- +based approach is representative of the rationalist approach, which holds that a large part of people’s +language knowledge is innate and determined by genetics. Data-driven approaches include statistical +machine learning methods and neural network methods (or deep learning methods). This method +assumes that the complex language structure of humans can be learned through acquired training. +1. Rule-based approach +Since natural language is essentially a symbolic system produced by human society due to the +need for communication, its rules and reasoning features are distinctive, so the early research on +natural language processing first adopted the rule method. This method obtains the understand- +ing of people’s language ability through the study of some representative sentences or language +phenomena and summarizes the laws of language use to analyze and infer the expected results of +the test samples. The rationale for rule-based approaches is presented here using the application +of context-free grammars to the problem of syntactic parsing as an example. For example, we +have the following context-free grammar: +11 + +G = (Vn, Vt, S, P) +Vn = S, NP, V P, N, V, Det +Vt = one, kid, chases, a, cat +S = S +P : S → NP V P, NP → Det N, V P → V NP, Det → one, Det → a, N → kid, N → cat, V → +chases +Based on the above rules, the syntax tree can be constructed by the top-down or bottom-up +analysis method. Here, we use the top-down analysis method as an example for illustration. +First, start from the initial symbol S, search from top to bottom, select the applicable rule in +the grammar P to replace the search target, match the word in the sentence with the right part +of the grammar rule, and erase the word if the match is successful. Then, the search continues +on the remaining part of the input sentence until the end of the sentence. Specifically, in the +above example, the search steps are: +(a) Select rule S → NP V P, no matching word, left sentence ”one kid chases a cat”. +(b) Select rule NP → Det N, no matching word, left sentence ”one kid chases a cat”. +(c) Select rule Det → one, matching word ”one”, left sentence ”kid chases a cat”. +(d) Select rule N → kid, matching word ”kid”, left sentence ”chases a cat”. +(e) Select rule V P → V NP, no matching word, left sentence ”chases a cat”. +(f) Select rule V → chases, matching word ”chases”, left sentence ”a cat”. +(g) Select rule NP → Det N, no matching word ”a cat”. +(h) Select rule Det → a, matching word ”a”, left sentence ”cat”. +(i) Select rule N → cat, matching word ”cat”, no left sentence. +According to such a search process, we can express the syntax tree of the sentence ”a child +chasing a cat”, as shown in Figure 2. +S +NP +VP +Det +N +NP +V +Det +N +one +kid +chases +a +cat +Figure 2: Example of a syntax tree +2. Data-driven approach +Human language is not a formal language for rules and patterns often exist implicitly in the +language, making the rules difficult. In addition, the complexity of natural language makes it +difficult for rules to cover all linguistic phenomena without conflicts. The data-driven method +saves the burden of manually compiling rules and automatically generates features or evaluates +the weight of features in model generation, which has good robustness. Generally, data-driven +methods include statistical methods and neural network methods, and the following two methods +are introduced. +Statistical-based language computation methods use large-scale language data and often need +artificial help (labeling data and screening features, etc.). They use statistical methods to discover +the law of language use and its probability which are then be used to calculate the possible outputs +of testing data. Here, we take the language model as an example to introduce the basic principles +of statistical methods in which the representative one is the n-gram model. +12 + +The goal of the language model is to calculate the probability of a string appearing as a sentence, +that is, to calculate the product of the probabilities of each word in the sentence s in different +historical situations: P(s) = p(w1)×p(w2|w1)×p(w3|w1w2)...p(wl|w1...wl−1). Among them, the +probability of generating the i-th word wi is determined by the generated i−1 word w1w2...wi−1. +When the above method is simplified and only the previous n − 1 words are used to predict +the next vocabulary, it is an n-gram model. +For example, the bigram model calculates the +probability p(a child chasing a cat) = p(one|bos)×p(child|one)×p(chasing|kid)×p(one| (cat|one). +Among them, bosis the start character. Generally, maximum likelihood estimation is adopted to +calculate the conditional probability of P(wi|wi−1) = +c(wi|wi−1) +� +wi c(wi, wi−1). +Methods based on neural networks mainly study how to design task-related neural network struc- +tures and optimize neural network parameters. Here, we take the text representation problem +as an example and use the recurrent neural network model as an example to introduce the ba- +sic principles of neural network-based methods. A recurrent neural network is a kind of neural +network that is good at processing sequence information and long-distance dependencies. For +example, the corpus has the following content: ”A child chases a cat ...”. To represent the above +text, the sentences in the text need to be sequentially input into the cyclic neural network. As +shown in Figure 3, the cyclic neural network model processes each vocabulary in the sentence in +turn, encodes the corresponding vocabulary wt into a real-valued vector xt at time t, and com- +bines it with the hidden layer vector ht−1 generated at the previous time to obtain the hidden +layer vector at this moment: ht := σ(Wxhxt + Whhht−1 + b), + one +kid +chases +a +cat +one kid +chases +a +cat + +h0 +h1 +x1 +h2 +x2 +h3 +x3 +h4 +x4 +h5 +x5 +x0 +Figure 3: Example of a recurrent neural network +Among them, the symbol ”:=” means ”defined as”; σ(z) = 1/(1 + exp(−z)); Wxh, Whh and b +are the model parameters. The objective function of the model is to maximize the probability of +predicting the following vocabulary above. The commonly used objective function is to minimize +the cross-entropy loss of the model: Loss= − � wtlogyt. wt is the real following vocabulary, and +yt is the following vocabulary predicted by the hidden layer vector ht : yt = Softmax(Vhyht +c), +where Vhy and c are model parameters. +The above objective function is used to learn the parameters of the model on large-scale unlabeled +texts so that the model can predict the following words through the above words. In this process, +the language model obtains the representation vector of the vocabulary. At the same time, the +representation vector of the sentence can be the hidden layer vector obtained at the last moment +of the model or the maximum or average pooling result of all hidden layer vectors. +3.5 +Limitations of existing research +Existing language-computation models are far from being able to understand language similar to +humans. The main problem currently faced is that the model structure lacks a theoretical basis and +parameter training relies on large-scale computing resources, which can be summarized as the following +four points: +1. Single textual representations +Existing text representation models do not distinguish between different types of information +(such as vocabulary of different parts of speech, text units of different granularities) and uniformly +13 + +encode them into dense vectors of the same dimension. This encoding method is very efficient +when constructing a neural network model, but it ignores the size of different types of texts. +Therefore, to encode all information, it is necessary to uniformly use the encoding method with +the largest amount of information for all types of texts. This method adopts larger numbers +of parameters instead of designing model structures and contains a large number of redundant +parameters. +2. Lack of interpretability +Although the deep learning method has greatly improved the performance of numerous tasks in +natural language processing, the meaning of the vector dimension cannot be explained. There- +fore, it is impossible to analyze what operation each unit in the network performs on the language +input. Moreover, it is impossible to give the reason the model obtained the wrong result. This +makes the results unreliable and hinders the design of the model structure and further improve- +ment of performance. +In contrast, if the model is interpretable and can give the reason for reaching a certain conclusion +like a human, then the model structure can be improved in a targeted manner, thereby improving +the model performance. For example, when judging whether two sentences express the same +meaning, the model can give a yes or no conclusion and, at the same time, give which text +fragments in the two sentences significantly affect this conclusion. These can be used as the basis +for judging the rationality of the model. +3. Lack of ability to learn independently +Existing methods construct training datasets for each task, adopt the ”training-test” development +method, and cannot directly use new training samples to correct the trained model. Additionally, +the model trained in a certain task is difficult to apply to other middle tasks. Different language +tasks require the model to be capable of language representation and understanding, so new +tasks can use the already-trained model to learn new task-specific information on this basis. +Analogous to the self-evolving learning ability of humans–that is, learning from simple tasks and +continuously learning new tasks and correcting existing knowledge on this basis–an intelligent +system should also be capable of continuous learning to achieve the self-evolution of the model. +4. Rely on largescale single modality training data +Existing language-computation models rely heavily on the quality and scale of training samples, +making it hard to handle those words, language structure, and language expressions that have +not appeared in the training corpus. In contrast, when humans learn language concepts, they +often obtain information from multiple modal samples and can learn a new word or language +expression with only a few samples. For example, when learning the concept of ”giraffe,” we +might understand from a text that ”it is the tallest mammal in the world, and its distinctive +features are long neck” and so on. When viewing a picture of an animal with a ”long neck, long +legs, body” and ”a spotted pattern,” we can quickly recognize that it is a ”giraffe” even if we +have never seen one before. This shows that multiple-modal data complement and verify each +other. Thus, it is imperative to develop small-sample learning algorithms by comprehensively +utilizing multiple-modal information. +4 +Comparison between language cognition and language com- +putation +The above chapters demonstrate that the processes of human and machine language understanding +defined in the fields of language cognition and language computation are similar–that is, starting +from the encoding of basic language units, then combining them into larger-grained text units, and +finally connecting to the outside world. However, the experimental methods and focus of the research +questions in the two fields are quite different. +Language cognition research primarily adopts the following experimental method: ”1) proposing +hypothesis; 2) designing experiment; 3) hypothesis verification.” Language computation research pri- +marily adopts the method of ”1) collecting data; 2) building model; 3) model performance verification.” +14 + +In terms of research issues, language cognition research tends to reveal the reasons behind problems +whereas language computation research focuses on methods to solve these problems. Table 1 sum- +marizes and contrasts the main ideas and concepts of the two fields. Language cognition studies the +basic units and dimensions of language understanding, its spatial representation and time course in +the brain, and the regulation of world knowledge and cognitive functions on language understanding, +emphasizing ”what” and ”why” questions. Language computation studies how to allow computers to +efficiently segment and represent vocabulary, combine vocabulary representations into sentences and +discourse representations, and analyze the functions and relationships of each element in a text. The +main concern is the question of ”how to do it.” +Table 1: Main concepts in language cognition and language computation areas +Language cognition +Language computation +Language processing unit +Language is organized according to +different levels of structure (such as +morphemes, words, phrases, sentences, +discourses, etc.), with different levels +of information (phonetic, grammatical, +semantic, etc.) +Word is generally used as the basic +unit, and they are integrated through +language models such as neural net- +works +Word representation +The neurons or neural networks related +to the word are activated. It is gener- +ally believed that the neural represen- +tations of multiple words can be acti- +vated at the same time +Encoding words into a form that can +be processed by computers (such as +symbols, +vectors, +matrices, +graphs, +etc.) +Word form analysis +Morphology information encoding +Remove the affix to get the root (plays +→ play) or transform the complex form +of the word into the most basic form +(are → be) +Information integration +The +process +of +assigning +syntactic +structure and semantics to the corre- +sponding input words +Analyze the semantic and syntactic in- +formation and their relationships, then +combine word representations to form +the representation of sentences and dis- +courses through a composition func- +tion +Multimodal information +World knowledge (common sense) and +language understanding scenarios can +affect language processing, which may +involve the interaction of brain lan- +guage networks with sensory and mo- +tor networks +Associate language symbols with other +modality information +Task effect +Language comprehension is regulated +by cognitive functions — if cognitive +functions such as attention and work- +ing memory are adjusted through ex- +perimental tasks, cognitive neural pro- +cessing of language will be affected +The parameters of the computational +language model are determined by the +objective function — once the objec- +tive function of the task changes, the +parameters of the computational lan- +guage model will change accordingly +These differences carry substantial challenges and opportunities to the combination of language +cognition and computation. One challenge is that studies on linguistic cognition and computation +are in separate fields. +Research on language cognition has mainly focused on the cognitive laws +behind human language and its neural basis. Owing to the limitations of data collection and analysis +methods, such research still focuses on qualitative analysis and exploration of macroscopic laws. It +lacks comprehensive and quantitative laws and models that can be applied in practice. +An example is the representation of lexical semantics. Although many semantic dimensions have +been discovered in the field of language cognition, their general neural connections have been revealed, +and some theoretical models have been initially established. However, owing to technical limitations, it +remains impossible to accurately record the neural activations of a single word. Therefore, it is difficult +to reveal the underlying neural representation rules and calculation methods of semantic information +directly. +Similarly, computational linguists have focused on the effectiveness of practical applications when +choosing research questions. They often ignore the most essential laws of language; they only consider +the performance in downstream tasks when constructing a language-computation model and ignore +interpretability and human-like features. Therefore, as a ”black box” that lacks human character- +15 + +istics, the language-computation model is difficult to use in the modeling of human brain language +understanding. +To illustrate these challenges further, let us consider the word representation problem as an exam- +ple. Regarding the problem of lexical (concept) representation in the human brain, Wang et al. [107] +examined the representation of abstract words. By constructing two different types of lexical repre- +sentation (statistical co-occurrence- and semantic feature-based), they tested two classical cognitive +theories of abstract concept representations in the brain. These two theories highlight that abstract +word representations are expressed in linguistic symbols through contextual associations or semantic +features. The experimental results show that both types of lexical representation have significant ef- +fects on the brain. The difference is that corpus-based abstract lexical representations are associated +with brain areas responsible for advanced language processing whereas semantic feature-based abstract +lexical representations present distributed representational features that are associated with multiple +brain regions. Therefore, the researchers concluded that the representation of abstract words in the +brain is divided into two modes and is responsible for different brain networks. +This study revealed that the semantic dimension of abstract vocabulary encoded by the brain may +include both linguistic and semantic features. This is very important for further analysis of the brain’s +understanding of language. However, this analysis is too macroscopic to directly guide the construction +of computational models. Computer scientists want to understand the specific semantic dimensions +and forms of encoding that the brain needs to encode lexical concepts. For example, which neurons in +the brain encode the concept of ”knowledge”? What information do these neurons represent? What +is the relationship between these neurons? What are the rules for their connection? The answers to +these questions can directly inspire the development of new lexical representation model architectures. +In contrast, when the language-computation model encodes the meaning of words, it represents +the words as dimension-agnostic real-valued vectors and encodes the information of the words through +the relationship between the vectors. The purpose is to allow computers to process language symbols +efficiently to complete various language tasks. Although this uninterpretable representation method is +crucial for computational models to complete language tasks, it does not reveal the laws of language +itself or directly shed light on how the brain encodes lexical concepts. What cognitive scientists want to +understand is which features can be used to construct semantic vectors to explain human behavioral +data? +What information is encoded in the computational model, and what kind of operation is +performed to lead to its excellent performance in downstream tasks? +Can the calculation process +explain the human language-processing mechanism? +Which calculation module in the calculation +model is essential for language modeling, and is there a corresponding processing module in the human +brain? +In summary, the fields of language cognition and language computation differ in their research +contents and ways of thinking. However, we believe that such differences can bring new insights into +both fields. For example, in the process of language comprehension, the human brain not only com- +bines words into sentences and discourse from the bottom up but also uses cognitive functions such +as attention and working memory to regulate the language process from top to bottom. In contrast, +language computational modeling is a static process unaffected by the external environment and com- +putes a fixed encoding result for a piece of text. However, if we can learn from the dynamic encoding +mechanism of the human brain and integrate modules such as human brain memory and attention to +construct a new language-computation model, the model may acquire more general knowledge and be +easier to transfer to other tasks. +As another example, experiments have shown that, in the process of machine language under- +standing, addition is a very effective combination when integrating vocabulary representations into +phrase representations. This automatic learning of language combination rules from data may provide +new ideas for the study of the word combination process in the human brain using the underlying +calculation method. Additional ideas on combining the two are introduced in Sections 5 and 6. +5 +Convergence of language cognition and language computa- +tion +Recent years have seen increasing attention to cross-disciplinary research in the fields of cognitive and +computer science. The following section introduces related work in the fields of language cognition +16 + +and computing that inspire and merge with each other. +5.1 +Language cognition experiments using language computation methods +In recent years, an increasing number of researchers have begun using language computation meth- +ods to study the process of understanding human language. This method shows great potential for +studying brain representations at the single-word level. Furthermore, it can be used to analyze both +traditional experimental data and natural language-processing data. Specifically, this type of method +collects neural activity data of words, sentences, or chapters; uses language-computation models to +encode experimental stimuli; and uses the encoded stimuli to study the problem of brain language +understanding. Such methods typically work as follows. +Mitchell et al. [46] published an article in ”Science” in 2008 regarding the issue of how the brain +represents conceptual semantics. They found that fMRI data of people reading nouns can be modeled +using the statistical laws of certain action words. Specifically, they collected fMRI data when research +participants read 60 noun stimuli (pictures + lexical texts) and calculated the representation vectors +of these 60 nouns by using their co-occurrence with 25 sensory-motor-related representative verbs +(e.g., ”see,” ”listen, ”speak,” ”eat). These representation vectors were then trained to predict fMRI +data using a leave-two-out cross-validation method; each cross-validation predicted fMRI data for +two test words and compared them with the real fMRI data as test accuracy. The regression model +had a significantly higher classification accuracy than the random value for brain activation patterns +evoked by nouns. This suggests that there is a predictable relationship between word representations +and fMRI data. +Moreover, it provides evidence that the brain represents the semantics of nouns +that are significantly dependent on sensorimotor properties. +This work has changed the previous +experimental paradigm of examining concept representation only through the comparison between +semantic categories, opened a new data-driven method for studying a single lexical concept, and +inspired many subsequent studies. +Another representative work is an article published in ”Nature” by Huth et al. [47] in 2016. They +used language-computation models to comprehensively study how different semantic information is +encoded in the brain. Specifically, they collected fMRI data when the participants listened to more +than 2 h of narrative stories (including a total of 10,470 different words) and selected 985 basic words +describing different topics in the corpus as different semantic attributes. Then, they constructed a +985-dimensional vector for each word in the stimuli by counting their co-occurrence with the 985 basic +words in a large-scale corpus. +Next, they trained a ridge regression model such that 985-dimensional word vectors predicted each +voxel in the fMRI data. Among them, the parameter matrix with the size of ”985 number of voxels” +obtained in the model was the brain representation of 985 semantic attributes. The results showed +that the brain semantic representations were very similar across participants, and different semantic +features were encoded in specific brain regions. In contrast to the study by Mitchell et al. on the +representation of 25 sensorimotor attributes in the brain, Huth et al. comprehensively studied the +representation mode of 985 semantic features in different voxels of the whole brain for the first time. +This once again proved the usefulness of the language-computation model in studying the brain’s +language understanding. +Regarding the brain’s computational mechanism for language processing, Brennan et al. +[108] +studied whether the brain uses a linear or hierarchical structure to process sentences. +They first +collected fMRI data while participants listened to the first chapter of ”Alice in Wonderland.” They then +used a series of grammatical models (including linear and hierarchical grammar models) to calculate +the syntactic complexity of each word in the stimulus. +Finally, regression analysis was performed +between the syntax complexity index and the fMRI data. +Syntactic complexity can be calculated from the probability of words appearing in a certain gram- +matical structure. The hypothesis is that, if people use hierarchical structures to comprehend stories, +then the complexity indicators calculated using hierarchical grammar should have stronger correlations +with the fMRI data compared to the linear grammar model. The results show that the linear effect is +widely distributed in the language network of the brain while only a specific area of the left temporal +lobe of the brain is responsible for processing hierarchical structure information. This suggests that +the temporal lobe processes information in a hierarchical structure when comprehending languages. +Research on language cognition also draws on the operating mechanisms of language-computation +models to propose hypotheses. +A representative example of this type of work is that of Li et al. +17 + +[109]. They used a cognitive model and a neural network model to study the neural mechanism of the +brain when understanding pronouns. Different languages have different expressions of pronouns; for +example, the pronunciation of Chinese pronouns is gender-neutral (”ta”) while English pronouns are +pronounced differently depending on gender (”she”, ”he”, ”it”). Thus, to explore whether the human +brain adopts a general parsing strategy that is not affected by language, Li et al. collected fMRI +data from Chinese and English native speakers while listening to the full text of ”The Little Prince” +in Chinese or English. A generalized linear model, commonly used in cognitive science, was used to +calculate the brain regions related to pronoun resolution. Both Chinese and English listening materials +significantly activated the left anterior middle temporal gyrus, left posterior middle temporal gyrus, +and anterior and angular gyrus brain regions. +To further explore the computing mechanism of the brain when parsing the relationship of reference, +the researchers first constructed five computing models for pronoun reference resolution: the Hobbs +model based on syntactic theory, the Centering model based on discourse theory, the ACT model based +on memory theory, and the pronoun resolution model of ELMo and BERT based on neural networks. +Next, they calculated the reference probabilities of each pronoun using the above models and correlated +them with fMRI data. Only the ACT-R model based on memory theory could significantly predict +neural activation data corresponding to the Chinese and English experimental materials, indicating +that the brain adopts a language-independent general memory retrieval strategy when parsing the +pronoun reference relationship. +Another examples is Wehbe et al. [110], who proposed an analogy between the recurrent neural +network language model (RNNLM) and the working mechanism of the reading brain. They found that +the way the human brain works when reading a story is somewhat similar to how RNNLMs work when +processing sentences. Additionally, Schrimpf et al. [111] compared the association of 43 state-of-the-art +neural network models with various neural activity datasets. They found that the model based on the +language model and the transformer network structure can significantly predict the neural response, +behavioral data, and neural response of the next word, indicating that the language system of the +brain is optimized for predictive processing. See more recent work at survey [112]. +5.2 +Language computation methods inspired by language cognition +The deep-learning method based on neural networks has been highly praised in recent years. In a sense, +it simulates the cognitive function of the biological brain. However, this method is not a mathematical +model based on the working mechanism of the brain; thus, it is difficult to eliminate its dependence on +largescale training samples. A large gap remains between language-computation models and human in- +telligence in terms of generalization and learning ability. The language-computation model inspired by +the cognitive mechanism proposed in this article aims to study the language cognitive mechanism of the +brain, analyze the relationship between the cognitive mechanism and machine language computation, +and design a more intelligent language-computation model to complete various language-processing +tasks. +Since the current research on the mechanism of language understanding in the brain is far less +in-depth than other cognitive functions, most computational methods inspired by cognition are con- +centrated in the fields of visual cognition and machine learning, and less work has been done in the field +of language. In this paper, existing cognitive-inspired language-computation methods are summarized +into the following four categories: +1. Cognitive function-inspired models +To improve model performance when processing downstream tasks, we could borrow ideas of +cognitive mechanisms such as brain representation, learning, attention, and memory to build +new or improve existing computational models so that (part of) the model has a structure +similar to the brain. +For example, inspired by humans selectively looking at or skipping certain words when reading +sentences, Wang et al. [113] proposed a sentence-representation model inspired by the human +attention mechanism. This method utilizes the predictors of eye-movement signals (i.e., lexical +surprisal and part-of-speech labels) to build attention modules. It introduces their results as +weights into the sentence representation learning model. The results show that the attention +module assigns higher attention weights to important words, and the weight results are signifi- +18 + +cantly correlated with human reading time. In addition, the attention module can significantly +improve the performance of sentence representation on several downstream tasks. +In addition, Liu et al. [114] used the human attention mechanism to improve the performance of +image description generation models. Sun et al. [115] proposed a small data word representation +learning method based on memory enhancement. Finally, Han et al. [116] proposed a continuous +learning approach based on episodic memory activation and memory consolidation. +2. Cognitive data-enhanced models +We can use brain neural activity, neuroimaging, or behavioral data as an additional modality, +which can provide different information than the existing data. Therefore, fusing these two data +during model training could improve model performance. +For example, Klerke et al. [117] proposed a multitask learning approach to incorporate eye- +tracking data into a sentence-compression task. They utilized a three-layer bidirectional recurrent +network model with the bottom layer predicting eye-movement timing and the top layer predict- +ing sentence compression. The results show that this multitask learning method can effectively +introduce eye-movement data into the sentence-compression task and improve the performance of +the model. Tiwalayo et al. [118] fused the probability distribution of the next word predicted by +humans with that of the next word predicted by the language model, which effectively improved +the performance of the language model. +In addition, Malmaud et al. [119] introduced predicted eye-movement time into reading-comprehension +tasks. Barrett et al. [120] added eye-movement data as a feature to part-of-speech tagging and +named entity recognition models. Mishra et al. [121] applied eye-tracking data to improve the +quality of sentiment-analysis models. Additionally, Fereidoni et al. [122] introduced fMRI data +into vocabulary representation learning, and Roller et al. [123] and Wang et al. [124] introduced +human behavior data (lexical association score) into multimodal vocabulary representation learn- +ing. +3. Build models by simulating neurons +Another way to build a more intelligent computing model is to simulate the structure and working +mechanism of biological neurons or neural circuits from the underlying architecture of the model. +For example, the fruit fly brain uses Kenyon cells to receive information from multiple sensory +modalities. Specific neurons control the activation and inhibition states of these cells, so the fruit +fly brain is a sparse high-dimensional representation of input information. +Liang et al. [125] formalized this information-encoding process and applied it to the task of word- +representation learning. The experimental results indicate that the network can learn the static +and context-dependent semantic representation of words, and its performance is comparable to +other representation-learning methods. This method also represents the word as a sparse binary +hash code, which requires fewer computational resources than other methods. +In addition, this cognitively inspired method is commonly found in the research of general com- +puting methods [126], such as memory networks [127], neural Turing machines [128], capsule +networks [129], and plastic weight consolidation (Elastic Weight Consolidation) algorithms [130]. +4. Borrow methods from cognitive science to interpret models +We could learn from or directly use research methods from cognitive science to interpret infor- +mation encoded by neural network models. +For example, Chien et al. [131] used the timescale-mapping method commonly used in the field +of neuroscience to study the information encoded by each neuron in the long short-term memory +(LSTM) model. They inferred the neuron’s function by observing the activation value of each +neuron in the model for the following sentence when the normal sentence and the above fragment +were randomly replaced. The logic behind it is that, if the function of a neuron encodes short- +time-scale language information, then, when a small piece of text is replaced, its activation value +change in the following sentence should be greater than that of neurons that encode long-term- +scale language information. The study found that approximately 15% of neurons are used to +encode long-term scale information. Such neurons can be divided into two types: the controller +19 + +responsible for connecting each neuron and the integrator (integrator) responsible for integrating +long-distance information. +In addition, inspired by neuroscience’s approach to studying neuron encoding mechanisms, +Lakretz et al. [132] studied the working mechanism of each neuron in the LSTM model when +completing tasks. Ivanva et al. [133] drew on the design methods of neuroscience probe tasks +and proposed guidelines for designing probe tasks in machine-learning methods. +6 +Discussion +At this stage, researchers have achieved preliminary results in the two directions of using language- +computation models to predict brain-activity data and inspiring language-computation models through +the language understanding mechanism of the human brain. However, there is still a lack of granular +and systematic research on the combination of language cognition and computation. For example, +in terms of language comprehension in the human brain, it is not clear how people start from the +most basic language unit and gradually build larger units until they finally understand the language. +Moreover, there is still a lack of systematic and effective modeling methods to address this question. +In terms of machine language understanding, the computation model has achieved super-human +accuracy in many language-processing tasks. However, it is still far from human intelligence regarding +common sense reasoning ability, autonomous learning ability, generalization ability, learning efficiency, +interpretability, and reliability. There is no clear solution for how to learn from a deeper-brain language- +understanding mechanism to build a more intelligent language-computation system. +In the future, the author believes that computational theory-driven language-comprehension cogni- +tive experiments are promising for the study of human language comprehension, as shown in Figure 4.. +In other words, research hypotheses are proposed based on the structure or results of computational +models, and behavioral or brain activity data are verified. There are five important research directions. +Figure 4: Schematic diagram of the cognitive experiment for natural language understanding driven +by computational theory +1. Collection of multilingual and multimodal neural activity data +Most existing research on language cognition is limited to using a single data-collection method +(such as fMRI or MEG) to study the specific language phenomenon of a certain language. This +often leads to the problems of low robustness and poor repeatability of the conclusions drawn. +Therefore, future language-cognition research should conduct verifications using multiple lan- +guages and multiple types of data [134, 135]. Especially for studies combining computational +models, the scale and quality of data directly determine the reliability of the results. Therefore, +it is crucial to use both invasive and noninvasive tools to collect largescale high-quality neural +activity data for different languages. +At the same time, the opening and sharing of data is +gradually becoming a trend, which will greatly promote the study of language cognition. +20 + +Build a computational model +Collectbrainactivationdata +Propose a hypothesis +Test a hypothesis +Bee +Bee +Bees are social insects. +Bees are social insects. +Language computational model +Experimental stimulus +Modelresult +Brainactivationdata +ExperimentalStimulus2. Inspired new cognitive mechanism hypotheses +The operation process of the language-computation model is transparent and global to a cer- +tain extent, and its calculation process is also visible. For instance, the vocabulary represen- +tation learned by the model, the calculation method of combining vocabulary representations +into phrases and sentence representations, and the prediction and inference of certain calculation +steps of the result are all observable. Explaining the working principle of the brain from the +level of the computing mechanism is an important task of cognitive science. The author believes +that, in the future, we can deeply explore whether the representation and computing modules +in the computing model can indeed explain the neural activities of some brain regions in the +process of language processing. If the neural activity of a brain region can be explained by a +computational model, then the brain region can be considered to perform the computational +functions clearly visible in the model. In other words, we can regard each module in different +language computation models as a hypothesis of a brain computing mechanism and use cognitive +science experiments to verify it. +3. Correlating multiple linguistic variables and cognitive function +The process of language comprehension is very complex, not only involving the processing of +multiple language variables, such as morphology, syntax, and semantics, but also closely related to +multiple cognitive functions, such as executive control, attention, and memory. Previous studies +often eliminated the influence of other language variables and cognitive functions by strictly +controlling experimental variables and only studied the effect of a certain language variable or +cognitive function in an experiment. The author believes that language-cognition experiments +combined with computational models can eliminate the research limitations above. For example, +using computational models can separate different experimental variables and study the role of +different language variables and cognitive functions based on neural activity data collected from +natural texts[136, 137, 138]. With the continuous improvement of the performance of language- +computation methods based on neural network methods, it is increasingly accurate to use models +to separate different language features so that the visual and auditory perception, multimodal +information fusion, and language in different regions of the brain can be calculated on the same +batch of data. Other functional mechanisms in understanding become possible. +4. Analyzing the underlying computing mechanism of brain language understanding +Most of the existing research on language cognition is based on linguistic theory, but there +is a large gap between linguistics and neuroscience research. For example, regarding how the +brain manipulates the most basic language units, linguistics mainly studies phrase structure +and semantic combination while neuroscience focuses on neural oscillations and synchrony. This +has led to the lack of a neural basis in the current research on language cognition, which cannot +match the conclusions of neuroscience findings. With the continuous development of spike neural +networks (spike neural networks) and oscillating neural networks (oscillatory neural networks), +future computing models must be able to integrate the conclusions of neuroscience to simulate +the working mode of underlying neurons. Manipulating language units to complete the task of +language understanding provides a new solution for research linking linguistics and neuroscience. +5. Exploring the mechanism of language learning and evolution +As early as the 1980s, cognitive scientists used the connectionist model to explore what kind of +model and what kind of data can simulate the human language-acquisition process. However, +the computing power of the connectionist model at that time was quite limited, and it could +only solve some case-specific and simple language tasks. Today, the language-processing ability +of deep neural networks has made a qualitative leap compared to the 1980s, and there are more +corpus records in the process of infant language acquisition. Therefore, it is possible to try to use +computational models to explore the mechanism of language learning and evolution. It is even +possible to explore in depth whether the cyclic connection, convolution operation, dot product +attention mechanism, backpropagation algorithm, and so on in the language-computation model +are also necessary links in the human brain’s language understanding and computing process. +With regard to machine language comprehension, as shown in Figure 5, research on language cog- +nition suggests that the human brain may have many efficient processing methods. Thus, it has great +21 + +potential to inspire the construction of a new generation of language-computation models regarding the +cognitive mechanisms of representation, learning, and memory and the working mechanisms of neu- +rons. The author believes that the following five aspects will become important development directions +in the future. +Figure 5: Schematic diagram of a language computational model inspired by the cognitive function of +the human brain +1. Representation and combination of text +When encoding the meaning of concepts, the brain uses different representations for different +types. For example, when reading nouns and verbs, the brain activates different brain networks, +showing the characteristics of distributed coding. When observing a specific or familiar person, a +specific neuron in the brain is activated. When encoding the syntactic structure, the brain will use +different combination methods for different types of phrases, use hierarchical encoding methods +such as tree structures to guide the combination sequence of words, and use parallel processing to +encode multiple levels of language unit (words, phrases, sentences, etc.) information. With this +encoding method, the human brain stores and calculates the meaning of the text very efficiently. +It is also closely related to the ability of humans to ”infer other cases from one instance” and learn +quickly. Future language computation models can learn from this mechanism; combine symbolic +and distributed representation methods; and adopt a combination of diversity, hierarchy, and +parallelism to learn text representation and combination models. +2. Continuous language learning +Humans have the ability of continuous and small-sample learning in childhood. For example, +if a 2-year-old child is shown a picture of a giraffe, the child can recognize other pictures of +giraffes. This ability can be transferred to tasks such as recognizing pictures of other animals +and finding the text corresponding to the picture. +This learning ability is closely related to +the human memory system, and the result of learning new information is memory. Different +types of information are processed and stored by different memory systems, and information +such as sounds and images are stored by sensory memory and maintained in a short time. The +complex and structured memory system of the human brain ensures the efficient organization of +massive data and rapid extraction when necessary. All these mechanisms could be learned by +computation models to improve performance. +3. Interactive learning of language +The best existing general-purpose language-computation models use predicting the next word as +an objective function. They are trained in massive texts and achieve excellent performance in +multiple tasks. The difference is that humans often learn language by interacting with others, +which is a more effective way to learn and improve language ability than analyzing and mem- +orizing language structures. Drawing on this interactive learning method, in addition to using +text information as a supervisory signal, future language-computation models can also obtain +feedback from the structure or output results of other models and learn in continuous interaction +with each other. +4. Multimodal information fusion +Closely related to interactive learning is the comprehensive processing of multiple-modality infor- +mation. The human language-learning environment is a multimodal system. Humans are better +22 + +Cognitivefunctionsofhumanbrain +Buildacomputationalmodel +Mechanisms ofrepresentation, +Dialogue systems +learning, memory, et al. +Machinetranslation +Working mechanisms of neurons +Emotion analysis +Brain activationandbehaviordata +Specch recognition +Brain activation data +Language computational modelat processing multimodal than single-modal information, and the processing speed is faster for +multimodal information than for single-modal information [139]. Therefore, the author believes +that, in the fusion of multimodal information, brain-inspired computing models are an impor- +tant research direction in the future. For example, according to the ”Hub and Spoke” theory +[140], concepts are represented by multimodal information such as vision, hearing, smell, and +somatosensory information, and there is a semantic center that encodes modal-independent in- +formation to associate different modalities of information. The information between different +modalities is complementary and mutually verifiable and, when combined, can represent more +abundant information. The fusion mechanism of multimodal information can also learn from the +abovementioned ”central” mechanism and design a modality-independent module to integrate +and correlate different types of information. +5. Interpretability of computation models +Cognitive science designs experiments to analyze the working mechanism of the human brain. +This research method and cognitive experimental data can also be used to analyze or evaluate +the working mechanism of language-computation models and inspire new model interpretability +methods. For example, referring to the comparative analysis method often used in language- +cognition experiments, two groups of experimental materials are designed so that they differ only +in a certain language attribute. For example, groups of sentences with high and low syntactic +complexity that are basically the same in terms of sentence length, sentence meaning, and so on +are input into the calculation model, and we observe whether the effect of the model is consistent +with the syntactic complexity degree correlation. If the activation of some nodes in the network is +significantly stronger when encoding high- than low-complexity sentences, then these neurons are +responsible for encoding syntactic information; otherwise, they are not responsible for encoding +syntactic information. +7 +Conclusion +Language cognition is one of the core issues in cognitive and brain sciences. 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Zong, “Probing word syntactic representations in +the brain by a feature elimination method,” 2022. +30 + +[138] X. Zhang, S. Wang, and C. Zong, “How does the experimental setting affect the conclusions of +neural encoding models?,” in Proceedings of the Thirteenth Language Resources and Evaluation +Conference, pp. 6397–6404, 2022. +[139] J. Holler and S. C. Levinson, “Multimodal language processing in human communication,” +Trends in Cognitive Sciences, vol. 23, no. 8, pp. 639–652, 2019. +[140] K. Patterson and M. A. L. Ralph, “The hub-and-spoke hypothesis of semantic memory,” in +Neurobiology of language, pp. 765–775, Elsevier, 2016. +31 + diff --git a/l9E3T4oBgHgl3EQf6Qta/content/tmp_files/load_file.txt b/l9E3T4oBgHgl3EQf6Qta/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..65782477b4b376da51d787bdc1084993e3cb8139 --- /dev/null +++ b/l9E3T4oBgHgl3EQf6Qta/content/tmp_files/load_file.txt @@ -0,0 +1,1800 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf,len=1799 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='04788v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='CL] 12 Jan 2023 Language Cognition and Language Computation Human and Machine Language Understanding∗ Shaonan Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Nai Ding3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Nan Lin5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Jiajun Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Chengqing Zong1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='2 1National Laboratory of Pattern Recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Institute of Automation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' CAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' China 2School of Artificial Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' China 3Key Laboratory for Biomedical Engineering of Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' College of Biomedical Engineering and Instrument Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' China 4Zhejiang Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' China 5CAS Key Laboratory of Behavioural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Institute of Psychology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' China 6Department of Psychology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' China Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Email: shaonan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='wang@nlpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='cn, ding nai@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='cn Abstract Language understanding is a key scientific issue in the fields of cognitive and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, the two disciplines differ substantially in the specific research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Cognitive sci- ence focuses on analyzing the specific mechanism of the brain and investigating the brain’s response to language;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' few studies have examined the brain’s language system as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' By contrast, com- puter scientists focus on the efficiency of practical applications when choosing research questions but may ignore the most essential laws of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Given these differences, can a combination of the disciplines offer new insights for building intelligent language models and studying language cognitive mechanisms?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In the following text, we first review the research questions, history, and methods of language understanding in cognitive and computer science, focusing on the current progress and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' We then compare and contrast the research of language understanding in cognitive and computer sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Finally, we review existing work that combines insights from language cognition and language computation and offer prospects for future development trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 1 Introduction Language is a multilevel symbolic system that includes multiple levels: phonetics, morphology, syntax, semantics, and pragmatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The most basic language symbols can be combined to form more complex and endless symbol sequences to allow flexible expression of meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' As such, language is also considered the carrier of human thought and the most natural tool through which humans exchange ideas and express emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Because of the diverse and flexible characteristics of language, it is difficult to study the mech- anism of human language understanding and to build a computation model that can understand language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In the early days of computer science, language research pioneers attempted to conduct cross-disciplinary research in computer science, linguistics, and cognitive science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They aimed to es- tablish connections between human language-understanding mechanisms and language-computation models [1, 2, 3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, owing to the complexity of the problem, interdisciplinary research has gradually become separated over the decades, forming subfields such as natural language under- standing in computer science, psycholinguistics in cognitive psychology, and neurobiology of language research in cognitive neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In this paper, ”cognitive science” mainly refers to the two fields of cognitive psychology and cognitive neuroscience, particularly the branches of psycholinguistics and the cognitive neuroscience of language [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Figure 1 shows the relationship between cognitive and computer science in the direction of language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' There are substantial differences in the research questions and methods adopted in the two fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Computer scientists primarily adopt rationalist (represented by rule-based methods) ∗This paper is originally written in Chinese and published in SCIENTIA SINICA Informationis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Here we translate it into English with an extension of recent work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 1 Language cognition ♦Human language understanding � Build a computational framework to analyze natural languages Analyzing the working mechanisms of the brain language understanding Research methods Cognitive Science, Psychology, Neuroscience Language behavior and neural response analysis (Behavior experiment, fMRI, MEG, EEG, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=') Research problems Foundations Computational modeling (Using computational methods to model the mechanisms of brain language understanding) Rule-based approach (Rationalist model) Data-based approach (Empiricism model) Language computation ♦Machine language understanding � Linguistics Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Statistics Figure 1: Connections between cognitive sciences and computer sciences on the language understanding problem and empirical methods (data-driven modeling methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' represented by statistical machine learning and neural network methods) and pay more attention to applied research–that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' how to build an intelligent system to understand natural language to complete various practical applications (such as machine translation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' dialogue system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' and automatic summarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The ”understanding” here refers, more precisely, to application-oriented ”processing”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' thus, in many cases, ”natural language processing” is commonly used to collectively refer to this research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' On the other hand, cognitive scientists adopt neuroimaging and behavioral analysis methods and pay more attention to the psychological and neural basis of human language understanding, such as the functions of each brain region in language understanding and how neural activities encode different levels of language information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The commonality between these two fields is that both use linguistics as the subject basis and computational modeling as tools for analysis in terms of language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In general, language cognition and language computation have achieved fruitful results in their respective directions, and new theories and methods have been continuously proposed and successfully applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In recent years, improvements in computing resources and deep learning algorithms have led to the rapid development of artificial intelligence and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Computers have defeated professional human players in tasks such as chess, quizzes, Go, and video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In the field of natural language understanding, automatic dialogue systems and question answering systems, such as Siri and Watson, have also emerged as well as more practical machine translation systems [8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, the current artificial intelligence system relies on largescale training data that lack basic common-sense knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' thus, a large gap remains between human and artificial intelligence in terms of learning and generalization abilities [11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, the human brain, as the only example of the realization of intelligence, has once again attracted the attention of computer scientists, and the development of brain-like intelligence by drawing on the neural and cognitive behavior mechanisms of the human brain has become a hotspot at the forefront of research worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Simultaneously, cognitive science, including cognitive neuroscience and cognitive psychology, has developed rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Noninvasive and real-time monitoring of language processing in the brain has become possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Through various sophisticated experimental designs, researchers in the field of cog- nitive science have made valuable discoveries regarding the neurological basis of language processing [14, 15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Traditional cognitive science experiments rely on strict experimental controls, lead- ing to significant deficiencies in terms of ecology and globality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, in recent years, an increasing 2 number of studies have begun to adopt high ecological paradigms and use advanced data-analysis methods to analyze the information-processing mechanism of the human brain under high ecological validity tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In terms of language research, researchers have gradually begun to use computational modeling methods to study the language understanding process of the human brain under experimental stimulation conditions of natural texts [19, 20, 21, 22, 23, 24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' With the rapid accumulation of interdisciplinary research in the fields of cognitive science and computer science in recent years, researchers have begun to summarize the changes that computer science methods, especially deep learning models, can contribute to research in the field of cognitive science [27, 28, 29, 30] and how the conclusions of cognitive science discoveries can help in building artificial intelligence models [31, 32, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The above work explored the possibility of combining the two fields at the macro level but did not discuss how to combine the two fields to carry out work on subdivided issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' To provide a reference for cognitive and computer scientists to conduct interdisciplinary research in the direction of language understanding, this paper summarizes and looks forward to the existing interdisciplinary research work on language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In summary, in terms of language comprehension, computational models can help cognitive sci- entists to quantitatively study and model the brain while understanding brain mechanisms can help computer scientists build smarter language and computation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, to promote a new round of development in human and machine language understanding research, it is imperative to conduct interdisciplinary research that combines cognitive and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The following sections first introduce the definitions, main research issues, research status, and research methods of human language cognition (Section 2) and machine language computation (Section 3) as well as the limitations of existing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' We then compare the main ideas and concepts of language cognition and language computation (Section 4) and analyze the similarities and differences between the two at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Section 5 summarizes the existing work on combining language cognition and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' On this basis, the limitations of the existing combination methods are analyzed, and feasible future research directions are proposed (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Finally, we present the main conclusions of this study (Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2 Research on language cognition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='1 Definition of language cognition The language cognition mentioned in this article refers to the human brain’s understanding of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Specifically, it refers to the process of extracting abstract symbolic information from auditory, visual, and other sensory information when an individual receives information, such as speech and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Language cognition is a complex process with varying structures and mechanisms of different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Moreover, the brain networks on which they depend are also very complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, in the process of speech comprehension, the auditory system must encode the basic acoustic features of speech and then follow multiple steps, such as vocabulary recognition, syntax construction, and semantic analysis, before finally realizing language comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='2 Main research questions Language is a complex sequence, and the human brain is a complex system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This makes it very challenging to study the language-processing mechanism of the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' On one hand, a language contains units of different sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' which language unit should be used as the starting point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Does the brain use a certain unit as the most important unit in language processing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' What about the core processing unit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This is the first research question presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' On the other hand, information processing in the brain is very complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Different types of information are processed by different brain areas in a certain order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Research on the related brain regions and time courses of language processing is summarized below in the second and third research questions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Ultimately, both the observation of the brain area and the processing timing are only phenomeno- logical descriptions of language processing in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' What cognitive and computational mechanisms underlie these phenomena?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This is the fourth research question introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Units and dimensions of language cognition 3 Linguists have defined many language units of different sizes and types such as phonemes, sylla- bles, morphemes, words, phrases, and sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' A key concern in the study of language cognition is whether these language units are merely concepts proposed by linguists for the convenience of research or truly processing units on which the brain relies for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In the normal process of language understanding, what type of language unit does the brain analyze?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Does the brain construct different neural representations for different types of language informa- tion (such as phonetics, grammar, and semantics)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These are the concerns of language cognition research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, Liberman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [35] believe that the phoneme is the basic unit of speech processing and that phoneme recognition is the function of the brain motor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, Greenberg [36] and Hickok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [37] hold that the syllable is the more central processing unit and that phoneme processing is possible only for certain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For another example, Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [38] believe that large phrases or even sentences are the basic units of semantic under- standing, but many connectionists think that words are the basic processing units of the brain and that phrase and sentence structures have little effect on the brain’s language processing [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Brain networks that localize different types of language information Language is a function of the brain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' but which parts of the brain are crucial for this function?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Neuroscience research has found that the brain can be divided structurally and functionally, and the earliest evidence for functional division of the brain comes from studies of aphasia, which will be introduced in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Aphasia research and modern neuroimaging research have found that language is not a single function but includes many functional modules [40, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, current language cognition research pays more attention to the brain network involved in locating specific functional modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Time course and control of language information processing What is the processing order of different modules for language understanding in the brain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, will the brain parse the grammatical structure first and then process the meaning [43]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' How long does it take to process each step?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Can different features in a vocabulary be identified for a long time [44]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Are the steps and sequences of brain processing language automatic and invariable, or do they require the influence and regulation of cognitive functions such as attention and working memory [45]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These are also the focus of research on language cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Neural coding and computational mechanism of language information Studies of brain regions and time courses have focused on describing language processing phe- nomenologically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' How do these phenomena arise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' From a computing point of view, what are the ”data structures” for computing in the brain, and what algorithms are used to operate these data structures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Research on processing mechanisms will inevitably involve mathematical mod- els, which also presents difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' At present, one type of model seeks to directly explain the neural response of the brain [46, 47, 48], and the other type simulates language behavior, such as by simulating the language-acquisition process [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='3 Development of language cognition Early research focused primarily on aphasia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These studies emerged around the middle of the nine- teenth century and mainly analyzed the relationship between brain damage and language behav- ior in patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In the past 50 years, the maturity of technologies, including electroencephalogram (EEG), magnetoencephalography (MEG), positron emission computed tomography (PET), and func- tional magnetic resonance imaging (fMRI), has provided powerful tools for studying the language function of the normal brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Studies based on behavioral or neuroscience experiments have achieved many results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Combined with the research questions in the previous section, we introduce four studies as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The first study focused on units of language comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' An extreme view is that sentences (or large phrases) form the basis for processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In this unit, the process of listening or reading a sentence is simply a process of acquiring information, and the information is processed and integrated when it reaches the boundary of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Another extreme perspective is that language processing is carried out in real time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' that is, the brain will process the current information at every moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Thus, information obtained at all times can be fully processed, and there is no need for centralized processing 4 at some important language boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Evidence for the first view is that language understanding is heavily dependent on context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' George Miller found that, in noisy environments, the same word can be better recognized if it is placed in a sentence [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' As another example, in the process of listening to an article, if the article is suddenly interrupted and the listener is asked to recall what he heard before, the listener can only accurately recall the vocabulary in the current sentence [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The second view is supported by considerable evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, if you play a voice and ask people to read along, some people can follow at a speed of approximately 300 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This means that the voice spoken to the reader is only approximately one word slower than the voice heard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In this case, if you heard the word ”tomorrane,” but, according to the contextual information, the word should mean ”tomorrow,” you would have a higher probability of saying ”tomorrow” instead of ”tomorrane.” However, when contextual information is lacking, the reader will not perform this type of correction [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In this case, the reader integrated the above information in real time rather than waiting to process it until the end of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Another study found that, if a person saw four objects on a screen (such as a horse, an apple, a table, and a newspaper), when they heard ”this kid is riding,” the person’s gaze would often fall to the ”horse object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This indicates that people’s language processing is predictive and that people instantly generate expectations based on the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Combining these two perspectives, one can argue that the brain performs both immediate predictive processing and additional integration at sentence or phrase boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The second study concerned the modules included in language processing and the corresponding regions or networks in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Early research on aphasia found that, when certain brain regions are damaged by trauma or disease, language function is impaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' More importantly, language is not a single function but a complex system of functions in different brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, patients with Broca’s aphasia cannot produce language but can understand it, and patients with Wernicke’s aphasia can speak language but cannot understand it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These two aphasias suggest that language production and comprehension are in separate brain areas and, therefore, can be selectively impaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' More detailed studies have found that some patients have impaired recognition of nouns but pre- served recognition of verbs while others have the opposite, suggesting that verbs and nouns are pro- cessed differently in the brain [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Similarly, some patients have an impaired ability to distinguish phonemes (such as being unable to distinguish /ba/ and /da/) but normal auditory word compre- hension while others have the opposite, which shows that phoneme discrimination and auditory word recognition involve different brain regions [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' All these phenomena indicate that language com- prehension involves many modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' thus, damaging specific brain areas affects only part of language function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Studies based on fMRI methods, which observe the activation of different brain regions in processing different information or performing different tasks in people with typical brain functioning, have also revealed this functional division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Moreover, they have found that different brain areas are activated by grammatical and semantic processing [55, 56, 57], and different brain regions are activated by different word categories (such as tools versus seeing animals) [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In general, aphasia studies have found that damage to some key brain areas can affect a certain function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' however, MRI studies have generally found that this function actually involves a more widespread brain network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, traditional aphasia studies generally hold that temporal lobe damage is more likely to cause noun comprehension problems, and frontal lobe damage is more likely to cause verb comprehension problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, recent MRI studies have shown that processing verbs and nouns involves very complex brain networks in which internal connection properties are related to the processing of two types of words [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The third study examined vocabulary recognition and processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Numerous studies have shown that word recognition is a parallel process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, when hearing the English syllable ”/kp/,” it is generally believed that the brain will activate all the words at the beginning of this syllable in parallel, such as cap, captain, and caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Among these words, those with a higher word frequency or that are more consistent with the context have stronger activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' How do psychologists conclude that many words are activated simultaneously?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Classic experiments used the cross-modal priming effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These experiments found that, after hearing ”/kp/,” people recognize visual presentation of words including ”captains” and ”captions” faster than after hearing other syllables (such as ”/da/”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Moreover, after hearing a word, vocabulary related to the semantics of the word is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, after hearing ”captains,” people recognize words like ”ships” more quickly [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' From the perspective of neuroscience, vocabulary induces an EEG response N400 with a latency 5 of approximately 400 ms, and the amplitude of N400 is closely related to word frequency and the previous context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, the amplitude of N400 induced by ”ship” depends on the preceding word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' If the preceding word is ”captain,” the amplitude of N400 induced by ”ship” is relatively small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' if the preceding word is an irrelevant word (such as ”apple), the amplitude of N400 induced by ship is relatively large [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Vocabulary recognition is generally believed to be a parallel process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' After hearing part of the vocabulary information, a large amount of vocabulary is activated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' however, as the information in- creases, the vocabulary that does not match the new information is suppressed until the brain finally determines a possible vocabulary [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The brain processes possible words in parallel because language is full of ambiguity, and information is presented very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' If you wait until all ambiguity is resolved before starting processing, not only may the reaction speed be too slow, but the information that has already exceeded the brains working memory capacity may be forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Similar problems are more common in sentence processing, such as the sentence ”The horse raced past the fence fell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Before seeing the last word, we will think that ”raced” is the predicate verb of the sentence, but after seeing the last word, we will find that the previous understanding was wrong;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' ”fell” is the predicate verb of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Sentences that contain ambiguity so that the analysis of sentence structure changes during the comprehension process are called ”garden path” sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' At present, some theories suggest that the human brain will also construct a variety of possible sentence structures at the same time and then continue to screen, but others suggest that the brain will first construct the most likely sentence structure and reanalyze if the structure is found to be wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The fourth study focuses on speech understanding in complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In the 1950s, British scientist Colin Cherry discovered that attention plays a crucial role in speech understanding in complex environments [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Cherry’s and subsequent studies found that, if two different speeches (speeches from different speakers or different spatial orientations) were played simultaneously in the experiment and the listener was asked to focus on one of the speeches, they could understand the speech that they paid attention to very well but could not recall the content of the other speech they did not pay attention to afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Psychologists also quantitatively analyze the impact of various factors on speech recognition, such as measuring the speech recognition rate under different noise intensities, and then draw the psychological curve of the speech recognition rate changing with noise intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' American scientist George Miller found that speech recognition rate is related not only to noise intensity and listener attention but also to the listener’s prior language knowledge [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In noisy environments, humans can recognize grammatical sentences better than random word strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Cognitive neuroscience experiments have further shown that both attention and prior knowledge can directly regulate the processing of the acoustic features of speech in the auditory cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Under this mediation, the neural activity of the auditory cortex mainly encodes the attended speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These studies demonstrate the important roles of attention and prior knowledge in language comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The research on the above four aspects shows that predictive processing, language structure pro- cessing, parallel processing, attention, and prior knowledge are all important characteristics of human language cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='4 Research methods Studies in cognitive and life sciences can be divided into hypothesis- and data-driven research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Ac- cordingly, linguistic studies can also be roughly divided into these two types of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Hypothesis-driven research Most language cognition research is hypothesis-driven;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' that is, researchers will clarify the hypoth- esis to be verified by the experiment (generally referred to as H1) and its opposite hypothesis (H0) before the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They will also clarify how the experimental results are consistent with the expectations of H1 or H0 unanimously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' If the final experimental result is consistent with the expectation of H1 and inconsistent with the expectation of H0, then H0 is falsified (that is, the findings support H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In contrast, if the final experimental result is consistent with the expectation of H0 and inconsistent with the expectation of H1, then H1 is proven false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, suppose researchers wanted to test the hypothesis that the auditory cortex encodes not only the acoustic features of speech but also phonemic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Based on this hypothesis, the researchers designed experiments to distinguish acoustic features from phonemic features and 6 then analyzed whether the latter affected the responses of the auditory cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Two specific examples below illustrate hypothesis-driven research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' As mentioned above, the unit of brain processing language is a controversial issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The hypothesis proposed by one study is (H1) that the brain can encode multiple levels of language units in parallel and that the neural activity encoding a language unit should be synchronized in time with the unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' that is, when a language unit appears, the neural activity also occurs, and when a language unit ends, so does the corresponding neural activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The counter hypothesis (H0) is that the brain processes only according to a single level (such as words) or that different levels of language units do not show neural responses synchronized with language units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' If this assumption holds, then the update rates of neural activity encoding language units of different sizes, such as syllables, words, phrases, and sentences, will be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For instance, if there are four syllables in speech per second, the response to the syllables will also change four times per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' If every two syllables are combined into a phrase and every two phrases form a sentence, then the neural response of words and phrases should be updated two times per second, and the neural response of sentences should be updated one time per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, the researchers designed the experiment according to the above ideas and found that the MEG/EEG response of the human brain in the process of listening to speech does contain 4 Hz, 2 Hz, and 1 Hz components, corresponding to the neural responses of hypothetical syllables, phrases, and sentences [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This experimental result supports H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Another example is the study of word meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The semantics of a word can be learned in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' One way is to directly establish the relationship between the word and the objective entity it refers to, such as seeing the fruit ”guava” and being told it is a ”guava.” This learning method directly establishes the relationship between the language symbol ”guava” and the sen- sory characteristics (visual, taste, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=') of the object it refers to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Another way is to describe the meaning of the acquired vocabulary through words, such as reading in the dictionary, ”Guava is a plant of the myrtle family, and the fruit is edible.” The hypothesis (H1) here is that the semantics acquired through the above two methods are encoded in different regions of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The opposite hypothesis (H0) is that the representation of a word in the brain is independent of the acquisition pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' To distinguish between these two modes of acquisition, the study compared the processing of colors by sighted people and congenitally blind people;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' both groups can acquire color concepts through language, but only sighted people can directly establish color words and visual correspondence between color information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The study found that some brain regions encode color in the same way in both groups, but others encode color in a way that is only present in sighted people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This result also supports H1 hypothesis [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Data-driven research Hypothesis-driven research is often highly targeted research–that is, experiments specifically designed to test a particular hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The opposite of hypothesis-driven research is data- driven research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Data-driven research is exploratory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' it does not put forward a hypothesis first but explores possible results by collecting experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The purpose of hypothesis-driven research experiments is clear, so it is easier to obtain stable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, to verify the hypothesis that neural activity and hierarchical language structure are synchronized in the first experiment above, a constant rate was used to play speech to simplify data analysis (the researchers only needed to analyze the response of a specific frequency in the frequency spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Moreover, to distinguish the two methods of acquiring word meaning in the second experiment, two groups of people were selected for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, it is often difficult to strictly distinguish between hypothesis- and data-driven research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Without exploratory research, it is difficult to formulate a hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' without a hypothesis, it is difficult to determine which aspect of the data to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, the above two hypothesis- driven studies also contain data-driven components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The first study did not determine which MEG/EEG channels can obtain responses, and the second did not assume in advance the brain region in which the experimental phenomenon would be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='5 Limitations of existing research Language cognition research initially revealed some patterns of human language understanding, but much more is needed to truly analyze the mechanism of language understanding in the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Currently, the main problem is that, in theory, the existing research focuses on the qualitative expla- nation of local problems and relies on small samples and strict experimental control, which leads to a lack of ecological and global research conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The overview is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Lack of discussion on the quantitative mechanism Most cognitive science research is described at the phenomenon level, and even the discussion of the mechanism is often qualitative and subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, as mentioned before, when the brain processes a word, it produces an EEG response of N400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Many of studies have investigated this response, clarifying how the previous contextual information of various properties affects the magnitude of the N400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, these studies only show that the previous contextual information can affect the N400 response and do not answer what computational mechanism this effect reflects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Cognitive science literature discusses multiple mechanisms for the generation of N400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' One hypothesis is that N400 represents the current word that can be predicted by the brain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' that is, words that can be predicted produce smaller N400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Another hypothesis suggests that N400 represents the ease of integration of a current word with previous context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' that is, N400 is smaller if it is easy to integrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, these hypotheses are qualitative language descriptions, and it is not clear how the brain’s prediction and integration reflect the computational mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Targeting specific linguistic phenomena Cognitive science experiments often use strictly controlled experimental designs to study specific, even very detailed language phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Due to the strict control of experimental variables, the corpus in the experiment tends to be consistent, so the experimental conclusions are likely to be applicable only to the highly consistent corpus involved in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Poeppel and Embick [67] identified a mismatch of research scales between linguistics and neuroscience research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Linguistics is often concerned with very fine-grained issues (such as the usage of a word and how the syntactic structure of a sentence should be divided) while neuroscience is concerned with relatively macro issues (such as which part of the brain processes grammar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, even neurolinguistics studies often only use a relatively consistent and typical corpus for research, so the universality of the conclusions is not strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Research conclusions are difficult to integrate Closely related to the previous study, tightly controlled experiments lead to fragmentation of research with one study only concerned with one particular linguistic phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' If each study focuses on one linguistic phenomenon and language contains a limited number of linguistic phe- nomena, then an overall conclusion can be drawn by integrating different local studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, because the language is too complex to be uniformly divided into several basic phenomena and the experimental methods are too diverse, it is very difficult to integrate various research conclu- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For instance, cognitive experimental studies have found that different types of language materials can activate different brain regions and induce different types of EEG responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' How- ever, if these studies are combined, can they tell us how the brain understands even a simple sentence step-by-step?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In fact, they cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3 Research on language computation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='1 Definition of language computation The language computation mentioned in this article refers to the process of machine understanding of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Taking Chinese as an example, the process of language computation includes the recogni- tion and representation of characters, the structure and semantic analysis of texts (including words, phrases, sentences, and discourses), and the analysis of the association between text symbols and the external world and finally achieves the goal of enabling machines to understand language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The 8 language computation we refer to below can be compared with the basic research problems in nat- ural language processing (also called natural language analysis), such as lexical, syntactic, semantic and discourse analysis, knowledge representation, and computing, and does not involve application technology research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='2 Main research questions To make a machine understand natural language, we must first encode the information in a language into a form that can be processed by the computer, which is called the text representation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' To further analyze the information in a text, it is necessary to analyze its structure and semantic information–that is, to perform structural analysis and semantic analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Thus far, the machine has known the relationship between different language symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' It is necessary to further associate language symbols with the external world and knowledge to understand natural languages like humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The main research questions are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Text representation method Language is composed of small elements hierarchically and recursively, which in turn form words, phrases, sentences, and discourses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In language communication, word is the most basic semantic unit, and the combination of words needs to be based on specific rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These limited rules can combine different concepts to construct endless text units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' How to represent lexical semantics, such as by using symbols [68], functions [69], vectors [70], and tensors [71]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' How to build efficient lexical semantics learning methods[72, 73, 74]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' How to combine lexical meaning to form the meaning of larger-grained text units [75, 76, 77, 78]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These are key issues in linguistic computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Structural Analysis Methods Structural analysis is generally divided into syntactic structure analysis and discourse structure analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Among them, syntactic structure analysis studies the combination and depen- dence relationship between words in a sentence, and discourse structure analysis studies the combination and dependence relationship between sentences in a paragraph or discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These two types of analysis can resolve the ambiguity of the structure in the input text, analyze the in- ternal structure of the input text, and provide structural information for the semantic analysis of the text, which is considered to be an important part of language understanding [79, 80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The main research issues in this direction include how to design or select formal rules for grammars and how to design automatic analysis algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Representative of this kind of work are the rule-based phrase structure analysis method proposed by Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [82] and the dependency structure analysis method based on neural networks proposed by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Semantic analysis method For different language units, the task of semantic analysis is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For word, semantic analysis focuses on how to disambiguate the meaning of words and how to identify the semantic relationship between words (including antonyms, synonyms, part-whole and event relations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' for sentences, semantic analysis includes semantic role labeling, semantic parsing, calculation of semantic similarity between texts, and identification of implication relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' for discourses, semantic analysis includes how to resolve references and identify inter sentence relations in texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The identification and calculation of the above-mentioned semantic information and semantic relationship is the basis for understanding the meaning of a text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' It is also a difficult problem in language computation to build an efficient semantic analysis method [84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Knowledge representation and symbol association method Knowledge in this paper refers to world knowledge, historical knowledge, commonsense knowl- edge, and professional knowledge of various disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Knowledge representation is a description of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The current model represents knowledge in the form of symbols [86] or distributed vectors [87] and realizes the association between language symbols and knowledge through re- trieval or mapping to a unified representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Among them, how to design the encoding form of knowledge and automatically learn the representation of knowledge is the key to this type of research [88, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition, no matter it is a human or a machine, to understand the meaning 9 encoded in language symbols, it is necessary to associate it with world knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Otherwise, as the ”Chinese room” described, the people in the room do not know Chinese and cannot truly un- derstand the received Chinese information, but he can make Chinese native speakers think that he can speak Chinese fluently, creating an intelligent impression [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, how to associate language symbols and knowledge is also the core issue of language computation research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='3 Developments of language cognition Language is a serialized and structured symbolic expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' How to represent the meaning of text and automatically analyze its semantics and structure is a crucial step in research on language computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Moreover, this has always been a major challenge in machine language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Almost all natural language processing tasks, such as machine translation, question answering, and dialogue systems, rely on semantic representation and computation of input language sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Amidst the decades of the development of natural language processing, text-representation methods have undergone a systematic transformation from discrete symbol representation to continuous vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' With discrete symbol representation, words are regarded as discrete symbols, and each word can be expressed as a one-hot vector whose dimensions are equal to the size of the vocabulary, where one dimension is 1 and the other dimensions are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In this representation system, sentences and discourses are usually represented by a bag-of-words model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In 1954, Harris proposed the concept of a bag of words in the article ”Distributional Structure.” In the following decades, the bag of words model has been the mainstream model of text representation [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This text representation method, based on discrete symbols, can only use string matching to extract features and calculate the similarity between language units, which easily leads to data sparsity problems and cannot capture the semantic similarity between words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' On the other hand, distributed continuous vector representation is convenient for semantic cal- culation and measurement and can theoretically solve the problem of semantic gaps between words, sentences, and discourses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Harris and Firth proposed and clarified the distributed hypothesis of words in 1954 and 1957, respectively, in which the semantics of a word are determined by its context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' that is, words with similar contexts have similar semantics [91, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Matrix decomposition and neural networks are the two main models for learning the distributed vector representations of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Among these, neural networks have been the mainstream model for learning distributed vector representations in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In 2003, Yoshua Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [93] proposed a neural network language model that uses a low- dimensional continuous real number vector to represent each word and learns an n-gram grammar model based on this, marking the beginning of distributed text representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Tomas Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [70] proposed the Word2Vec method, including two models of CBOW and Skip-gram in 2013, which greatly simplifies the distributed vector learning method of words so that it can make full use of massive unlabeled text data to learn words efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In 2017, the transformer model proposed by Google [94] combined the semantics of vocabulary more efficiently through pairwise calculations between words to obtain a semantic representation of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Since then, largescale pre-training models based on transformer architectures, such as BERT [95], TransformerXL [96], GPT3 [97], PaLM [98], and ChatGPT 1, have been developed for various language processing applications, further establishing the dominance of distributed vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The distributed text representation model greatly facilitates the representation and calculation of natural language, thus becoming the cornerstone of deep learning applied to natural language processing tasks and further promoting the breakthrough development of applications such as text understanding and machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, existing methods lack the modeling of fine-grained text semantics and structural information and cannot effectively deal with linguistic phenomena such as lexical ambiguity, antonyms, extended meanings, and structural ambiguities of sentences and texts [99, 100, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' On the other hand, semantic, syntactic, and discourse analysis methods study how to represent the meaning, combination, and dependency of language units and provide structural information for text representation, which may be helpful for existing text representation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Semantic parsing is the most representative task in semantic analysis, which studies how to parse natural language into a complete semantic representation (such as logical expressions) that can be recognized or calculated by machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Specifically, it includes how to obtain the semantics of words in 1https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='com/blog/chatgpt/ 10 a sentence, the semantic relationship between words, and how to combine word representations into sentence semantic representations [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Similar to most natural language processing tasks, semantic analysis has undergone a shift from rule-based, statistical, and neural-network approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Although the performance of the neural-network-based semantic analysis model has made great progress, this method still strongly relies on a large amount of manually labeled data and cannot truly realize semantic understanding by learning statistical laws for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition to the abovementioned semantic analysis, understanding the meaning of a text is inseparable from the analysis of its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' At the sentence level, structural analysis refers to the analysis of the input word sequence in the syntactic structure of a grammatical sentence, which is mainly divided into phrase and dependency structure analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Phrase structure analysis analyzes sentences in a hierarchical phrase structure tree based on phrase structure grammar, which is part of the theory of language transformation and generation created by Chomsky [103] in 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Dependency structure grammar is mainly used to describe the semantic dependencies between words and was first proposed by Tesniere in 1959 [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' At the discourse level, structural analysis studies the combination and dependency relationship between sentences in paragraphs or discourses with the goal of analyzing discourse structure as a whole to understand discourse meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Computational linguists generally believe that syntax and discourse analyses are necessary to realize language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Only by analyzing the structure of the text correctly can we understand its meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, the results of various language processing tasks in recent years have shown that explicitly modeling text structures does not significantly improve model performance [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This leads researchers to rethink the role of text structure analysis, whether existing language models can implicitly learn certain syntactic rules, and whether artificially defined syntactic structures are necessary for machine-language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition to expressing and analyzing shallow meaning units such as words, phrases, and sen- tences, language understanding also requires higher-level expressive abilities, such as the connection with ”knowledge,” the ability to perceive context, and reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Traditional knowledge representa- tion methods use a computational framework based on logical symbols, such as a logical reasoning language based on first-order predicate knowledge representation represented by Prolog and a prob- abilistic graphical model represented by a Markov logic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These methods are good at precise reasoning and computation but lack generalization and approximate semantic computation capabilities under incomplete knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In recent years, with the development of deep learning, knowledge repre- sentation has gradually adopted neural-network-based methods to learn the representation of entities and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This method based on representation learning has better generalization performance and is more conducive to the widespread uncertainty calculation problems in information processing, but it cannot perform precise semantic calculations and reasoning [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, the combination of precise semantic computing based on symbolic logic and approximate semantic computing based on representation learning is the future research trend of knowledge representation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='4 Research methods Consistent with most research directions in the field of computer science, natural language methods are mainly divided into two camps: rationalist and empiricist (or a data-driven approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The rule- based approach is representative of the rationalist approach, which holds that a large part of people’s language knowledge is innate and determined by genetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Data-driven approaches include statistical machine learning methods and neural network methods (or deep learning methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This method assumes that the complex language structure of humans can be learned through acquired training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Rule-based approach Since natural language is essentially a symbolic system produced by human society due to the need for communication, its rules and reasoning features are distinctive, so the early research on natural language processing first adopted the rule method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This method obtains the understand- ing of people’s language ability through the study of some representative sentences or language phenomena and summarizes the laws of language use to analyze and infer the expected results of the test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The rationale for rule-based approaches is presented here using the application of context-free grammars to the problem of syntactic parsing as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, we have the following context-free grammar: 11 G = (Vn, Vt, S, P) Vn = S, NP, V P, N, V, Det Vt = one, kid, chases, a, cat S = S P : S → NP V P, NP → Det N, V P → V NP, Det → one, Det → a, N → kid, N → cat, V → chases Based on the above rules, the syntax tree can be constructed by the top-down or bottom-up analysis method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Here, we use the top-down analysis method as an example for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' First, start from the initial symbol S, search from top to bottom, select the applicable rule in the grammar P to replace the search target, match the word in the sentence with the right part of the grammar rule, and erase the word if the match is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Then, the search continues on the remaining part of the input sentence until the end of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Specifically, in the above example, the search steps are: (a) Select rule S → NP V P, no matching word, left sentence ”one kid chases a cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' (b) Select rule NP → Det N, no matching word, left sentence ”one kid chases a cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' (c) Select rule Det → one, matching word ”one”, left sentence ”kid chases a cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' (d) Select rule N → kid, matching word ”kid”, left sentence ”chases a cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' (e) Select rule V P → V NP, no matching word, left sentence ”chases a cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' (f) Select rule V → chases, matching word ”chases”, left sentence ”a cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' (g) Select rule NP → Det N, no matching word ”a cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' (h) Select rule Det → a, matching word ”a”, left sentence ”cat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' (i) Select rule N → cat, matching word ”cat”, no left sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' According to such a search process, we can express the syntax tree of the sentence ”a child chasing a cat”, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' S NP VP Det N NP V Det N one kid chases a cat Figure 2: Example of a syntax tree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Data-driven approach Human language is not a formal language for rules and patterns often exist implicitly in the language, making the rules difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition, the complexity of natural language makes it difficult for rules to cover all linguistic phenomena without conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The data-driven method saves the burden of manually compiling rules and automatically generates features or evaluates the weight of features in model generation, which has good robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Generally, data-driven methods include statistical methods and neural network methods, and the following two methods are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Statistical-based language computation methods use large-scale language data and often need artificial help (labeling data and screening features, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They use statistical methods to discover the law of language use and its probability which are then be used to calculate the possible outputs of testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Here, we take the language model as an example to introduce the basic principles of statistical methods in which the representative one is the n-gram model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 12 The goal of the language model is to calculate the probability of a string appearing as a sentence, that is, to calculate the product of the probabilities of each word in the sentence s in different historical situations: P(s) = p(w1)×p(w2|w1)×p(w3|w1w2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='p(wl|w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='wl−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Among them, the probability of generating the i-th word wi is determined by the generated i−1 word w1w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='wi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' When the above method is simplified and only the previous n − 1 words are used to predict the next vocabulary, it is an n-gram model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, the bigram model calculates the probability p(a child chasing a cat) = p(one|bos)×p(child|one)×p(chasing|kid)×p(one| (cat|one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Among them, bosis the start character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Generally, maximum likelihood estimation is adopted to calculate the conditional probability of P(wi|wi−1) = c(wi|wi−1) � wi c(wi, wi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Methods based on neural networks mainly study how to design task-related neural network struc- tures and optimize neural network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Here, we take the text representation problem as an example and use the recurrent neural network model as an example to introduce the ba- sic principles of neural network-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' A recurrent neural network is a kind of neural network that is good at processing sequence information and long-distance dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, the corpus has the following content: ”A child chases a cat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='..”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' To represent the above text, the sentences in the text need to be sequentially input into the cyclic neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' As shown in Figure 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' the cyclic neural network model processes each vocabulary in the sentence in turn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' encodes the corresponding vocabulary wt into a real-valued vector xt at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' and com- bines it with the hidden layer vector ht−1 generated at the previous time to obtain the hidden layer vector at this moment: ht := σ(Wxhxt + Whhht−1 + b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' one kid chases a cat one kid chases a cat h0 h1 x1 h2 x2 h3 x3 h4 x4 h5 x5 x0 Figure 3: Example of a recurrent neural network Among them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' the symbol ”:=” means ”defined as”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' σ(z) = 1/(1 + exp(−z));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Wxh, Whh and b are the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The objective function of the model is to maximize the probability of predicting the following vocabulary above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The commonly used objective function is to minimize the cross-entropy loss of the model: Loss= − � wtlogyt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' wt is the real following vocabulary, and yt is the following vocabulary predicted by the hidden layer vector ht : yt = Softmax(Vhyht +c), where Vhy and c are model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The above objective function is used to learn the parameters of the model on large-scale unlabeled texts so that the model can predict the following words through the above words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In this process, the language model obtains the representation vector of the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' At the same time, the representation vector of the sentence can be the hidden layer vector obtained at the last moment of the model or the maximum or average pooling result of all hidden layer vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='5 Limitations of existing research Existing language-computation models are far from being able to understand language similar to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The main problem currently faced is that the model structure lacks a theoretical basis and parameter training relies on large-scale computing resources, which can be summarized as the following four points: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Single textual representations Existing text representation models do not distinguish between different types of information (such as vocabulary of different parts of speech, text units of different granularities) and uniformly 13 encode them into dense vectors of the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This encoding method is very efficient when constructing a neural network model, but it ignores the size of different types of texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, to encode all information, it is necessary to uniformly use the encoding method with the largest amount of information for all types of texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This method adopts larger numbers of parameters instead of designing model structures and contains a large number of redundant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Lack of interpretability Although the deep learning method has greatly improved the performance of numerous tasks in natural language processing, the meaning of the vector dimension cannot be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' There- fore, it is impossible to analyze what operation each unit in the network performs on the language input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Moreover, it is impossible to give the reason the model obtained the wrong result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This makes the results unreliable and hinders the design of the model structure and further improve- ment of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In contrast, if the model is interpretable and can give the reason for reaching a certain conclusion like a human, then the model structure can be improved in a targeted manner, thereby improving the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, when judging whether two sentences express the same meaning, the model can give a yes or no conclusion and, at the same time, give which text fragments in the two sentences significantly affect this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These can be used as the basis for judging the rationality of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Lack of ability to learn independently Existing methods construct training datasets for each task, adopt the ”training-test” development method, and cannot directly use new training samples to correct the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Additionally, the model trained in a certain task is difficult to apply to other middle tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Different language tasks require the model to be capable of language representation and understanding, so new tasks can use the already-trained model to learn new task-specific information on this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Analogous to the self-evolving learning ability of humans–that is, learning from simple tasks and continuously learning new tasks and correcting existing knowledge on this basis–an intelligent system should also be capable of continuous learning to achieve the self-evolution of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Rely on largescale single modality training data Existing language-computation models rely heavily on the quality and scale of training samples, making it hard to handle those words, language structure, and language expressions that have not appeared in the training corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In contrast, when humans learn language concepts, they often obtain information from multiple modal samples and can learn a new word or language expression with only a few samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, when learning the concept of ”giraffe,” we might understand from a text that ”it is the tallest mammal in the world, and its distinctive features are long neck” and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' When viewing a picture of an animal with a ”long neck, long legs, body” and ”a spotted pattern,” we can quickly recognize that it is a ”giraffe” even if we have never seen one before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This shows that multiple-modal data complement and verify each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Thus, it is imperative to develop small-sample learning algorithms by comprehensively utilizing multiple-modal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 4 Comparison between language cognition and language com- putation The above chapters demonstrate that the processes of human and machine language understanding defined in the fields of language cognition and language computation are similar–that is, starting from the encoding of basic language units, then combining them into larger-grained text units, and finally connecting to the outside world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, the experimental methods and focus of the research questions in the two fields are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Language cognition research primarily adopts the following experimental method: ”1) proposing hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2) designing experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3) hypothesis verification.” Language computation research pri- marily adopts the method of ”1) collecting data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2) building model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3) model performance verification.” 14 In terms of research issues, language cognition research tends to reveal the reasons behind problems whereas language computation research focuses on methods to solve these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Table 1 sum- marizes and contrasts the main ideas and concepts of the two fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Language cognition studies the basic units and dimensions of language understanding, its spatial representation and time course in the brain, and the regulation of world knowledge and cognitive functions on language understanding, emphasizing ”what” and ”why” questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Language computation studies how to allow computers to efficiently segment and represent vocabulary, combine vocabulary representations into sentences and discourse representations, and analyze the functions and relationships of each element in a text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The main concern is the question of ”how to do it.” Table 1: Main concepts in language cognition and language computation areas Language cognition Language computation Language processing unit Language is organized according to different levels of structure (such as morphemes, words, phrases, sentences, discourses, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' ), with different levels of information (phonetic, grammatical, semantic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=') Word is generally used as the basic unit, and they are integrated through language models such as neural net- works Word representation The neurons or neural networks related to the word are activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' It is gener- ally believed that the neural represen- tations of multiple words can be acti- vated at the same time Encoding words into a form that can be processed by computers (such as symbols, vectors, matrices, graphs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=') Word form analysis Morphology information encoding Remove the affix to get the root (plays → play) or transform the complex form of the word into the most basic form (are → be) Information integration The process of assigning syntactic structure and semantics to the corre- sponding input words Analyze the semantic and syntactic in- formation and their relationships,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' then combine word representations to form the representation of sentences and dis- courses through a composition func- tion Multimodal information World knowledge (common sense) and language understanding scenarios can affect language processing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' which may involve the interaction of brain lan- guage networks with sensory and mo- tor networks Associate language symbols with other modality information Task effect Language comprehension is regulated by cognitive functions — if cognitive functions such as attention and work- ing memory are adjusted through ex- perimental tasks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' cognitive neural pro- cessing of language will be affected The parameters of the computational language model are determined by the objective function — once the objec- tive function of the task changes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' the parameters of the computational lan- guage model will change accordingly These differences carry substantial challenges and opportunities to the combination of language cognition and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' One challenge is that studies on linguistic cognition and computation are in separate fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Research on language cognition has mainly focused on the cognitive laws behind human language and its neural basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Owing to the limitations of data collection and analysis methods, such research still focuses on qualitative analysis and exploration of macroscopic laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' It lacks comprehensive and quantitative laws and models that can be applied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' An example is the representation of lexical semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Although many semantic dimensions have been discovered in the field of language cognition, their general neural connections have been revealed, and some theoretical models have been initially established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, owing to technical limitations, it remains impossible to accurately record the neural activations of a single word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, it is difficult to reveal the underlying neural representation rules and calculation methods of semantic information directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Similarly, computational linguists have focused on the effectiveness of practical applications when choosing research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They often ignore the most essential laws of language;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' they only consider the performance in downstream tasks when constructing a language-computation model and ignore interpretability and human-like features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, as a ”black box” that lacks human character- 15 istics, the language-computation model is difficult to use in the modeling of human brain language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' To illustrate these challenges further, let us consider the word representation problem as an exam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Regarding the problem of lexical (concept) representation in the human brain, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [107] examined the representation of abstract words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' By constructing two different types of lexical repre- sentation (statistical co-occurrence- and semantic feature-based), they tested two classical cognitive theories of abstract concept representations in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These two theories highlight that abstract word representations are expressed in linguistic symbols through contextual associations or semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The experimental results show that both types of lexical representation have significant ef- fects on the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The difference is that corpus-based abstract lexical representations are associated with brain areas responsible for advanced language processing whereas semantic feature-based abstract lexical representations present distributed representational features that are associated with multiple brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, the researchers concluded that the representation of abstract words in the brain is divided into two modes and is responsible for different brain networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This study revealed that the semantic dimension of abstract vocabulary encoded by the brain may include both linguistic and semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This is very important for further analysis of the brain’s understanding of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, this analysis is too macroscopic to directly guide the construction of computational models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Computer scientists want to understand the specific semantic dimensions and forms of encoding that the brain needs to encode lexical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, which neurons in the brain encode the concept of ”knowledge”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' What information do these neurons represent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' What is the relationship between these neurons?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' What are the rules for their connection?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The answers to these questions can directly inspire the development of new lexical representation model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In contrast, when the language-computation model encodes the meaning of words, it represents the words as dimension-agnostic real-valued vectors and encodes the information of the words through the relationship between the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The purpose is to allow computers to process language symbols efficiently to complete various language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Although this uninterpretable representation method is crucial for computational models to complete language tasks, it does not reveal the laws of language itself or directly shed light on how the brain encodes lexical concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' What cognitive scientists want to understand is which features can be used to construct semantic vectors to explain human behavioral data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' What information is encoded in the computational model, and what kind of operation is performed to lead to its excellent performance in downstream tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Can the calculation process explain the human language-processing mechanism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Which calculation module in the calculation model is essential for language modeling, and is there a corresponding processing module in the human brain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In summary, the fields of language cognition and language computation differ in their research contents and ways of thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, we believe that such differences can bring new insights into both fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, in the process of language comprehension, the human brain not only com- bines words into sentences and discourse from the bottom up but also uses cognitive functions such as attention and working memory to regulate the language process from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In contrast, language computational modeling is a static process unaffected by the external environment and com- putes a fixed encoding result for a piece of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, if we can learn from the dynamic encoding mechanism of the human brain and integrate modules such as human brain memory and attention to construct a new language-computation model, the model may acquire more general knowledge and be easier to transfer to other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' As another example, experiments have shown that, in the process of machine language under- standing, addition is a very effective combination when integrating vocabulary representations into phrase representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This automatic learning of language combination rules from data may provide new ideas for the study of the word combination process in the human brain using the underlying calculation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Additional ideas on combining the two are introduced in Sections 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 5 Convergence of language cognition and language computa- tion Recent years have seen increasing attention to cross-disciplinary research in the fields of cognitive and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The following section introduces related work in the fields of language cognition 16 and computing that inspire and merge with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='1 Language cognition experiments using language computation methods In recent years, an increasing number of researchers have begun using language computation meth- ods to study the process of understanding human language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This method shows great potential for studying brain representations at the single-word level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Furthermore, it can be used to analyze both traditional experimental data and natural language-processing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Specifically, this type of method collects neural activity data of words, sentences, or chapters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' uses language-computation models to encode experimental stimuli;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' and uses the encoded stimuli to study the problem of brain language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Such methods typically work as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [46] published an article in ”Science” in 2008 regarding the issue of how the brain represents conceptual semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They found that fMRI data of people reading nouns can be modeled using the statistical laws of certain action words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Specifically, they collected fMRI data when research participants read 60 noun stimuli (pictures + lexical texts) and calculated the representation vectors of these 60 nouns by using their co-occurrence with 25 sensory-motor-related representative verbs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=', ”see,” ”listen, ”speak,” ”eat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' These representation vectors were then trained to predict fMRI data using a leave-two-out cross-validation method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' each cross-validation predicted fMRI data for two test words and compared them with the real fMRI data as test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The regression model had a significantly higher classification accuracy than the random value for brain activation patterns evoked by nouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This suggests that there is a predictable relationship between word representations and fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Moreover, it provides evidence that the brain represents the semantics of nouns that are significantly dependent on sensorimotor properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This work has changed the previous experimental paradigm of examining concept representation only through the comparison between semantic categories, opened a new data-driven method for studying a single lexical concept, and inspired many subsequent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Another representative work is an article published in ”Nature” by Huth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [47] in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They used language-computation models to comprehensively study how different semantic information is encoded in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Specifically, they collected fMRI data when the participants listened to more than 2 h of narrative stories (including a total of 10,470 different words) and selected 985 basic words describing different topics in the corpus as different semantic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Then, they constructed a 985-dimensional vector for each word in the stimuli by counting their co-occurrence with the 985 basic words in a large-scale corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Next, they trained a ridge regression model such that 985-dimensional word vectors predicted each voxel in the fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Among them, the parameter matrix with the size of ”985 number of voxels” obtained in the model was the brain representation of 985 semantic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The results showed that the brain semantic representations were very similar across participants, and different semantic features were encoded in specific brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In contrast to the study by Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' on the representation of 25 sensorimotor attributes in the brain, Huth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' comprehensively studied the representation mode of 985 semantic features in different voxels of the whole brain for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This once again proved the usefulness of the language-computation model in studying the brain’s language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Regarding the brain’s computational mechanism for language processing, Brennan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [108] studied whether the brain uses a linear or hierarchical structure to process sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They first collected fMRI data while participants listened to the first chapter of ”Alice in Wonderland.” They then used a series of grammatical models (including linear and hierarchical grammar models) to calculate the syntactic complexity of each word in the stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Finally, regression analysis was performed between the syntax complexity index and the fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Syntactic complexity can be calculated from the probability of words appearing in a certain gram- matical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The hypothesis is that, if people use hierarchical structures to comprehend stories, then the complexity indicators calculated using hierarchical grammar should have stronger correlations with the fMRI data compared to the linear grammar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The results show that the linear effect is widely distributed in the language network of the brain while only a specific area of the left temporal lobe of the brain is responsible for processing hierarchical structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This suggests that the temporal lobe processes information in a hierarchical structure when comprehending languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Research on language cognition also draws on the operating mechanisms of language-computation models to propose hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' A representative example of this type of work is that of Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 17 [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They used a cognitive model and a neural network model to study the neural mechanism of the brain when understanding pronouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Different languages have different expressions of pronouns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' for example, the pronunciation of Chinese pronouns is gender-neutral (”ta”) while English pronouns are pronounced differently depending on gender (”she”, ”he”, ”it”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Thus, to explore whether the human brain adopts a general parsing strategy that is not affected by language, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' collected fMRI data from Chinese and English native speakers while listening to the full text of ”The Little Prince” in Chinese or English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' A generalized linear model, commonly used in cognitive science, was used to calculate the brain regions related to pronoun resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Both Chinese and English listening materials significantly activated the left anterior middle temporal gyrus, left posterior middle temporal gyrus, and anterior and angular gyrus brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' To further explore the computing mechanism of the brain when parsing the relationship of reference, the researchers first constructed five computing models for pronoun reference resolution: the Hobbs model based on syntactic theory, the Centering model based on discourse theory, the ACT model based on memory theory, and the pronoun resolution model of ELMo and BERT based on neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Next, they calculated the reference probabilities of each pronoun using the above models and correlated them with fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Only the ACT-R model based on memory theory could significantly predict neural activation data corresponding to the Chinese and English experimental materials, indicating that the brain adopts a language-independent general memory retrieval strategy when parsing the pronoun reference relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Another examples is Wehbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [110], who proposed an analogy between the recurrent neural network language model (RNNLM) and the working mechanism of the reading brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They found that the way the human brain works when reading a story is somewhat similar to how RNNLMs work when processing sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Additionally, Schrimpf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [111] compared the association of 43 state-of-the-art neural network models with various neural activity datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They found that the model based on the language model and the transformer network structure can significantly predict the neural response, behavioral data, and neural response of the next word, indicating that the language system of the brain is optimized for predictive processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' See more recent work at survey [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='2 Language computation methods inspired by language cognition The deep-learning method based on neural networks has been highly praised in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In a sense, it simulates the cognitive function of the biological brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, this method is not a mathematical model based on the working mechanism of the brain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' thus, it is difficult to eliminate its dependence on largescale training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' A large gap remains between language-computation models and human in- telligence in terms of generalization and learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The language-computation model inspired by the cognitive mechanism proposed in this article aims to study the language cognitive mechanism of the brain, analyze the relationship between the cognitive mechanism and machine language computation, and design a more intelligent language-computation model to complete various language-processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Since the current research on the mechanism of language understanding in the brain is far less in-depth than other cognitive functions, most computational methods inspired by cognition are con- centrated in the fields of visual cognition and machine learning, and less work has been done in the field of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In this paper, existing cognitive-inspired language-computation methods are summarized into the following four categories: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Cognitive function-inspired models To improve model performance when processing downstream tasks, we could borrow ideas of cognitive mechanisms such as brain representation, learning, attention, and memory to build new or improve existing computational models so that (part of) the model has a structure similar to the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, inspired by humans selectively looking at or skipping certain words when reading sentences, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [113] proposed a sentence-representation model inspired by the human attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This method utilizes the predictors of eye-movement signals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=', lexical surprisal and part-of-speech labels) to build attention modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' It introduces their results as weights into the sentence representation learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The results show that the attention module assigns higher attention weights to important words, and the weight results are signifi- 18 cantly correlated with human reading time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition, the attention module can significantly improve the performance of sentence representation on several downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [114] used the human attention mechanism to improve the performance of image description generation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [115] proposed a small data word representation learning method based on memory enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Finally, Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [116] proposed a continuous learning approach based on episodic memory activation and memory consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Cognitive data-enhanced models We can use brain neural activity, neuroimaging, or behavioral data as an additional modality, which can provide different information than the existing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, fusing these two data during model training could improve model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, Klerke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [117] proposed a multitask learning approach to incorporate eye- tracking data into a sentence-compression task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They utilized a three-layer bidirectional recurrent network model with the bottom layer predicting eye-movement timing and the top layer predict- ing sentence compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The results show that this multitask learning method can effectively introduce eye-movement data into the sentence-compression task and improve the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Tiwalayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [118] fused the probability distribution of the next word predicted by humans with that of the next word predicted by the language model, which effectively improved the performance of the language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition, Malmaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [119] introduced predicted eye-movement time into reading-comprehension tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Barrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [120] added eye-movement data as a feature to part-of-speech tagging and named entity recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [121] applied eye-tracking data to improve the quality of sentiment-analysis models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Additionally, Fereidoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [122] introduced fMRI data into vocabulary representation learning, and Roller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [123] and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [124] introduced human behavior data (lexical association score) into multimodal vocabulary representation learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Build models by simulating neurons Another way to build a more intelligent computing model is to simulate the structure and working mechanism of biological neurons or neural circuits from the underlying architecture of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, the fruit fly brain uses Kenyon cells to receive information from multiple sensory modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Specific neurons control the activation and inhibition states of these cells, so the fruit fly brain is a sparse high-dimensional representation of input information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [125] formalized this information-encoding process and applied it to the task of word- representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The experimental results indicate that the network can learn the static and context-dependent semantic representation of words, and its performance is comparable to other representation-learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This method also represents the word as a sparse binary hash code, which requires fewer computational resources than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition, this cognitively inspired method is commonly found in the research of general com- puting methods [126], such as memory networks [127], neural Turing machines [128], capsule networks [129], and plastic weight consolidation (Elastic Weight Consolidation) algorithms [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Borrow methods from cognitive science to interpret models We could learn from or directly use research methods from cognitive science to interpret infor- mation encoded by neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, Chien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [131] used the timescale-mapping method commonly used in the field of neuroscience to study the information encoded by each neuron in the long short-term memory (LSTM) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They inferred the neuron’s function by observing the activation value of each neuron in the model for the following sentence when the normal sentence and the above fragment were randomly replaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The logic behind it is that, if the function of a neuron encodes short- time-scale language information, then, when a small piece of text is replaced, its activation value change in the following sentence should be greater than that of neurons that encode long-term- scale language information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The study found that approximately 15% of neurons are used to encode long-term scale information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Such neurons can be divided into two types: the controller 19 responsible for connecting each neuron and the integrator (integrator) responsible for integrating long-distance information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In addition, inspired by neuroscience’s approach to studying neuron encoding mechanisms, Lakretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [132] studied the working mechanism of each neuron in the LSTM model when completing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Ivanva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' [133] drew on the design methods of neuroscience probe tasks and proposed guidelines for designing probe tasks in machine-learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 6 Discussion At this stage, researchers have achieved preliminary results in the two directions of using language- computation models to predict brain-activity data and inspiring language-computation models through the language understanding mechanism of the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, there is still a lack of granular and systematic research on the combination of language cognition and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, in terms of language comprehension in the human brain, it is not clear how people start from the most basic language unit and gradually build larger units until they finally understand the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Moreover, there is still a lack of systematic and effective modeling methods to address this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In terms of machine language understanding, the computation model has achieved super-human accuracy in many language-processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, it is still far from human intelligence regarding common sense reasoning ability, autonomous learning ability, generalization ability, learning efficiency, interpretability, and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' There is no clear solution for how to learn from a deeper-brain language- understanding mechanism to build a more intelligent language-computation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In the future, the author believes that computational theory-driven language-comprehension cogni- tive experiments are promising for the study of human language comprehension, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content='. In other words, research hypotheses are proposed based on the structure or results of computational models, and behavioral or brain activity data are verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' There are five important research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Figure 4: Schematic diagram of the cognitive experiment for natural language understanding driven by computational theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Collection of multilingual and multimodal neural activity data Most existing research on language cognition is limited to using a single data-collection method (such as fMRI or MEG) to study the specific language phenomenon of a certain language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This often leads to the problems of low robustness and poor repeatability of the conclusions drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, future language-cognition research should conduct verifications using multiple lan- guages and multiple types of data [134, 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Especially for studies combining computational models, the scale and quality of data directly determine the reliability of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, it is crucial to use both invasive and noninvasive tools to collect largescale high-quality neural activity data for different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' At the same time, the opening and sharing of data is gradually becoming a trend, which will greatly promote the study of language cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 20 Build a computational model Collectbrainactivationdata Propose a hypothesis Test a hypothesis Bee Bee Bees are social insects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Bees are social insects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Language computational model Experimental stimulus Modelresult Brainactivationdata ExperimentalStimulus2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Inspired new cognitive mechanism hypotheses The operation process of the language-computation model is transparent and global to a cer- tain extent, and its calculation process is also visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For instance, the vocabulary represen- tation learned by the model, the calculation method of combining vocabulary representations into phrases and sentence representations, and the prediction and inference of certain calculation steps of the result are all observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Explaining the working principle of the brain from the level of the computing mechanism is an important task of cognitive science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The author believes that, in the future, we can deeply explore whether the representation and computing modules in the computing model can indeed explain the neural activities of some brain regions in the process of language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' If the neural activity of a brain region can be explained by a computational model, then the brain region can be considered to perform the computational functions clearly visible in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' In other words, we can regard each module in different language computation models as a hypothesis of a brain computing mechanism and use cognitive science experiments to verify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Correlating multiple linguistic variables and cognitive function The process of language comprehension is very complex, not only involving the processing of multiple language variables, such as morphology, syntax, and semantics, but also closely related to multiple cognitive functions, such as executive control, attention, and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Previous studies often eliminated the influence of other language variables and cognitive functions by strictly controlling experimental variables and only studied the effect of a certain language variable or cognitive function in an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The author believes that language-cognition experiments combined with computational models can eliminate the research limitations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, using computational models can separate different experimental variables and study the role of different language variables and cognitive functions based on neural activity data collected from natural texts[136, 137, 138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' With the continuous improvement of the performance of language- computation methods based on neural network methods, it is increasingly accurate to use models to separate different language features so that the visual and auditory perception, multimodal information fusion, and language in different regions of the brain can be calculated on the same batch of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Other functional mechanisms in understanding become possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Analyzing the underlying computing mechanism of brain language understanding Most of the existing research on language cognition is based on linguistic theory, but there is a large gap between linguistics and neuroscience research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, regarding how the brain manipulates the most basic language units, linguistics mainly studies phrase structure and semantic combination while neuroscience focuses on neural oscillations and synchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This has led to the lack of a neural basis in the current research on language cognition, which cannot match the conclusions of neuroscience findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' With the continuous development of spike neural networks (spike neural networks) and oscillating neural networks (oscillatory neural networks), future computing models must be able to integrate the conclusions of neuroscience to simulate the working mode of underlying neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Manipulating language units to complete the task of language understanding provides a new solution for research linking linguistics and neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Exploring the mechanism of language learning and evolution As early as the 1980s, cognitive scientists used the connectionist model to explore what kind of model and what kind of data can simulate the human language-acquisition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' However, the computing power of the connectionist model at that time was quite limited, and it could only solve some case-specific and simple language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Today, the language-processing ability of deep neural networks has made a qualitative leap compared to the 1980s, and there are more corpus records in the process of infant language acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, it is possible to try to use computational models to explore the mechanism of language learning and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' It is even possible to explore in depth whether the cyclic connection, convolution operation, dot product attention mechanism, backpropagation algorithm, and so on in the language-computation model are also necessary links in the human brain’s language understanding and computing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' With regard to machine language comprehension, as shown in Figure 5, research on language cog- nition suggests that the human brain may have many efficient processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Thus, it has great 21 potential to inspire the construction of a new generation of language-computation models regarding the cognitive mechanisms of representation, learning, and memory and the working mechanisms of neu- rons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The author believes that the following five aspects will become important development directions in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Figure 5: Schematic diagram of a language computational model inspired by the cognitive function of the human brain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Representation and combination of text When encoding the meaning of concepts, the brain uses different representations for different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, when reading nouns and verbs, the brain activates different brain networks, showing the characteristics of distributed coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' When observing a specific or familiar person, a specific neuron in the brain is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' When encoding the syntactic structure, the brain will use different combination methods for different types of phrases, use hierarchical encoding methods such as tree structures to guide the combination sequence of words, and use parallel processing to encode multiple levels of language unit (words, phrases, sentences, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=') information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' With this encoding method, the human brain stores and calculates the meaning of the text very efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' It is also closely related to the ability of humans to ”infer other cases from one instance” and learn quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Future language computation models can learn from this mechanism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' combine symbolic and distributed representation methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' and adopt a combination of diversity, hierarchy, and parallelism to learn text representation and combination models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Continuous language learning Humans have the ability of continuous and small-sample learning in childhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, if a 2-year-old child is shown a picture of a giraffe, the child can recognize other pictures of giraffes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This ability can be transferred to tasks such as recognizing pictures of other animals and finding the text corresponding to the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This learning ability is closely related to the human memory system, and the result of learning new information is memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Different types of information are processed and stored by different memory systems, and information such as sounds and images are stored by sensory memory and maintained in a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The complex and structured memory system of the human brain ensures the efficient organization of massive data and rapid extraction when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' All these mechanisms could be learned by computation models to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Interactive learning of language The best existing general-purpose language-computation models use predicting the next word as an objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' They are trained in massive texts and achieve excellent performance in multiple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The difference is that humans often learn language by interacting with others, which is a more effective way to learn and improve language ability than analyzing and mem- orizing language structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Drawing on this interactive learning method, in addition to using text information as a supervisory signal, future language-computation models can also obtain feedback from the structure or output results of other models and learn in continuous interaction with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Multimodal information fusion Closely related to interactive learning is the comprehensive processing of multiple-modality infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The human language-learning environment is a multimodal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Humans are better 22 Cognitivefunctionsofhumanbrain Buildacomputationalmodel Mechanisms ofrepresentation, Dialogue systems learning, memory, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Machinetranslation Working mechanisms of neurons Emotion analysis Brain activationandbehaviordata Specch recognition Brain activation data Language computational modelat processing multimodal than single-modal information, and the processing speed is faster for multimodal information than for single-modal information [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Therefore, the author believes that, in the fusion of multimodal information, brain-inspired computing models are an impor- tant research direction in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, according to the ”Hub and Spoke” theory [140], concepts are represented by multimodal information such as vision, hearing, smell, and somatosensory information, and there is a semantic center that encodes modal-independent in- formation to associate different modalities of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The information between different modalities is complementary and mutually verifiable and, when combined, can represent more abundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' The fusion mechanism of multimodal information can also learn from the abovementioned ”central” mechanism and design a modality-independent module to integrate and correlate different types of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Interpretability of computation models Cognitive science designs experiments to analyze the working mechanism of the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' This research method and cognitive experimental data can also be used to analyze or evaluate the working mechanism of language-computation models and inspire new model interpretability methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, referring to the comparative analysis method often used in language- cognition experiments, two groups of experimental materials are designed so that they differ only in a certain language attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' For example, groups of sentences with high and low syntactic complexity that are basically the same in terms of sentence length, sentence meaning, and so on are input into the calculation model, and we observe whether the effect of the model is consistent with the syntactic complexity degree correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' If the activation of some nodes in the network is significantly stronger when encoding high- than low-complexity sentences, then these neurons are responsible for encoding syntactic information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' otherwise, they are not responsible for encoding syntactic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' 7 Conclusion Language cognition is one of the core issues in cognitive and brain sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' It is of great significance not only to reveal the basis of language intelligence and the working mechanisms of the brain but also to help promote the development of brain-inspired language intelligence technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E3T4oBgHgl3EQf6Qta/content/2301.04788v1.pdf'} +page_content=' Simultaneously, new ideas and technologies in language-computation research can also provide important 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index 0000000000000000000000000000000000000000..c94ee338a931c0bedea13990c1f6fde0bddd2666 --- /dev/null +++ b/n9E2T4oBgHgl3EQffAcI/content/tmp_files/2301.03921v1.pdf.txt @@ -0,0 +1,629 @@ +arXiv:2301.03921v1 [gr-qc] 10 Jan 2023 +Black String solutions in Rainbow Gravity +R. Dárlla∗,1 F. A. Brito†,1 and J. Furtado‡2 +1∗Universidade Federal de Campina Grande, Departamento de Física, 63500-000, Campina Grande-PB, Brazil. +2∗Universidade Federal do Cariri, Centro de Ciência e Tecnologia, 63048-080, Juazeiro do Norte-CE, Brazil. +(Dated: January 11, 2023) +In this paper we study black string solutions under the consideration of rainbow gravity. We have analytically +obtained the solution for four-dimensional black strings in terms of the functions f (E/Ep) and g(E/Ep) that sets +the energy scale where the rainbow gravity becomes relevant. We have also obtained the Hawking temperature +for the black string, from which we could see that the rainbow functions play the role of increasing or decreasing +the Hawking temperature for a given horizon radius depending on the choice of such rainbow functions. We +have computed the entropy, specific heat and free energy for the black string. The entropy and specific heat +exhibit a rainbow dependence, while the free energy is not modified by the rainbow functions. Finally we have +studied the effects of the rainbow gravity in the orbits of massive and massless particles around a black string. +We could verify that neither massive nor massless particles exhibit stable orbits around a black string in the +scenario of rainbow gravity, for any configuration of rainbow functions. +I. +INTRODUCTION +There are models of quantum gravity known as Doubly +Special Relativity (DSR) that suggests the existence of an in- +variant energy scale (or lenght) independent of the observer. +In such models the dispersion relation is modified when we +consider energies near to Planck energy scale [1–4], which +implies that the speed of light is not the only relativistic invari- +ant. Moreover, these models suggest that exists more possible +consistent modifications to general relativity other than the in- +troduction of quantum corrections in the Einstein-Hilbert ac- +tion. +The approaches that receive the name of rainbow’s grav- +ity suggest that the usual energy-momentum dispersion re- +lation is deformed close to the Planck scale and that space- +time is also modified due to the non-linear representation of +Lorentz transformations, so that its geometry changes accord- +ing to the energy of the test particle in it. This means that +particles with different energies distort spacetime differently +in a type of spacetime backreaction leading to the mentioned +modification of the relativistic energy-momentum dispersion +relation [9]. These approaches are studied in several scenar- +ios such as string field theory [5], loop quantum gravity [6], +and non-commutative geometry [7]. Some theoretical propos- +als suggest corrections both in the action and in the dispersion +relation as in [8]. +Some phenomena can be explained through this semi- +classical approach, such as the ultra-high-energy cosmic rays +that are currently observed but still have unknown origin, sug- +gesting that the dispersion relation is indeed modified. In as- +trophysics, the influence of the rainbow’s gravity on the prop- +erties of a black hole has been studied in several scenarios, +including its thermodynamics [10–20], and also in the study +of cosmic strings [21–23]. In addition, to understand the early +universe, in which the energies involved were close to the +∗E-mail:robertadarlla2@gmail.com +†E-mail:fabrito@df.ufcg.edu.br +‡E-mail:job.furtado@ufca.edu.br +Planck scale, such a modified theory of gravity plays an im- +portant role to avoid an initial singularity [24–29]. Finally, +in general field theory there is a lot of recent developments +regarding rainbow gravity in the context of Bose-Einstein +condensation [30], Klein-Gordon oscillation [29], Landau- +Aharonov-Casher effect [31], particle production [32], among +others. +In this paper we study black string solutions under the con- +sideration of rainbow gravity. We have analytically obtained +the solution for four-dimensional black strings in terms of +the functions f(E/Ep) and g(E/Ep) that sets the energy scale +where the rainbow gravity becomes relevant. We have also +obtained the Hawking temperature for the black string, from +which we could see that for a give horizon radius the rainbow +gravity contribution promotes an increasing in the Hawking +temperature. +II. +RAINBOW GRAVITY REVIEW +The rainbow gravity was first studied in the context of Dou- +ble Special Relativity (DSR) and it emerges as a generaliza- +tion to curved spacetime of the deformed Lorentz symmetry +group (locally). One of its consequences is the arising of a +modified energy–momentum dispersion relation. Such modi- +fication is usually written in the form [3, 33, 34] +E2 f 2(E/EP) − p2c2g2(E/EP) = m2c4, +(1) +where f(E/EP) and g(E/EP) are the so called rainbow func- +tions, being E the energy of the probe particle and EP the +Planck energy. In the low-energy limit the rainbow functions +converges to unit, restoring the standard dispersion relation. +However in the high-energy limit the rainbow functions end +up violating the usual energy-momentum dispersion relation. +The modification of this latter corresponds to a change in the +metric, according to [34], so that the Minkowski spacetime +becomes +ds2 = +dt2 +f 2(E/EP) − +1 +g2(E/EP)δi jdxidx j. +(2) + +2 +ϵ=0.2 +ϵ=0.5 +ϵ=0.8 +Black String +4 +6 +8 +10 +r +-40 +-20 +20 +40 +A(r) +(a) +ϵ=0.2 +ϵ=0.5 +ϵ=0.8 +Black String +4 +6 +8 +10 +r +-20 +20 +40 +A(r) +(b) +Figure 1: Black string solution in rainbow gravity. For this plot we have considered Ep = 1, s = 1, ξ = 0.4, α = 0.5 and µ = 0.7. +In (a) we are considering the case I for the rainbow functions while in (b) we are considering the case II. +In order to study the rainbow gravity effects on the +Friedmann-Robertson-Walker (FRW) universe [35, 36], the +following rainbow functions were considered (case I) +f(E/EP) = 1, g(E/EP) = +� +1 − ξ(E/EP)s, +(3) +where s > 1 and ξ is a dimensionless free parameter of the +model, which we will consider the same as the other rain- +bow functions to facilitate comparison between the employed +models. +Another interesting choice for the rainbow functions is the +following (case II), +f(E/EP) = g(E/EP) = +1 +1 − ξ(E/EP). +(4) +Such rainbow functions were considered in [3, 33] (and refer- +ences therein) in studying possible nonsingular universe solu- +tions and in [34], since it assures a constant light velocity, it +may provides a solution for the horizon problem. +A last choice of rainbow functions of great interest is given +by (case III) +f(E/EP) = eξ(E/EP) − 1 +ξ(E/EP) , g(E/EP) = 1. +(5) +This choice of the rainbow functions was originally consid- +ered in [9] in the context of Gamma Ray Bursts. Later, this +same choice was also addressed in [35, 37] in connection with +FRW solutions. +III. +BLACK STRING SOLUTION IN RAINBOW GRAVITY +Let us consider the following line element for the black +string +ds2 = − +A(r) +f(E/Ep)dt2 + +1 +g(E/Ep)A(r)dr2 + +r2 +g(E/Ep)dφ2 ++ +α2r2 +g(E/Ep)dz2, +(6) +where t ∈ (−∞, ∞), the radial coordinate r ∈ [0, ∞), the +angular coordinate φ ∈ [0, 2π) and the axial coordinate z ∈ +(−∞, ∞). The α parameter is considered as α2 = −Λ/3. +A(r) = +α2r2 +[g(E/Ep)]2 − 4µ +αr , +(7) +The above solution for the black string in the rainbow gravity +scenario recovers the usual black string solution [38] when +g(E/Ep) = 1, i.e., +A(r) = +� +α2r2 − 4µ +αr +� +. +(8) +For the black string the Einstein-Hilbert effective action re- +quires the cosmological constant contribution, so that, +S u = +1 +2κ2 +� +d4x √−g (R − 2Λ). +(9) +where κ = 8πG and R is the Ricci scalar. The EFE for this +ansatz gives us +Gt +t − 3α2 = [g(E/Ep)]2 +�1 +r +dA(r) +dr ++ A(r) +r2 +� +− 3α2, (10) +We can see that the energy-momentum tensor for the +ansatz of equation (6) is T µ +ν += +−ρ(r) diag(1, 1, 0, 0) + +pl(r) diag(0, 0, 1, 1), where pl = pφ = pz. This way we can +find A(r) by solving Gt +t − 3α2 = −κ2ρ(r), so that we find +The behaviour of the black string solution in rainbow grav- +ity is depicted in figure (1) for the cases I and II. Note that +the case III for the rainbow functions does not gives us any +modification in the black string solution, since g(E/Ep) = 1. +Let us discuss briefly the role played by the rainbow gravity +scenario in the black string solution. As we can see in (1), as +we increase the value of the energy E, approaching the Planck +energy scale, we also increase the value of the horizon radius +for the cases I and II of the rainbow functions. + +3 +Black String +ϵ += 0.2 +ϵ +� 0.5 +ϵ +� 0.8 +4 +6 +8 +10 +rh +0.2 +0.4 +0.6 +0.8 +1.0 +T + +H +(a) +Black String +ϵ = 0.2 +ϵ = 0.5 +ϵ = 0.8 +4 +6 +8 +10 +rh +0.2 +0.4 +0.6 +0.8 +1.0 +T + +H +(b) +Figure 2: Hawking temperature for black string solution in rainbow gravity. For this plot we have considered Ep = 1, s = 1, +ξ = 0.4, α = 0.5 and µ = 0.7. In (a) we are considering the case I for the rainbow functions while in (b) we are considering the +case II. +IV. +BLACK STRING THERMODYNAMICS IN RAINBOW +GRAVITY +Our black string solution in the rainbow gravity scenario +has the horizon curves defined by A( ˜rh) = 0, so that the linear +mass can be written as +µ = +α3 ˜rh3 +4g(E/Ep)2 . +(11) +Here ˜rh = rh [g(E/EP)]2/3, where rh is the horizon radius +of the usual General Relativity solution for the black string. +Then, the expression (11) becomes +µ = α3r3 +h +4 . +(12) +Thus, this linear mass has no modification due the rainbow +gravity. +In possession of the solution for the static black string in the +rainbow gravity scenario given by (7), we are able to study the +thermodynamics of the black string by computing the Hawk- +ing’s temperature by means of TH = A′(˜rh) +4π . Thus we obtain +˜TH = +3α2 rh +4π [g(E/EP)]4/3 . +(13) +The behaviour of the Hawking temperature for the cases I +and II is depicted in the figure (2). For both cases (I and II) +the same linear behaviour of the usual Hawking temperature +for black strings is present. However some slight differences +between the cases must be highlighted. For the case I (fig(2a)) +we can see that for a given horizon radius the Hawking tem- +perature is greater when we consider the effect of rainbow +gravity. The opposite occurs for the case II (fig(2a)), where +for a given horizon radius the Hawking temperature is smaller +when we consider the effect of rainbow gravity. +In order to properly understand the thermodynamics of the +black string in the rainbow gravity context it is necessary to +compute the entropy, specific heat and free energy. The en- +tropy can computed directly from the expression dS = dµ +˜TH , in +which we get +˜S = π α r2 +h [g(E/EP)]4/3 +2 +. +(14) +Cleary, this recovers the usual black string result S = 1 +2παr2 +h +when g(E/EP) = 1. As we can see in figure (3), for both +cases we have the same quadratic dependence of the horizon +radius that the usual black string entropy exhibit. However, +differently from the Hawking temperature, the case I promotes +a decreasing in the entropy for a given horizon radius while +the case II promotes an increasing in the entropy for a given +horizon radius. +The specific heat can be calculated by ˜Cv = +dµ +d ˜TH from which +we obtain +˜Cv = παr2 +h [g(E/EP)]4/3 +(15) +Similar to entropy, in case I for a given horizon radius the +specific heat is smaller when we consider the effect of rainbow +gravity. The opposite happens for case II. When g(E/EP) = 1 +we get Cv = παr2 +h, i.e. the usual black string specific heat +in General Relativity. The behaviour of the specific heat for +the cases I and II of the rainbow functions is depicted in (5). +As it is widely known, the thermodynamical stability of black +holes (black strings for our case) is directly related to the sign +of the heat capacity. A positive heat capacity indicates that +the system is thermodynamically stable, while its negativity +imply a thermodynamical instability. Therefore, the result for +the specific heat in the context of rainbow gravity indicates a +thermodynamically stable black string. +On the other hand, the rainbow gravity presents no modifi- +cation in the free energy F = µ − THS , yielding therefore the +usual black string result +F = −α3r3 +h +8 . +(16) + +4 +Black String +ϵ = 0.2 +ϵ = 0.5 +ϵ = 0.8 +rh +0.2 +0.4 +0.6 +0.8 +1.0 +S + +(a) +Black String +ϵ = 0.2 +ϵ = 0.5 +ϵ = 0.8 +rh +0.2 +0.4 +0.6 +0.8 +1.0 +S + +(b) +Figure 3: Entropy for black string solution in rainbow gravity. For this plot we have considered Ep = 1, s = 1, ξ = 0.4, α = 0.5 +and µ = 0.7. In (a) we are considering the case I for the rainbow functions while in (b) we are considering the case II. +Black String +ϵ = 0.2 +ϵ = 0.5 +ϵ = 0.8 +rh +0.2 +0.4 +0.6 +0.8 +1.0 +C + +v +(a) +Black String +ϵ = 0.2 +ϵ = 0.5 +ϵ = 0.8 +rh +0.2 +0.4 +0.6 +0.8 +1.0 +C + +v +(b) +Figure 4: Especific Heat for black string solution in rainbow gravity. For this plot we have considered Ep = 1, s = 1, ξ = 0.4, +α = 0.5 and µ = 0.7. In (a) we are considering the case I for the rainbow functions while in (b) we are considering the case II. +V. +GEODESICS AND CIRCULAR ORBITS +The particle’s geodesic in orbit around a static black string +is given by +˙r2 = ω2 − A(r) +�L2 +r2 + m2 +� +, +(17) +where ω is the particle’s energy, L is the angular momentum +and m is the particle’s mass. Thus the effective potential is +defined as +Vr = A(r) +�L2 +r2 + m2 +� +. +(18) +The circular geodesics occur at the points rc satisfying +1 +2 ˙r2 +c = 0 and V′ +r(rc) = 0. In Fig. (5) we depict the effec- +tive potential of massless and massive particles for the black +string in the rainbow gravity scenario. It is shown that there is +no case where circular orbits are stable, similarly to the usual +black string solution. Therefore, the rainbow gravity does not +modify significantly the results for geodesics and circular or- +bits in comparison to the usual black string. +VI. +CONCLUSION +In this paper we study black string solutions under the con- +sideration of rainbow gravity. We have analytically obtained +the solution for four-dimensional black strings in terms of +the functions f(E/Ep) and g(E/Ep) that sets the energy scale +where the rainbow gravity becomes relevant. We could ver- +ify that the black string solution depends only on the function +g(E/Ep), and consequently, all the thermodynamic parame- +ters will depend only on g(E/Ep). We have plotted the be- +haviour of the black string solution in (1), and we could see +that as we increase the value of the energy E, approaching the +Planck energy scale, we also increase the value of the horizon +radius for the cases I and II of the rainbow functions. +We have also obtained the Hawking temperature for the +black string, from which we could see that the rainbow func- +tions play the role of increasing or decreasing the Hawk- +ing temperature for a given horizon radius depending on the +choice of such rainbow functions. We have computed the en- +tropy, specific heat and free energy for the black string. The +entropy and specific heat exhibit a rainbow dependence, while +the free energy is not modified by the rainbow functions. + +5 +ϵ=0.2 +ϵ=0.4 +ϵ=0.6 +4 +6 +8 +10 +r +-0.0010 +-0.0005 +0.0005 +0.0010 +V[r] +(a) +ϵ=0.2 +ϵ=0.4 +ϵ=0.6 +4 +6 +8 +10 +r +-0.002 +-0.001 +0.001 +0.002 +V[r] +(b) +Figure 5: Effective potential (a) for massless particles and (b) for massive particles. For this plot we have considered Ep = 1, +s = 1, ξ = 0.4, α = 0.5, µ = 0.7 and L = 0.1. +Finally we have studied the effects of the rainbow grav- +ity in the orbits of massive and massless particles around a +black string. We could verify that neither massive nor mass- +less particles exhibit stable orbits around a black string in the +scenario of rainbow gravity, for any configuration of rainbow +functions. +Acknowledgements +FAB +would +like +to +thank +CNPq +and +CNPq/PRONEX/FAPESQ-PB +(Grant +nos. +165/2018 +and 312104/2018-9), for partial financial support. JF would +like to thank the Fundação Cearense de Apoio ao Desenvolvi- +mento Científico e Tecnológico (FUNCAP) under the grant +PRONEM PNE0112-00085.01.00/16 for financial support. +[1] G. Amelino-Camelia, Relativity in space-times with short dis- +tance structure governed by an observer independent (Planck- +ian) length scale, Int. J. Mod. Phys. D11 (2002) 35–60, +[gr-qc/0012051]. +[2] G. Amelino-Camelia, Doubly Special Relativity, Nature 418, +34 (2002). +[3] J. Magueijo and L. Smolin, Lorentz invariance with an in- +variant energy scale, Phys. Rev. Lett. 88 (2002) 190403, +[hep-th/0112090]. +[4] Galan and G. A. Mena Marugan, Quantum time uncertainty in +a gravity’s rainbow formalism, Phys. Rev. D70 (2004) 124003, +[gr-qc/0411089]. +[5] V. A. Kostelecky and S. Samuel, Spontaneous breaking of +Lorentz symmetry in string theory, Phys. Rev. D39, 683 (1989). +[6] R. Gambini and J. Pullin, Nonstandard optics from quantum +spacetime, Phys. Rev. D59, 124021 (1999). +[7] S. M. Carroll, J. 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B 353, 46-51 (1995) + diff --git a/n9E2T4oBgHgl3EQffAcI/content/tmp_files/load_file.txt b/n9E2T4oBgHgl3EQffAcI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2501f5d11d1a5eaef3dfe6b17a10eca996cc4eff --- /dev/null +++ b/n9E2T4oBgHgl3EQffAcI/content/tmp_files/load_file.txt @@ -0,0 +1,491 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf,len=490 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='03921v1 [gr-qc] 10 Jan 2023 Black String solutions in Rainbow Gravity R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Dárlla∗,1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Brito†,1 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Furtado‡2 1∗Universidade Federal de Campina Grande, Departamento de Física, 63500-000, Campina Grande-PB, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' 2∗Universidade Federal do Cariri, Centro de Ciência e Tecnologia, 63048-080, Juazeiro do Norte-CE, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (Dated: January 11, 2023) In this paper we study black string solutions under the consideration of rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have analytically obtained the solution for four-dimensional black strings in terms of the functions f (E/Ep) and g(E/Ep) that sets the energy scale where the rainbow gravity becomes relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have also obtained the Hawking temperature for the black string, from which we could see that the rainbow functions play the role of increasing or decreasing the Hawking temperature for a given horizon radius depending on the choice of such rainbow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have computed the entropy, specific heat and free energy for the black string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The entropy and specific heat exhibit a rainbow dependence, while the free energy is not modified by the rainbow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Finally we have studied the effects of the rainbow gravity in the orbits of massive and massless particles around a black string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We could verify that neither massive nor massless particles exhibit stable orbits around a black string in the scenario of rainbow gravity, for any configuration of rainbow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' INTRODUCTION There are models of quantum gravity known as Doubly Special Relativity (DSR) that suggests the existence of an in- variant energy scale (or lenght) independent of the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In such models the dispersion relation is modified when we consider energies near to Planck energy scale [1–4], which implies that the speed of light is not the only relativistic invari- ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Moreover, these models suggest that exists more possible consistent modifications to general relativity other than the in- troduction of quantum corrections in the Einstein-Hilbert ac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The approaches that receive the name of rainbow’s grav- ity suggest that the usual energy-momentum dispersion re- lation is deformed close to the Planck scale and that space- time is also modified due to the non-linear representation of Lorentz transformations, so that its geometry changes accord- ing to the energy of the test particle in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' This means that particles with different energies distort spacetime differently in a type of spacetime backreaction leading to the mentioned modification of the relativistic energy-momentum dispersion relation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' These approaches are studied in several scenar- ios such as string field theory [5], loop quantum gravity [6], and non-commutative geometry [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Some theoretical propos- als suggest corrections both in the action and in the dispersion relation as in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Some phenomena can be explained through this semi- classical approach, such as the ultra-high-energy cosmic rays that are currently observed but still have unknown origin, sug- gesting that the dispersion relation is indeed modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In as- trophysics, the influence of the rainbow’s gravity on the prop- erties of a black hole has been studied in several scenarios, including its thermodynamics [10–20], and also in the study of cosmic strings [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In addition, to understand the early universe, in which the energies involved were close to the ∗E-mail:robertadarlla2@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='com †E-mail:fabrito@df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='ufcg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='br ‡E-mail:job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='furtado@ufca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='br Planck scale, such a modified theory of gravity plays an im- portant role to avoid an initial singularity [24–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Finally, in general field theory there is a lot of recent developments regarding rainbow gravity in the context of Bose-Einstein condensation [30], Klein-Gordon oscillation [29], Landau- Aharonov-Casher effect [31], particle production [32], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In this paper we study black string solutions under the con- sideration of rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have analytically obtained the solution for four-dimensional black strings in terms of the functions f(E/Ep) and g(E/Ep) that sets the energy scale where the rainbow gravity becomes relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have also obtained the Hawking temperature for the black string, from which we could see that for a give horizon radius the rainbow gravity contribution promotes an increasing in the Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' RAINBOW GRAVITY REVIEW The rainbow gravity was first studied in the context of Dou- ble Special Relativity (DSR) and it emerges as a generaliza- tion to curved spacetime of the deformed Lorentz symmetry group (locally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' One of its consequences is the arising of a modified energy–momentum dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Such modi- fication is usually written in the form [3, 33, 34] E2 f 2(E/EP) − p2c2g2(E/EP) = m2c4, (1) where f(E/EP) and g(E/EP) are the so called rainbow func- tions, being E the energy of the probe particle and EP the Planck energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In the low-energy limit the rainbow functions converges to unit, restoring the standard dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' However in the high-energy limit the rainbow functions end up violating the usual energy-momentum dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The modification of this latter corresponds to a change in the metric, according to [34], so that the Minkowski spacetime becomes ds2 = dt2 f 2(E/EP) − 1 g2(E/EP)δi jdxidx j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (2) 2 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 Black String 4 6 8 10 r 40 20 20 40 A(r) (a) ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 Black String 4 6 8 10 r 20 20 40 A(r) (b) Figure 1: Black string solution in rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' For this plot we have considered Ep = 1, s = 1, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In (a) we are considering the case I for the rainbow functions while in (b) we are considering the case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In order to study the rainbow gravity effects on the Friedmann-Robertson-Walker (FRW) universe [35, 36], the following rainbow functions were considered (case I) f(E/EP) = 1, g(E/EP) = � 1 − ξ(E/EP)s, (3) where s > 1 and ξ is a dimensionless free parameter of the model, which we will consider the same as the other rain- bow functions to facilitate comparison between the employed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Another interesting choice for the rainbow functions is the following (case II), f(E/EP) = g(E/EP) = 1 1 − ξ(E/EP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (4) Such rainbow functions were considered in [3, 33] (and refer- ences therein) in studying possible nonsingular universe solu- tions and in [34], since it assures a constant light velocity, it may provides a solution for the horizon problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' A last choice of rainbow functions of great interest is given by (case III) f(E/EP) = eξ(E/EP) − 1 ξ(E/EP) , g(E/EP) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (5) This choice of the rainbow functions was originally consid- ered in [9] in the context of Gamma Ray Bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Later, this same choice was also addressed in [35, 37] in connection with FRW solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' BLACK STRING SOLUTION IN RAINBOW GRAVITY Let us consider the following line element for the black string ds2 = − A(r) f(E/Ep)dt2 + 1 g(E/Ep)A(r)dr2 + r2 g(E/Ep)dφ2 + α2r2 g(E/Ep)dz2, (6) where t ∈ (−∞, ∞), the radial coordinate r ∈ [0, ∞), the angular coordinate φ ∈ [0, 2π) and the axial coordinate z ∈ (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The α parameter is considered as α2 = −Λ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' A(r) = α2r2 [g(E/Ep)]2 − 4µ αr , (7) The above solution for the black string in the rainbow gravity scenario recovers the usual black string solution [38] when g(E/Ep) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=', A(r) = � α2r2 − 4µ αr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (8) For the black string the Einstein-Hilbert effective action re- quires the cosmological constant contribution, so that, S u = 1 2κ2 � d4x √−g (R − 2Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (9) where κ = 8πG and R is the Ricci scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The EFE for this ansatz gives us Gt t − 3α2 = [g(E/Ep)]2 �1 r dA(r) dr + A(r) r2 � − 3α2, (10) We can see that the energy-momentum tensor for the ansatz of equation (6) is T µ ν = −ρ(r) diag(1, 1, 0, 0) + pl(r) diag(0, 0, 1, 1), where pl = pφ = pz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' This way we can find A(r) by solving Gt t − 3α2 = −κ2ρ(r), so that we find The behaviour of the black string solution in rainbow grav- ity is depicted in figure (1) for the cases I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Note that the case III for the rainbow functions does not gives us any modification in the black string solution, since g(E/Ep) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Let us discuss briefly the role played by the rainbow gravity scenario in the black string solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' As we can see in (1), as we increase the value of the energy E, approaching the Planck energy scale, we also increase the value of the horizon radius for the cases I and II of the rainbow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' 3 Black String ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 ϵ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 4 6 8 10 rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0 T \uf02d H (a) Black String ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 4 6 8 10 rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0 T \uf02d H (b) Figure 2: Hawking temperature for black string solution in rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' For this plot we have considered Ep = 1, s = 1, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In (a) we are considering the case I for the rainbow functions while in (b) we are considering the case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' BLACK STRING THERMODYNAMICS IN RAINBOW GRAVITY Our black string solution in the rainbow gravity scenario has the horizon curves defined by A( ˜rh) = 0, so that the linear mass can be written as µ = α3 ˜rh3 4g(E/Ep)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (11) Here ˜rh = rh [g(E/EP)]2/3, where rh is the horizon radius of the usual General Relativity solution for the black string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Then, the expression (11) becomes µ = α3r3 h 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (12) Thus, this linear mass has no modification due the rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In possession of the solution for the static black string in the rainbow gravity scenario given by (7), we are able to study the thermodynamics of the black string by computing the Hawk- ing’s temperature by means of TH = A′(˜rh) 4π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Thus we obtain ˜TH = 3α2 rh 4π [g(E/EP)]4/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (13) The behaviour of the Hawking temperature for the cases I and II is depicted in the figure (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' For both cases (I and II) the same linear behaviour of the usual Hawking temperature for black strings is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' However some slight differences between the cases must be highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' For the case I (fig(2a)) we can see that for a given horizon radius the Hawking tem- perature is greater when we consider the effect of rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The opposite occurs for the case II (fig(2a)), where for a given horizon radius the Hawking temperature is smaller when we consider the effect of rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In order to properly understand the thermodynamics of the black string in the rainbow gravity context it is necessary to compute the entropy, specific heat and free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The en- tropy can computed directly from the expression dS = dµ ˜TH , in which we get ˜S = π α r2 h [g(E/EP)]4/3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (14) Cleary, this recovers the usual black string result S = 1 2παr2 h when g(E/EP) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' As we can see in figure (3), for both cases we have the same quadratic dependence of the horizon radius that the usual black string entropy exhibit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' However, differently from the Hawking temperature, the case I promotes a decreasing in the entropy for a given horizon radius while the case II promotes an increasing in the entropy for a given horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The specific heat can be calculated by ˜Cv = dµ d ˜TH from which we obtain ˜Cv = παr2 h [g(E/EP)]4/3 (15) Similar to entropy, in case I for a given horizon radius the specific heat is smaller when we consider the effect of rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The opposite happens for case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' When g(E/EP) = 1 we get Cv = παr2 h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' the usual black string specific heat in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The behaviour of the specific heat for the cases I and II of the rainbow functions is depicted in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' As it is widely known, the thermodynamical stability of black holes (black strings for our case) is directly related to the sign of the heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' A positive heat capacity indicates that the system is thermodynamically stable, while its negativity imply a thermodynamical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Therefore, the result for the specific heat in the context of rainbow gravity indicates a thermodynamically stable black string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' On the other hand, the rainbow gravity presents no modifi- cation in the free energy F = µ − THS , yielding therefore the usual black string result F = −α3r3 h 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (16) 4 Black String ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0 S \uf02d (a) Black String ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0 S \uf02d (b) Figure 3: Entropy for black string solution in rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' For this plot we have considered Ep = 1, s = 1, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In (a) we are considering the case I for the rainbow functions while in (b) we are considering the case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Black String ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0 C \uf02d v (a) Black String ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 rh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0 C \uf02d v (b) Figure 4: Especific Heat for black string solution in rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' For this plot we have considered Ep = 1, s = 1, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In (a) we are considering the case I for the rainbow functions while in (b) we are considering the case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' GEODESICS AND CIRCULAR ORBITS The particle’s geodesic in orbit around a static black string is given by ˙r2 = ω2 − A(r) �L2 r2 + m2 � , (17) where ω is the particle’s energy, L is the angular momentum and m is the particle’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Thus the effective potential is defined as Vr = A(r) �L2 r2 + m2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (18) The circular geodesics occur at the points rc satisfying 1 2 ˙r2 c = 0 and V′ r(rc) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' (5) we depict the effec- tive potential of massless and massive particles for the black string in the rainbow gravity scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' It is shown that there is no case where circular orbits are stable, similarly to the usual black string solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Therefore, the rainbow gravity does not modify significantly the results for geodesics and circular or- bits in comparison to the usual black string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' CONCLUSION In this paper we study black string solutions under the con- sideration of rainbow gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have analytically obtained the solution for four-dimensional black strings in terms of the functions f(E/Ep) and g(E/Ep) that sets the energy scale where the rainbow gravity becomes relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We could ver- ify that the black string solution depends only on the function g(E/Ep), and consequently, all the thermodynamic parame- ters will depend only on g(E/Ep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have plotted the be- haviour of the black string solution in (1), and we could see that as we increase the value of the energy E, approaching the Planck energy scale, we also increase the value of the horizon radius for the cases I and II of the rainbow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have also obtained the Hawking temperature for the black string, from which we could see that the rainbow func- tions play the role of increasing or decreasing the Hawk- ing temperature for a given horizon radius depending on the choice of such rainbow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We have computed the en- tropy, specific heat and free energy for the black string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' The entropy and specific heat exhibit a rainbow dependence, while the free energy is not modified by the rainbow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' 5 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='6 4 6 8 10 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='0010 V[r] (a) ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='2 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4 ϵ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='6 4 6 8 10 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='002 V[r] (b) Figure 5: Effective potential (a) for massless particles and (b) for massive particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' For this plot we have considered Ep = 1, s = 1, ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='4, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='5, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='7 and L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Finally we have studied the effects of the rainbow grav- ity in the orbits of massive and massless particles around a black string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' We could verify that neither massive nor mass- less particles exhibit stable orbits around a black string in the scenario of rainbow gravity, for any configuration of rainbow functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' Acknowledgements FAB would like to thank CNPq and CNPq/PRONEX/FAPESQ-PB (Grant nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' 165/2018 and 312104/2018-9), for partial financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' JF would like to thank the Fundação Cearense de Apoio ao Desenvolvi- mento Científico e Tecnológico (FUNCAP) under the grant PRONEM PNE0112-00085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content='00/16 for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E2T4oBgHgl3EQffAcI/content/2301.03921v1.pdf'} +page_content=' [1] G.' 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in a semi-infinite geometry. Here, using extensive Monte Carlo simulations, we observe extraordinary- +log critical behavior on the plane defects of O(2) critical systems in an infinite geometry. In this extraordinary- +log critical phase, the large-distance two-point correlation G obeys the logarithmic finite-size scaling G ∼ +(lnL)−ˆq with the linear size L, having the critical exponent ˆq = 0.29(2). Meanwhile, the helicity modulus +Υ follows the scaling form Υ ∼ α(lnL)/L with the universal parameter α = 0.56(3). The values of ˆq and +α do not fall into any known universality class of critical phenomena, yet they conform to the scaling relation +of extraordinary-log universality. We also discuss the extension of current results to a quantum system that is +experimentally accessible. These findings reshape our understanding of extraordinary-log critical phenomena. +Introduction. Critical phenomena have been a subject of +long-standing interest, and universality is recognized as a pil- +lar of modern critical theory [1]. When critical phenomena +occur on lattices of special geometries [2–16], nontrivial sce- +narios of universality may emerge and be connected to a wide +range of modern concepts [17–24] +The O(2) criticality is a textbook illustration of critical +phenomena. +Two-dimensional (2D) and three-dimensional +(3D) O(2) critical phenomena have been found in diverse sys- +tems [25]. In two dimensions, the Kosterlitz-Thouless transi- +tion is driven by the unbinding of vortex-antivortex pairs, and +the quasi-long-range order can emerge in the low-temperature +phase [26]. In three dimensions, the O(2) criticality serves +as a testbed for various techniques and theories, such as the +“λ transition” of helium which has been the subject of exper- +iments in Earth orbit [27]. +Recently, a renormalization-group study predicted the +extraordinary-log universality (ELU) for the surface critical +behavior (SCB) of the O(n) model with 2 ≤ n < nc, where +nc is not precisely known [16]. A consequence of this predic- +tion is the logarithmic scaling form of the surface two-point +correlation g as a function of the distance r [16], which can +be reexpressed by the finite-size scaling (FSS) of the large- +distance correlation G = g(r → ∞) as +G ∼ (lnL)−ˆq, +(1) +where the exponent ˆq depends on n, and L denotes the linear +system size. It was also proposed that the helicity modulus Υ, +characterizing the response against a twist in boundary condi- +tions [28], scales as +Υ ∼ 2α +L (lnL), +(2) +where α is a universal renormalization-group parameter. Be- +sides, a scaling relation for ˆq and α is given by [16] +ˆq = n − 1 +2πα . +(3) +∗ Y.S. and M.H. contributed equally to this work. +† jplv2014@ahnu.edu.cn +W +K +0.1 +0.2 +0.3 +0.4 +0.5 +0.1 +0.3 +0.5 +0.7 +0.9 +1.1 +KT-like +Order +Disorder +Quasi Order +Extraordinary-log +3D O(2) +Kc +K +W +Figure 1. A three-dimensional lattice in infinite geometry with a +plane defect (left panel) and the phase diagram for plane defect of +Villain model (right panel). The W and K axes, which represent in- +teractions inside and outside the plane defect, respectively, span the +phase diagram. The phase diagram includes the quasi-long-range or- +dered, ordered, and disordered phases. It also contains, at the bulk +critical point Kc, the plane-defect critical phenomena in bulk O(2) +and extraordinary-log critical universality classes, for W = Kc and +W > Kc, respectively. A critical line of Kosterlitz-Thouless-like +(KT-like) transition is also included. +For the SCB, the existence of extraordinary-log critical phase +and scaling relation (3) were confirmed at n = 3 [29] and +n = 2 [30]. For n = 2, estimates of ˆq and α were also ob- +tained from distinct contexts of O(2) criticality [31–34][35] +(Table I). Hence, the long-standing contradiction about the ex- +traordinary phases of SCB in XY and Heisenberg models has +been reconciled [7, 16]. +It is unknown whether the ELU may be achieved in a wider +context of critical systems beyond SCB, despite the recent ad- +vances [16, 29–34]. Given that the SCB is associated with +systems in semi-infinite geometries, this question is essential. +In the context of O(2) critical systems, where plane defects +are present (Fig. 1), we address the problem. We demonstrate +the presence of ELU in cases where plane defects have strong +intra-plane coupling. The phase diagram shown in Fig. 1 il- +lustrates the main findings. +Plane-defect Villain Model. We start with a plane-defect +arXiv:2301.11720v1 [cond-mat.stat-mech] 27 Jan 2023 + +2 +Table I. For the surface and plane-defect critical phenomena of O(n) +critical systems in semi-infinite and infinite geometries, respectively, +two classes of extraordinary-log universality have been identified. +The critical exponent ˆq and the universal parameter α quantitatively +characterize the extraordinary-log universality. +Surface critical phenomena +Bulk universality +model +ˆq +α +year +O(3) +O(3) φ4 [29] +2.1(2) +0.15(2) +2020 +O(3) φ4 [31] +– +0.190(4) +2021 +O(2) +XY [30] +0.59(2) +0.27(2) +2021 +O(2) φ4 [31] +– +0.300(5) +2021 +Potts [32] +0.60(2) +– +2022 +clock [33] +0.59(1) +0.26(2) +2022 +Villain [36] +0.58(2) +0.28(1) +2022 +Plane-defect critical phenomena (this work) +Bulk universality +model +ˆq +α +year +O(2) +Villain +0.29(2) +0.56(3) +2023 +XY +Villain model with the Hamiltonian +H = 1 +2 +∆J =0 +� +⟨rr′⟩ +J 2 +rr′ +Crr′ +(4) +on simple-cubic lattices, where Crr′ is a variable for the pairs +of nearest-neighbor sites r and r′. Jrr′ ∈ {0, ±1, ±2, ...} pa- +rameterizes the directed flows along bonds. As in the standard +Villain model [37–42], the flows are non-divergent and consti- +tute closed directed loops. Periodic boundary conditions are +imposed in each of the [100] (x), [010] (y), and [001] (z) di- +rections. To involve a plane defect, we specify a plane that is +perpendicular to z direction and contains L2 sites. The values +of Crr′ obey +Crr′ = +� +W +r and r′ ∈ plane defect, +K +r or r′ /∈ plane defect. +(5) +By formulating a worm Monte Carlo algorithm based on the +original algorithm in Ref. [43], we simulate model (4). We de- +sign two update schemes, which are described in Appendix A, +based on biased random walks in an extended state space in- +volving the source and sink of flows. While an update scheme +can run on the entire simple-cubic lattice, the other update +scheme only works in the plane defect. The latter scheme al- +lows the sampling of some quantities that characterize plane- +defect critical phenomena. +Extraordinary-log Critical Phase. +We fix K at Kc = +0.333 067 04, the Villain model’s bulk critical point that was +previously estimated by two of us and our colleagues [44]. +With W = 0.5, 1, 3 and 7, we extensively simulate the strong- +W regime of plane defect. The system size L ranges from 4 +to 128. At L = 128, the total number of generated closed +loops for each W in the worm-algorithm simulations is about +2.9 × 1010. In the plane-defect update scheme, we extract the +two-point correlation g(δx, δy) [g(0, 0) ≡ 1] unbiasedly from +the distribution of the distances (δx, δy) between source and +sink. We define the large-distance two-point correlation G by +G = [g(0, L/2) + g(L/2, 0)]/2. +(6) +If Eq. (1) holds, a fitting ansatz of G reads +G = a0[ln(L/l0)]−ˆq, +(7) +with a0 a non-universal constant and l0 a reference length. +Throughout this paper, we test scaling ansatzes against Monte +Carlo data by least-squares fits. We monitor the evolution of +χ2 with the minimum size Lmin involved in fitting. In prin- +ciple, one searches the smallest Lmin relating to the χ2 per +degree of freedom (DoF) χ2/DoF = O(1), which does not +decrease drastically upon further increasing Lmin. Practically +one prefers the fits with χ2/DoF ≈ 1. We should not trust +a single fit—conclusions will be made by comparing all pre- +ferred fits. For each W, we find that the estimate of ˆq ex- +trapolates to ˆq ≈ 0.29. More precisely, for W = 0.5, 1, +3 and 7, we obtain ˆq = 0.308(2), 0.301(2), 0.289(5) and +0.28(1) as well as χ2/DoF ≈ 2.8, 0.1, 2.8 and 1.5 respec- +tively, with Lmin = 16. Based on these observations, by fix- +ing ˆq = 0.29, we obtain l0 = 3.15(2), 0.684(2), 0.0153(2) +and 0.0000100(3) as well as χ2/DoF ≈ 2.4, 1.4, 0.8 and 0.7, +with Lmin = 48, 32, 64 and 48, for W = 0.5, 1, 3 and 7, re- +spectively. Details of fits are given in Appendix B. Notice that +the estimates of ˆq from various W (W > Kc) are consistent, +indicating the universality of the logarithmic FSS form (7). +In the plane-defect update scheme, the susceptibility χs of +plane defect is sampled via χs = ⟨ns⟩, where ns is the number +of the movements of source and sink between consecutive hits +to the state space of model (4). Borrowing the insights into +FSS from the ELU of SCB, we have the scaling formula +χs = a1L2[ln(L/l0)]−ˆq, +(8) +with a1 a non-universal constant. For W = 0.5, the fit with +Lmin = 16 yields ˆq = 0.322(1) having a huge χ2/DoF +(χ2/DoF ≈ 7.0), which reduces to χ2/DoF ≈ 2.7 with +Lmin = 32 and ˆq = 0.309(3). For W = 1, 3 and 7, we obtain +ˆq = 0.310(1), 0.295(3) and 0.29(1) as well as χ2/DoF ≈ +1.7, 3.2 and 3.0 respectively, with Lmin = 16. +Further, +when ˆq = 0.29 is fixed, we obtain l0 = 4.08(3), 0.866(3), +0.01909(9) and 0.0000119(3) as well as χ2/DoF ≈ 1.1, 0.9, +1.4 and 0.3, with Lmin = 64, 48, 32 and 48, for W = 0.5, +1, 3 and 7, respectively. Similar to that found for G, as W +increases from 0.5 to 7.0, l0 decreases by several orders of +magnitude. Meanwhile, the estimates of ˆq from χs are close +to those from G. These observations confirm the scaling for- +mula (8), which corresponds to a logarithmic scaling of two- +point correlation. +From the FSS analyses of G and χs, we estimate the +value of ˆq as ˆq = 0.29(2). We plot G and χsL−2 versus +ln(L/l0) in Figs. 2(a) and (b) respectively, where l0 takes +above-mentioned values from preferred fits. +The mutually +consistent scaling forms and critical exponents are hence il- +lustrated. +We analyze the helicity modulus Υ, which is defined +through the fluctuations of winding number Wx of directed + +3 +2 +4 +6 +8 +10 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +0.56 +(c) Linear plot of ϒL vs. lnL +0.2 +0.3 +0.5 +0.8 +1.4 +0.8 +1.5 +3 +5 +10 +20 +-0.29 +(b) +Log-log plot of χsL-2 vs. ln(L/l0) +0.2 +0.3 +0.5 +0.8 +1.4 +0.8 +1.5 +3 +5 +10 +20 +-0.29 +(a) +Log-log plot of G vs. ln(L/l0) +Plane-defect Villain model + W = 0.5 + W = 1.0 + W = 3.0 + W = 7.0 +Figure 2. The two-point correlation G (a), the scaled susceptibility χsL−2 (b) and the scaled helicity modulus ΥL (c) for the extraordinary-log +critical phase of the plane-defect Villain model. In panels (a) and (b), the horizontal coordinates are written as ln(L/l0), where l0 are taken +from preferred least-squares fits, and the plots are further made on log-log scales. The slopes −0.29 and 0.56 of dashed lines stand for −ˆq and +α, respectively. +flows along a periodic direction (say x direction), +Υ = ⟨W2 +x⟩/L. +(9) +As indicated in Ref. [36], such an estimator of Υ is effective in +the analyses of ELU. For an extraordinary-log critical phase, +Υ is expected to scale as ΥL ∝ lnL. This behavior is roughly +illustrated by the Monte Carlo data of Υ in Fig. 2(c). Further, +we perform least-squares fits to +ΥL = α(lnL) + b + cL−ω, +(10) +where α is a universal parameter, whereas b and c are non- +universal. Unlike the scaling form ΥL ∼ 2α(lnL) [Eq. (2)], +which applies to the SCB that involves two open surfaces, the +prefactor 2 is now removed due to the uniqueness of plane +defect. ω denotes the exponent of leading finite-size correc- +tions. Accordingly, we estimate α by the least-squares fits to +Eq. (10). We adopt ω = 0.789, considering bulk irrelevant +fields [45, 46], and compare the fits with those using ω = 1. +We find that the inclusion of correction term stabilizes the fit- +ting. For W = 0.5, 1, 3 and 7, we obtain α = 0.555(3), +0.562(4), 0.580(6) and 0.57(1) as well as χ2/DoF ≈ 0.2, +0.7, 2.0 and 2.2 respectively, with Lmin = 16. Comparing the +fits, we estimate α = 0.56(3). +There is a lack of a known critical universality class that the +values of ˆq and α fall into. The scaling relation between ˆq and +α is then investigated. We test the scaling relation (3) with +n = 2, which was previously verified merely for SCB (Ta- +ble I). Figure 3 displays the above-obtained fitting results of ˆq +and α versus χ2/DoF. Next, from each of the estimates of ˆq +and α, via the equation αˆq = 1/(2π), we obtain an estimate +of α and ˆq, respectively. The estimates converted through the +equation are also included in Fig. 3. We note that the results +of ˆq and α from direct fits and conversions are consistent. For +χ2/DoF ≈ 1, such consistency is apparent. Hence, we obtain +strong evidence of the scaling relation αˆq = 1/(2π) for an +ELU of plane-defect critical phenomena. +0.1 +0.2 +0.3 +0.5 +0.7 +0.9 + 0.1 + 1 + 10 + 100 +Plane-defect Villain model +α = 0.56(3) +q^ = 0.29(2) +χ2/DoF +q^ = 1/(2πα) +α = 1/(2πq^) +Plane-defect Villain model +q^ +α +Figure 3. +Test of the scaling relation αˆq += +1/(2π) for the +extraordinary-log critical phase of the plane-defect Villain model. +Circles denote the results from least-squares fits to FSS formulae, +whereas squares denote the results converted from fitting results via +αˆq = 1/(2π). The estimated critical exponent ˆq = 0.29(2) and uni- +versal parameter α = 0.56(3) are denoted by the shadows centered +at the red and black lines, respectively. +Phase Diagram. We have demonstrated that ELU exists +in a strong-W regime with K = Kc. One also expects the +occurrence of a transition at W = Kc and K = Kc in the 3D +O(2) universality class. +We proceed to investigate critical phenomena on the plane +defect for K < Kc. +Figure 4(a) shows ΥL versus W at +K = 0.2. For W ⪆ 0.7, ΥL tends to be independent of L +in the L → ∞ limit and extrapolates to a W-dependent non- +trivial value. These observations indicate a critical phase for +the plane defect. As shown in Fig. 4(b), at W ≈ 0.73, G + +4 +2 +4 +6 +8 +10 +0.05 +0.15 +0.25 +0.35 +Kc +(c) W = 1.0 +ϒL +K +1.0 +1.1 +1.2 +0.05 +0.15 +0.25 +0.35 +Kc +(d) W = 1.0 + GL0.16 +K +0.4 +0.8 +1.2 +0.2 +0.5 +0.8 +1.1 +WKT +(a) +K = 0.2 +ϒL +W +0.89 +0.91 +0.93 +0.95 +0.97 +0.71 +0.73 +0.75 +WKT +(b) +K = 0.2 + GL1/4 +W +Plane-defect Villain model +L = 4 +L = 48 +L = 8 +L = 64 +L = 16 +L = 96 +L = 32 +L = 128 +Figure 4. Critical phenomena of the plane-defect Villain model with +K < Kc. The scaled helicity modulus ΥL (a) and the scaled two- +point correlation GL1/4 (b) versus W for K = 0.2. ΥL (c) and +GL0.16 (d) versus K for W = 1.0. Dashed lines represent phase +transition points. +scales as G ∝ L−η with η = 1/4, which is reminiscent of the +anomalous dimension of the Kosterlitz-Thouless transition. It +is natural to ask whether there is a critical phase that fea- +tures quasi-long-range order with 0 < η < 1/4. Figure 4(c) +demonstrates, with W = 1, that ΥL is invariant for K < Kc. +Meanwhile, the exponent η is nearly invariant (η ≈ 0.16) in +the parameter regime [Fig. 4(d)]. Our results suggest that, as +K → 0, the plane-defect criticality reduces drastically to an +essential 2D behavior. The phase diagram, shown in Fig. 1, is +then established. It calls for a sophisticated renormalization- +group study to reveal the distribution and properties of fixed +points. +Other O(2) Critical Systems. The relevance of O(2) crit- +icality to various systems of interest determines its signif- +icance [25]. +One may ask whether the current findings +hold true for a different O(2) critical system. +Here, we +study a plane-defect critical XY model and investigate the +extraordinary-log critical phase. +The model Hamiltonian +reads +H = − +� +⟨rr′⟩ +Crr′ ⃗Sr · ⃗Sr′ , +(11) +where ⃗Sr = (Sa +r , Sb +r) represents 2D vectors of unit length. +As defined by Eq. (5), Crr′ denotes the coupling strength for +sites r and r′. We set K = Kc, where Kc = 1/2.2018441 +is the bulk critical point obtained in Ref. [44]. +We simu- +late the model with W = 1, 3 and 7 up to the linear size +L = 128, using a cluster Monte Carlo algorithm [47]. We +sample the helicity modulus Υ = +1 +L3 (⟨E⟩ − ⟨T 2⟩) with E = +� +r Cr(r+ex)⃗Sr · ⃗Sr+ex and T = � +r Cr(r+ex)(Sa +r Sb +r+ex − +2 +7 +12 +17 +0.5 +0.7 +0.9 +2 +3 +4 +5 +1 +4 +7 +10 +G +ln(L/l +0 +) + Plane-defect XY model +W=1 +W=3 +W=7 +Log-log plot of G vs. ln(L/l +0 +) + -0.29 +(a) +(b) +�L +lnL +Linear plot of �L vs. lnL + 0.56 +Figure 5. The two-point correlation G (a) and the scaled helicity +modulus ΥL (b) for the extraordinary-log critical phase of the plane- +defect XY model. In panel (a), the horizontal coordinate is written +as ln(L/l0), where l0 = 1.240 (W = 1), 0.0219 (W = 3) and +0.0000146 (W = 7) are taken from preferred least-squares fits, and +the plot is further made on a log-log scale. The slopes −0.29 and +0.56 of dashed lines stand for −ˆq and α, respectively. +Sb +rSa +r+ex), as well as the two-point correlation G = ⟨⃗Sr · ⃗Sr′⟩ +with r′ − r = (L/2, 0). For G and Υ, we perform FSS anal- +yses according to Eqs. (7) and (10), respectively. Figure 5 +illustrates the estimates of ˆq and α, which are compatible with +those of the plane-defect Villain model. A specially designed +χ2 test for scaling relation of ˆq and α as well as the details of +fits are presented in Appendix C. +Discussion. In the context of O(2) critical systems, we find +that the ELU exists in plane defects of infinite-geometry lat- +tices. The conclusion is based on extensive Monte Carlo sim- +ulations of two models in a broad strong-coupling regime of +plane defects, where the extraordinary-log critical phase fea- +tures logarithmic scaling forms of two-point correlation and +helicity modulus. The critical exponent ˆq = 0.29(2) and the +universal parameter α = 0.56(3) quantitatively characterize +the scaling forms. Both ˆq and α are far from those of SCB, +indicating the existence of a new critical universality class. +Surprisingly, the scaling relation (3) is likely to be valid. As +a result, the current findings strongly suggest that ELU may +exist in a wider range of many-body systems than previously +thought. In comparison to SCB, the ˆq value of plane-defect +criticality is relatively small, leaving more room for the dis- +covery of ELU in O(n) models with n > 2. +Plane defects in O(2) critical bulks could serve as a platform +for exotic critical phenomena, inspiring numerous follow-up +studies. Here, we limit ourselves to quantum ELU. Theoret- +ically, a model of quantum ELU is precious. On the experi- +mental side, a realization of ELU is awaited. A close classical- +quantum mapping exists between the plane-defect Villain + +5 +model and the quantum rotor and Bose-Hubbard models. On +this basis, a possible pathway to realizing quantum ELU is +the 2D quantum Hamiltonian ˆH = − � +⟨rr′⟩ trr′(ˆΦ† +r ˆΦr′ + +ˆΦr ˆΦ† +r′) + U +2 +� +r ˆn2 +r, where ˆΦ† +r, ˆΦr and ˆnr are respectively +the bosonic creation, annihilation and particle number opera- +tors at site r, U represents onsite repulsion, and trr′ denotes +the bond-dependent hopping strength realizing an edge de- +fect. Recent advances in quantum simulations with ultra-cold +bosons in optical lattices [48–50] enable the experimental re- +alization of quantum ELU. +Note added. While we were finalizing this paper, an in- +dependent work appeared on the ArXiv [51]. +Based on a +renormalization-group theory, this work predicts the existence +of ELU for a plane defect in the O(n) model. +ACKNOWLEDGMENTS +This work has been supported by the National Natural Sci- +ence Foundation of China (Grant No. 12275002). +[1] H. E. 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A core part of the simulations relies on iteratively carrying out two update schemes, Algorithm 1 +and Algorithm 2, which run in the whole simple-cubic lattice and the plane defect, respectively. +Once the source-sink points collide (I = M), a closed loop of directed flows is superposed on the directed-flow configuration. +After producing a permanent quantity of closed loop(s), the update process shifts between Algorithm 1 and Algorithm 2. Even +if Algorithm 1 itself guarantees ergodicity, we include Algorithm 2 for the efficient simulations in plane defect as well as the +sampling of some quantities that characterize plane-defect critical phenomena. +Algorithm 1 Update in simple-cubic lattice +1. If I = M, randomly pick up a lattice site I′ in the simple-cubic lattice, with the probability 1/L3 for each site. Let I = M = I′, +sign(I) = 1, sign(M) = −1. +2. Interchange I ↔ M and sign(I) ↔ sign(M) with the probability 1/2. +3. Randomly pick up a neighbor In of I, with the probability 1/6 for each of the neighbors. +4. Propose to simultaneously move I → In and update the flow JIIn to J ′ +IIn: +J ′ +IIn = JIIn + df(I → In)sign(I), +where df(I → In) = ±1 is a fixed parameter, quantifying the direction of flow along bond-IIn. +5. Accept the proposed change with the probability +p = +� +� +� +min[1, e +−(J ′2 +IIn −J 2 +IIn ) +2W +] +I and In ∈ plane defect, +min[1, e +−(J ′2 +IIn −J 2 +IIn ) +2K +] +I or In /∈ plane defect. +Algorithm 2 Update in plane defect +1. If I = M, randomly pick up a lattice site I′ in the plane defect, with the probability 1/L2 for each site. Let I = M = I′, sign(I) = 1, +sign(M) = −1. +2. The same as Algorithm 1. +3. Randomly pick up a neighbor In of I in the plane defect, with the probability 1/4 for each of the neighbors. +4. The same as Algorithm 1. +5. Accept the proposed change with the probability +p = min[1, e +−(J ′2 +IIn −J 2 +IIn ) +2W +]. +Appendix B: Extraordinary-log critical phase of the plane-defect Villain model +We simulate the extraordinary-log critical phase of plane-defect Villain model using the worm algorithm, with the lattice sizes +L = 4, 8, 16, 32, 48, 64, 96 and 128. The number of generated closed loops ranges from 9.1 × 108 to 7.3 × 109 for L ≤ 32 and +ranges from 1.1 × 1010 to 2.9 × 1010 for 48 ≤ L ≤ 128. The first one sixth of the generated closed loops are utilized for the +thermalization in each Markov chain. +Here, we analyze the FSS for the extraordinary-log critical phase with W = 0.5, 1, 3 and 7 at K = Kc = 0.333 067 04. +According to a standard criterion, we prefer the fits with χ2/DoF ≈ 1 and conclude by comparing the fits that are stable against +gradually increasing Lmin, which is the size of the minimum system involved. To supplement the presentation in the main text, +we provide more details about the FSS analyses of G, χs, and Υ. The data of these quantities are fitted to the scaling formulae +G = a0[ln(L/l0)]−ˆq, +(S1) +χs = a1L2[ln(L/l0)]−ˆq, +(S2) +and +ΥL = α(lnL) + b + cL−ω, +(S3) + +2 +0.3 +0.5 +0.8 +1.1 +0.3 +0.7 +1.5 +3 +5 +10 +20 +Plane-defect Villain model +Log-log plot of G′ vs. ln(L/l0) + W = 0.5 + W = 1.0 + W = 3.0 + W = 7.0 + -0.29 +Figure S1. Two-point correlation G′ versus ln(L/l0) for the plane-defect Villain model, where l0 are taken from preferred least-squares fits. +The plot is further made on a log-log scale. The slope −0.29 of dashed lines stands for −ˆq. +respectively, where ˆq and α are universal parameters, and ω is a correction exponent. l0, a0, a1, b and c represent non-universal +constants. The results of least-squares fits are presented in Tables S1, S2 and S3, for G, χs and Υ, respectively. +In addition to the quantities discussed in the main text, we take into consideration the two-point correlation G′ for the distances +(δx, δy) = (L/4, 0) and (δx, δy) = (0, L/4), which is defined by +G′ = [g(0, L/4) + g(L/4, 0)]/2. +(S4) +For the extraordinary-log critical phase at W = 0.5, 1, 3 and 7, we perform FSS analyses according to +G′ = a0[ln(L/l0)]−ˆq. +(S5) +The results of fits are summarized in Table S4. Using the results of l0 from preferred fits, we plot G′ versus ln(L/l0) in Fig. S1. +The leading FSS behavior of G′ is similar to that of G. +Appendix C: Extraordinary-log critical phase of the plane-defect XY model +We simulate the extraordinary-log critical phase of plane-defect XY model using the Wolff cluster algorithm, with the lattice +sizes L = 8, 16, 32, 48, 64, 96 and 128. The total number of Wolff steps ranges from 1.6 × 108 to 6.4 × 108 for L ≤ 32 and +ranges from 2.0 × 108 to 9.6 × 108 for 48 ≤ L ≤ 128. The first one sixth of the Wolff Monte Carlo steps are utilized for the +thermalization in each Markov chain. +We consider the coupling strengths W = 1, 3 and 7 of plane defect at K = Kc = 1/2.2018441. The data for G and Υ are +fitted according to Eqs. (S1) and (S3), respectively. Tables S5 and S6 provide a summary of the outcomes of least-squares fits. +Using the results of ˆq and α from least-squares fits, a test of the scaling relation αˆq = 1/(2π) is given by Fig. S2. + +3 + Plane-defect XY model + + a + =1/(2pa) + a =1/(2p ) +0.01 +0.1 +1 +10 +100 +0.1 +0.6 +1.1 +q +q +q +q +c +2 +/DoF +a + +=0.56(3) +=0.29(2) +q +Figure S2. Test of the scaling relation αˆq = 1/(2π) for the extraordinary-log critical phase of the plane-defect XY model. Circles denote +the results from least-squares fits to FSS formulae, whereas squares denote the results converted from fitting results via αˆq = 1/(2π). The +estimated critical exponent ˆq = 0.29(2) and universal parameter α = 0.56(3) are denoted by the shadows centered at the red and black lines, +respectively. + +4 +Table S1. Fits of the two-point correlation G to G = a0[ln(L/l0)]−ˆq for the extraordinary-log critical phase of plane-defect Villain model at +W = 0.5, 1, 3 and 7. We investigate both cases where ˆq is free and fixed at 0.29. +W +Lmin +Lmax +χ2/DoF +a0 +l0 +ˆq +0.5 +4 +128 +3119.51/5 +0.5644(5) +1.535(4) +0.3728(5) +8 +114.42/4 +0.5116(8) +2.22(1) +0.3247(9) +16 +8.48/3 +0.494(2) +2.59(4) +0.308(2) +32 +7.97/2 +0.490(5) +2.7(1) +0.304(5) +48 +4.72/1 +0.47(1) +3.3(4) +0.29(1) +4 +47488.48/6 +0.49014(3) +2.2704(8) +0.29 +8 +2223.24/5 +0.47999(5) +2.815(3) +0.29 +16 +125.62/4 +0.47685(8) +3.016(5) +0.29 +32 +17.53/3 +0.4756(1) +3.10(1) +0.29 +48 +4.85/2 +0.4750(2) +3.15(2) +0.29 +64 +4.30/1 +0.4748(3) +3.16(3) +0.29 +1.0 +4 +128 +345.88/5 +1.001(1) +0.418(3) +0.3241(5) +8 +34.02/4 +0.968(2) +0.500(6) +0.3105(9) +16 +0.34/3 +0.945(4) +0.57(1) +0.301(2) +32 +0.14/2 +0.94(1) +0.59(4) +0.299(5) +48 +0.0001/1 +0.93(3) +0.6(1) +0.30(1) +4 +5471.78/6 +0.92571(5) +0.6099(4) +0.29 +8 +649.07/5 +0.92218(7) +0.6463(7) +0.29 +16 +42.88/4 +0.9198(1) +0.674(1) +0.29 +32 +4.16/3 +0.9189(2) +0.684(2) +0.29 +48 +0.23/2 +0.9186(3) +0.689(3) +0.29 +64 +0.04/1 +0.9184(4) +0.690(5) +0.29 +3.0 +4 +128 +9.97/5 +1.518(6) +0.0138(5) +0.295(1) +8 +9.36/4 +1.51(1) +0.0143(8) +0.293(2) +16 +8.47/3 +1.49(2) +0.016(2) +0.289(5) +32 +0.09/2 +1.40(3) +0.030(7) +0.268(8) +48 +0.09/1 +1.39(7) +0.03(2) +0.27(2) +4 +21.33/6 +1.4974(1) +0.01546(4) +0.29 +8 +11.58/5 +1.4970(2) +0.01557(5) +0.29 +16 +8.49/4 +1.4967(3) +0.01567(8) +0.29 +32 +6.86/3 +1.4964(3) +0.0158(1) +0.29 +48 +1.60/2 +1.4973(5) +0.0155(2) +0.29 +64 +0.81/1 +1.4978(8) +0.0153(2) +0.29 +7.0 +4 +128 +4.56/5 +1.89(3) +0.000025(5) +0.273(4) +8 +4.50/4 +1.88(5) +0.00003(1) +0.272(7) +16 +4.38/3 +1.91(8) +0.00002(1) +0.28(1) +32 +1.22/2 +1.6(1) +0.0002(2) +0.24(2) +48 +0.94/1 +1.8(3) +0.0001(2) +0.26(5) +4 +18.97/6 +2.0108(3) +0.0000109(1) +0.29 +8 +10.24/5 +2.0118(5) +0.0000106(1) +0.29 +16 +5.71/4 +2.0125(6) +0.0000104(2) +0.29 +32 +5.61/3 +2.0127(8) +0.0000103(2) +0.29 +48 +1.30/2 +2.014(1) +0.0000100(3) +0.29 +64 +0.31/1 +2.015(2) +0.0000096(4) +0.29 + +5 +Table S2. Fits of the susceptibility χs to χs = a1L2[ln(L/l0)]−ˆq for the extraordinary-log critical phase of plane-defect Villain model at +W = 0.5, 1, 3 and 7. +W +Lmin +Lmax +χ2/DoF +a1 +l0 +ˆq +0.5 +4 +128 +11202.68/5 +0.6290(5) +1.354(3) +0.4267(4) +8 +732.74/4 +0.5423(7) +2.252(9) +0.3550(6) +16 +21.14/3 +0.507(1) +2.95(3) +0.322(1) +32 +5.31/2 +0.493(4) +3.3(1) +0.309(3) +48 +3.76/1 +0.483(8) +3.7(3) +0.299(8) +4 +215291.53/6 +0.49828(2) +2.4563(5) +0.29 +8 +15937.48/5 +0.48279(4) +3.348(2) +0.29 +16 +866.04/4 +0.47686(6) +3.780(4) +0.29 +32 +38.33/3 +0.4746(1) +3.972(8) +0.29 +48 +5.28/2 +0.4738(2) +4.04(1) +0.29 +64 +1.10/1 +0.4734(3) +4.08(3) +0.29 +1.0 +4 +128 +2091.94/5 +1.077(1) +0.356(2) +0.3543(5) +8 +198.77/4 +1.007(2) +0.503(4) +0.3269(7) +16 +5.21/3 +0.964(3) +0.64(1) +0.310(1) +32 +3.18/2 +0.952(9) +0.69(4) +0.305(4) +48 +1.51/1 +0.93(2) +0.8(1) +0.295(8) +4 +27600.22/6 +0.92927(4) +0.7105(4) +0.29 +8 +3487.84/5 +0.92361(5) +0.7773(6) +0.29 +16 +244.87/4 +0.91951(9) +0.832(1) +0.29 +32 +20.79/3 +0.9178(1) +0.857(2) +0.29 +48 +1.85/2 +0.9172(2) +0.866(3) +0.29 +64 +1.85/1 +0.9172(3) +0.866(5) +0.29 +3.0 +4 +128 +146.26/5 +1.661(6) +0.0076(2) +0.323(1) +8 +25.48/4 +1.581(9) +0.0117(5) +0.308(2) +16 +9.46/3 +1.52(2) +0.016(1) +0.295(3) +32 +0.47/2 +1.44(3) +0.027(5) +0.277(6) +48 +0.41/1 +1.45(6) +0.03(1) +0.28(1) +4 +1088.78/6 +1.5013(1) +0.01733(3) +0.29 +8 +138.99/5 +1.4984(1) +0.01834(5) +0.29 +16 +12.08/4 +1.4967(2) +0.01894(7) +0.29 +32 +4.14/3 +1.4963(2) +0.01909(9) +0.29 +48 +0.93/2 +1.4967(3) +0.0189(1) +0.29 +64 +0.04/1 +1.4972(6) +0.0188(2) +0.29 +7.0 +4 +128 +21.84/5 +2.19(3) +0.0000039(8) +0.312(4) +8 +9.59/4 +2.03(5) +0.000011(3) +0.293(6) +16 +8.91/3 +1.99(7) +0.000015(8) +0.29(1) +32 +1.76/2 +1.7(1) +0.0002(2) +0.24(2) +48 +0.45/1 +1.9(3) +0.00002(6) +0.28(4) +4 +53.03/6 +2.0149(3) +0.00001168(8) +0.29 +8 +9.80/5 +2.0132(4) +0.0000122(1) +0.29 +16 +9.04/4 +2.0130(4) +0.0000123(1) +0.29 +32 +8.42/3 +2.0127(6) +0.0000124(2) +0.29 +48 +0.52/2 +2.0141(8) +0.0000119(3) +0.29 +64 +0.21/1 +2.015(1) +0.0000118(4) +0.29 + +6 +Table S3. Fits of the helicity modulus Υ to ΥL = α(lnL) + b + cL−ω for the extraordinary-log critical phase of plane-defect Villain model +at W = 0.5, 1, 3 and 7. The symbol ‘-’ means that the leading correction term is not involved. +W +Lmin +Lmax +χ2/DoF +α +b +c +ω +0.5 +16 +128 +805.87/4 +0.4780(5) +-0.004(2) +- +- +32 +55.14/3 +0.503(1) +-0.108(4) +- +- +48 +5.29/2 +0.512(2) +-0.148(7) +- +- +64 +0.16/1 +0.516(3) +-0.17(1) +- +- +4 +134.68/5 +0.5362(8) +-0.298(4) +1.254(9) +0.789 +8 +6.96/4 +0.549(1) +-0.358(6) +1.47(2) +0.789 +16 +0.52/3 +0.555(3) +-0.39(1) +1.60(6) +0.789 +32 +0.51/2 +0.554(7) +-0.38(4) +1.6(2) +0.789 +48 +0.35/1 +0.55(2) +-0.35(9) +1.3(6) +0.789 +4 +347.10/5 +0.5138(7) +-0.177(3) +1.326(9) +1 +8 +31.24/4 +0.530(1) +-0.250(5) +1.73(2) +1 +16 +0.69/3 +0.541(2) +-0.30(1) +2.12(7) +1 +32 +0.35/2 +0.544(6) +-0.32(3) +2.3(3) +1 +48 +0.30/1 +0.54(1) +-0.30(7) +2.1(9) +1 +1.0 +16 +128 +124.26/4 +0.5207(8) +0.633(3) +- +- +32 +13.83/3 +0.534(1) +0.578(6) +- +- +48 +0.69/2 +0.540(2) +0.55(1) +- +- +64 +0.65/1 +0.539(3) +0.55(2) +- +- +4 +8.56/5 +0.554(1) +0.462(5) +0.74(1) +0.789 +8 +3.69/4 +0.558(2) +0.44(1) +0.80(3) +0.789 +16 +2.12/3 +0.562(4) +0.42(2) +0.90(8) +0.789 +32 +2.01/2 +0.565(9) +0.41(5) +1.0(3) +0.789 +48 +0.68/1 +0.54(2) +0.5(1) +0.1(8) +0.789 +4 +31.92/5 +0.541(1) +0.532(4) +0.79(1) +1 +8 +7.88/4 +0.548(2) +0.502(8) +0.95(4) +1 +16 +2.31/3 +0.554(3) +0.47(2) +1.2(1) +1 +32 +1.89/2 +0.559(7) +0.45(4) +1.5(4) +1 +48 +0.68/1 +0.54(2) +0.54(9) +0.2(1.3) +1 +3.0 +16 +128 +32.79/4 +0.548(2) +2.611(7) +- +- +32 +3.59/3 +0.557(2) +2.574(9) +- +- +48 +3.54/2 +0.556(4) +2.58(2) +- +- +64 +1.01/1 +0.562(5) +2.55(2) +- +- +4 +15.00/5 +0.574(2) +2.47(1) +0.68(2) +0.789 +8 +6.32/4 +0.582(4) +2.43(2) +0.83(5) +0.789 +16 +6.10/3 +0.580(6) +2.45(3) +0.8(1) +0.789 +32 +3.56/2 +0.56(1) +2.56(8) +0.1(5) +0.789 +48 +2.33/1 +0.59(4) +2.4(2) +1.5(1.4) +0.789 +4 +26.34/5 +0.562(2) +2.539(8) +0.73(2) +1 +8 +5.69/4 +0.573(3) +2.49(1) +1.00(7) +1 +16 +5.67/3 +0.574(5) +2.49(3) +1.0(2) +1 +32 +3.57/2 +0.56(1) +2.57(6) +0.1(7) +1 +48 +2.27/1 +0.59(3) +2.4(2) +2.4(2.1) +1 +7.0 +16 +128 +10.00/4 +0.547(3) +6.64(1) +- +- +32 +8.71/3 +0.551(4) +6.63(2) +- +- +48 +0.85/2 +0.566(7) +6.56(3) +- +- +64 +0.56/1 +0.56(1) +6.58(5) +- +- +4 +6.63/5 +0.574(4) +6.51(2) +0.62(5) +0.789 +8 +6.63/4 +0.573(7) +6.51(3) +0.6(1) +0.789 +16 +6.52/3 +0.57(1) +6.53(6) +0.5(3) +0.789 +32 +2.09/2 +0.63(3) +6.2(2) +2.4(9) +0.789 +48 +0.82/1 +0.55(7) +6.6(4) +-0.5(2.7) +0.789 +4 +7.21/5 +0.563(4) +6.56(2) +0.66(5) +1 +8 +6.75/4 +0.566(6) +6.55(3) +0.7(1) +1 +16 +6.72/3 +0.56(1) +6.56(5) +0.7(4) +1 +32 +2.00/2 +0.61(2) +6.3(1) +3.5(1.3) +1 +48 +0.81/1 +0.55(6) +6.6(3) +-0.8(4.2) +1 + +7 +Table S4. Fits of the two-point correlation G′ to G′ = a0[ln(L/l0)]−ˆq for the extraordinary-log critical phase of plane-defect Villain model at +W = 0.5, 1, 3 and 7. +W +Lmin +Lmax +χ2/DoF +a0 +l0 +ˆq +0.5 +4 +128 +26009.53/5 +0.6413(4) +1.454(2) +0.4366(4) +8 +596.25/4 +0.5356(5) +2.638(8) +0.3471(5) +16 +12.53/3 +0.509(1) +3.23(3) +0.322(1) +32 +7.06/2 +0.502(3) +3.43(9) +0.315(3) +48 +3.72/1 +0.490(7) +3.9(3) +0.304(7) +4 +359734.02/6 +0.50273(2) +2.6074(4) +0.29 +8 +18915.26/5 +0.48562(3) +3.632(2) +0.29 +16 +1182.19/4 +0.47942(5) +4.107(4) +0.29 +32 +92.33/3 +0.47727(8) +4.293(7) +0.29 +48 +7.96/2 +0.4762(1) +4.39(1) +0.29 +64 +0.40/1 +0.4757(2) +4.44(2) +0.29 +1.0 +4 +128 +4332.24/5 +1.082(1) +0.415(2) +0.3568(5) +8 +57.61/4 +0.978(2) +0.684(5) +0.3153(7) +16 +2.24/3 +0.957(3) +0.77(1) +0.307(1) +32 +1.07/2 +0.949(9) +0.81(4) +0.303(4) +48 +0.34/1 +0.93(2) +0.9(1) +0.296(8) +4 +36548.61/6 +0.93147(4) +0.8081(4) +0.29 +8 +1746.29/5 +0.92372(6) +0.9088(7) +0.29 +16 +179.34/4 +0.92094(9) +0.950(1) +0.29 +32 +14.67/3 +0.9193(2) +0.976(2) +0.29 +48 +1.00/2 +0.9188(2) +0.985(3) +0.29 +64 +0.77/1 +0.9187(3) +0.987(5) +0.29 +3.0 +4 +128 +314.26/5 +1.611(5) +0.0121(3) +0.314(1) +8 +4.19/4 +1.490(7) +0.0234(9) +0.289(2) +16 +4.16/3 +1.49(1) +0.024(2) +0.288(3) +32 +3.23/2 +1.46(3) +0.028(5) +0.282(7) +48 +2.08/1 +1.53(8) +0.019(8) +0.30(2) +4 +904.58/6 +1.49952(9) +0.02125(3) +0.29 +8 +4.90/5 +1.4962(1) +0.02261(6) +0.29 +16 +4.47/4 +1.4963(2) +0.02257(8) +0.29 +32 +4.47/3 +1.4963(3) +0.0226(1) +0.29 +48 +2.25/2 +1.4967(4) +0.0224(2) +0.29 +64 +2.10/1 +1.4969(6) +0.0223(3) +0.29 +7.0 +4 +128 +29.95/5 +1.95(2) +0.000024(4) +0.281(3) +8 +12.69/4 +1.81(4) +0.00007(2) +0.261(5) +16 +5.42/3 +1.96(8) +0.00002(1) +0.28(1) +32 +1.99/2 +1.7(1) +0.0001(1) +0.25(2) +48 +0.22/1 +2.1(5) +0.00001(2) +0.30(6) +4 +35.54/6 +2.0106(3) +0.0000155(1) +0.29 +8 +35.45/5 +2.0105(4) +0.0000156(2) +0.29 +16 +5.87/4 +2.0121(5) +0.0000150(2) +0.29 +32 +5.82/3 +2.0120(7) +0.0000150(3) +0.29 +48 +0.28/2 +2.014(1) +0.0000143(4) +0.29 +64 +0.28/1 +2.014(1) +0.0000143(6) +0.29 + +8 +Table S5. Fits of the two-point correlation G to G = a0[ln(L/l0)]−ˆq for the extraordinary-log critical phase of plane-defect XY model at +W = 1, 3 and 7. +W +Lmin +Lmax +χ2/DoF +a0 +l0 +ˆq +1.0 +8 +128 +30.30/4 +0.7730(7) +1.059(5) +0.3022(4) +16 +10.31/3 +0.767(1) +1.10(1) +0.2990(8) +32 +0.53/2 +0.756(4) +1.20(3) +0.293(2) +48 +0.12/1 +0.76(1) +1.1(1) +0.297(7) +8 +935.43/5 +0.75322(4) +1.1966(7) +0.29 +16 +137.58/4 +0.75200(6) +1.222(1) +0.29 +32 +1.95/3 +0.75114(9) +1.240(2) +0.29 +48 +1.10/2 +0.7510(1) +1.242(3) +0.29 +64 +0.53/1 +0.7509(3) +1.247(7) +0.29 +3.0 +8 +128 +6.19/4 +1.429(4) +0.0264(6) +0.2842(9) +16 +6.19/3 +1.429(7) +0.027(1) +0.284(2) +32 +1.79/2 +1.39(2) +0.033(4) +0.276(4) +48 +0.03/1 +1.33(5) +0.05(2) +0.26(1) +8 +43.42/5 +1.45397(8) +0.02298(3) +0.29 +16 +17.98/4 +1.4544(1) +0.02281(5) +0.29 +32 +12.23/3 +1.4548(2) +0.02266(8) +0.29 +48 +5.41/2 +1.4553(3) +0.0224(1) +0.29 +64 +1.38/1 +1.4564(6) +0.0219(3) +0.29 +7.0 +8 +128 +4.47/4 +1.88(2) +0.000034(4) +0.274(2) +16 +2.97/3 +1.90(3) +0.000028(6) +0.278(4) +32 +2.23/2 +1.96(8) +0.00002(1) +0.29(1) +48 +1.89/1 +2.1(3) +0.00001(1) +0.30(4) +8 +52.38/5 +1.9919(1) +0.00001530(6) +0.29 +16 +12.04/4 +1.9928(2) +0.00001495(8) +0.29 +32 +2.38/3 +1.9936(3) +0.0000146(1) +0.29 +48 +2.07/2 +1.9938(5) +0.0000146(2) +0.29 +64 +1.09/1 +1.993(1) +0.0000149(4) +0.29 + +9 +Table S6. Fits of the helicity modulus Υ to ΥL = α(lnL) + b + cL−ω for the extraordinary-log critical phase of plane-defect XY model at +W = 1, 3 and 7. +W +Lmin +Lmax +χ2/DoF +α +b +c +ω +1.0 +8 +128 +1202.17/5 +0.4597(5) +0.510(1) +- +- +16 +36.92/4 +0.498(1) +0.393(4) +- +- +32 +4.22/3 +0.520(4) +0.31(1) +- +- +48 +2.55/2 +0.531(10) +0.26(4) +- +- +64 +0.02/1 +0.56(2) +0.14(9) +- +- +8 +1.83/4 +0.560(3) +0.11(1) +1.01(3) +0.789 +16 +1.81/3 +0.56(1) +0.10(5) +1.0(2) +0.789 +32 +1.72/2 +0.57(3) +0.1(2) +1.3(8) +0.789 +48 +0.31/1 +0.7(1) +-0.6(6) +5.7(38) +0.789 +8 +2.69/4 +0.543(2) +0.197(9) +1.14(3) +1 +16 +1.94/3 +0.550(9) +0.16(4) +1.3(2) +1 +32 +1.76/2 +0.56(3) +0.1(1) +1.8(12) +1 +48 +0.28/1 +0.66(9) +-0.4(5) +8.7(58) +1 +3.0 +8 +128 +585.80/5 +0.5086(5) +2.533(1) +- +- +16 +18.72/4 +0.534(1) +2.454(3) +- +- +32 +8.07/3 +0.546(4) +2.41(1) +- +- +48 +5.22/2 +0.562(10) +2.35(4) +- +- +64 +5.11/1 +0.56(2) +2.37(9) +- +- +8 +6.19/4 +0.575(3) +2.27(1) +0.66(3) +0.789 +16 +6.08/3 +0.57(1) +2.28(5) +0.6(2) +0.789 +32 +5.14/2 +0.60(3) +2.1(2) +1.4(8) +0.789 +48 +5.13/1 +0.6(1) +2.2(6) +1.1(38) +0.789 +8 +6.27/4 +0.564(2) +2.326(9) +0.74(3) +1 +16 +6.25/3 +0.565(9) +2.32(4) +0.8(2) +1 +32 +5.16/2 +0.59(3) +2.2(1) +1.9(11) +1 +48 +5.15/1 +0.58(9) +2.2(5) +1.5(57) +1 +7.0 +8 +128 +557.53/5 +0.5207(5) +6.530(1) +- +- +16 +13.86/4 +0.546(1) +6.453(3) +- +- +32 +8.58/3 +0.554(4) +6.42(1) +- +- +48 +1.31/2 +0.579(10) +6.32(4) +- +- +64 +1.28/1 +0.58(2) +6.34(9) +- +- +8 +7.18/4 +0.585(3) +6.27(1) +0.64(3) +0.789 +16 +6.11/3 +0.57(1) +6.32(5) +0.5(2) +0.789 +32 +2.15/2 +0.64(3) +6.0(2) +2.0(8) +0.789 +48 +1.09/1 +0.5(1) +6.6(6) +-1.7(37) +0.789 +8 +6.69/4 +0.575(2) +6.328(9) +0.72(3) +1 +16 +6.34/3 +0.570(9) +6.35(4) +0.6(2) +1 +32 +2.07/2 +0.62(3) +6.1(1) +2.9(11) +1 +48 +1.11/1 +0.54(9) +6.5(5) +-2.5(57) +1 + diff --git a/ntFKT4oBgHgl3EQfFi0Y/content/tmp_files/load_file.txt b/ntFKT4oBgHgl3EQfFi0Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf498b65be01d9d7a7f2a099020b11a2c189519e --- /dev/null +++ b/ntFKT4oBgHgl3EQfFi0Y/content/tmp_files/load_file.txt @@ -0,0 +1,1929 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf,len=1928 +page_content='Extraordinary-log Universality of Critical Phenomena in Plane Defects Yanan Sun,1, ∗ Minghui Hu,1, ∗ and Jian-Ping Lv1, 2, † 1Institute for Theoretical Physics, Anhui Normal University, Wuhu, Anhui 241000, China 2Key Laboratory of Functional Molecular Solids, Ministry of Education, Wuhu, Anhui 241000, China (Dated: January 30, 2023) There is growing evidence that extraordinary-log critical behavior emerges on the open surfaces of critical systems in a semi-infinite geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Here, using extensive Monte Carlo simulations, we observe extraordinary- log critical behavior on the plane defects of O(2) critical systems in an infinite geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In this extraordinary- log critical phase, the large-distance two-point correlation G obeys the logarithmic finite-size scaling G ∼ (lnL)−ˆq with the linear size L, having the critical exponent ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Meanwhile, the helicity modulus Υ follows the scaling form Υ ∼ α(lnL)/L with the universal parameter α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The values of ˆq and α do not fall into any known universality class of critical phenomena, yet they conform to the scaling relation of extraordinary-log universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We also discuss the extension of current results to a quantum system that is experimentally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' These findings reshape our understanding of extraordinary-log critical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Critical phenomena have been a subject of long-standing interest, and universality is recognized as a pil- lar of modern critical theory [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' When critical phenomena occur on lattices of special geometries [2–16], nontrivial sce- narios of universality may emerge and be connected to a wide range of modern concepts [17–24] The O(2) criticality is a textbook illustration of critical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Two-dimensional (2D) and three-dimensional (3D) O(2) critical phenomena have been found in diverse sys- tems [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In two dimensions, the Kosterlitz-Thouless transi- tion is driven by the unbinding of vortex-antivortex pairs, and the quasi-long-range order can emerge in the low-temperature phase [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In three dimensions, the O(2) criticality serves as a testbed for various techniques and theories, such as the “λ transition” of helium which has been the subject of exper- iments in Earth orbit [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Recently, a renormalization-group study predicted the extraordinary-log universality (ELU) for the surface critical behavior (SCB) of the O(n) model with 2 ≤ n < nc, where nc is not precisely known [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' A consequence of this predic- tion is the logarithmic scaling form of the surface two-point correlation g as a function of the distance r [16], which can be reexpressed by the finite-size scaling (FSS) of the large- distance correlation G = g(r → ∞) as G ∼ (lnL)−ˆq, (1) where the exponent ˆq depends on n, and L denotes the linear system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' It was also proposed that the helicity modulus Υ, characterizing the response against a twist in boundary condi- tions [28], scales as Υ ∼ 2α L (lnL), (2) where α is a universal renormalization-group parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Be- sides, a scaling relation for ˆq and α is given by [16] ˆq = n − 1 2πα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (3) ∗ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' † jplv2014@ahnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='cn W K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 KT-like Order Disorder Quasi Order Extraordinary-log 3D O(2) Kc K W Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' A three-dimensional lattice in infinite geometry with a plane defect (left panel) and the phase diagram for plane defect of Villain model (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The W and K axes, which represent in- teractions inside and outside the plane defect, respectively, span the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The phase diagram includes the quasi-long-range or- dered, ordered, and disordered phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' It also contains, at the bulk critical point Kc, the plane-defect critical phenomena in bulk O(2) and extraordinary-log critical universality classes, for W = Kc and W > Kc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' A critical line of Kosterlitz-Thouless-like (KT-like) transition is also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For the SCB, the existence of extraordinary-log critical phase and scaling relation (3) were confirmed at n = 3 [29] and n = 2 [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For n = 2, estimates of ˆq and α were also ob- tained from distinct contexts of O(2) criticality [31–34][35] (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Hence, the long-standing contradiction about the ex- traordinary phases of SCB in XY and Heisenberg models has been reconciled [7, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' It is unknown whether the ELU may be achieved in a wider context of critical systems beyond SCB, despite the recent ad- vances [16, 29–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Given that the SCB is associated with systems in semi-infinite geometries, this question is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In the context of O(2) critical systems, where plane defects are present (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 1), we address the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We demonstrate the presence of ELU in cases where plane defects have strong intra-plane coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The phase diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 1 il- lustrates the main findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Plane-defect Villain Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We start with a plane-defect arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='11720v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='stat-mech] 27 Jan 2023 2 Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For the surface and plane-defect critical phenomena of O(n) critical systems in semi-infinite and infinite geometries, respectively, two classes of extraordinary-log universality have been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The critical exponent ˆq and the universal parameter α quantitatively characterize the extraordinary-log universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Surface critical phenomena Bulk universality model ˆq α year O(3) O(3) φ4 [29] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='15(2) 2020 O(3) φ4 [31] – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='190(4) 2021 O(2) XY [30] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='59(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='27(2) 2021 O(2) φ4 [31] – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='300(5) 2021 Potts [32] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='60(2) – 2022 clock [33] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='59(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='26(2) 2022 Villain [36] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='58(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='28(1) 2022 Plane-defect critical phenomena (this work) Bulk universality model ˆq α year O(2) Villain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56(3) 2023 XY Villain model with the Hamiltonian H = 1 2 ∆J =0 � ⟨rr′⟩ J 2 rr′ Crr′ (4) on simple-cubic lattices, where Crr′ is a variable for the pairs of nearest-neighbor sites r and r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Jrr′ ∈ {0, ±1, ±2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='} pa- rameterizes the directed flows along bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' As in the standard Villain model [37–42], the flows are non-divergent and consti- tute closed directed loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Periodic boundary conditions are imposed in each of the [100] (x), [010] (y), and [001] (z) di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' To involve a plane defect, we specify a plane that is perpendicular to z direction and contains L2 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The values of Crr′ obey Crr′ = � W r and r′ ∈ plane defect, K r or r′ /∈ plane defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (5) By formulating a worm Monte Carlo algorithm based on the original algorithm in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' [43], we simulate model (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We de- sign two update schemes, which are described in Appendix A, based on biased random walks in an extended state space in- volving the source and sink of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' While an update scheme can run on the entire simple-cubic lattice, the other update scheme only works in the plane defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The latter scheme al- lows the sampling of some quantities that characterize plane- defect critical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Extraordinary-log Critical Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We fix K at Kc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='333 067 04, the Villain model’s bulk critical point that was previously estimated by two of us and our colleagues [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' With W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7, we extensively simulate the strong- W regime of plane defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The system size L ranges from 4 to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' At L = 128, the total number of generated closed loops for each W in the worm-algorithm simulations is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='9 × 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In the plane-defect update scheme, we extract the two-point correlation g(δx, δy) [g(0, 0) ≡ 1] unbiasedly from the distribution of the distances (δx, δy) between source and sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We define the large-distance two-point correlation G by G = [g(0, L/2) + g(L/2, 0)]/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (6) If Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (1) holds, a fitting ansatz of G reads G = a0[ln(L/l0)]−ˆq, (7) with a0 a non-universal constant and l0 a reference length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Throughout this paper, we test scaling ansatzes against Monte Carlo data by least-squares fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We monitor the evolution of χ2 with the minimum size Lmin involved in fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In prin- ciple, one searches the smallest Lmin relating to the χ2 per degree of freedom (DoF) χ2/DoF = O(1), which does not decrease drastically upon further increasing Lmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Practically one prefers the fits with χ2/DoF ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We should not trust a single fit—conclusions will be made by comparing all pre- ferred fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For each W, we find that the estimate of ˆq ex- trapolates to ˆq ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' More precisely, for W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7, we obtain ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='308(2), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='301(2), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='289(5) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='28(1) as well as χ2/DoF ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 respec- tively, with Lmin = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Based on these observations, by fix- ing ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29, we obtain l0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='15(2), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='684(2), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0153(2) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0000100(3) as well as χ2/DoF ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7, with Lmin = 48, 32, 64 and 48, for W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Details of fits are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Notice that the estimates of ˆq from various W (W > Kc) are consistent, indicating the universality of the logarithmic FSS form (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In the plane-defect update scheme, the susceptibility χs of plane defect is sampled via χs = ⟨ns⟩, where ns is the number of the movements of source and sink between consecutive hits to the state space of model (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Borrowing the insights into FSS from the ELU of SCB, we have the scaling formula χs = a1L2[ln(L/l0)]−ˆq, (8) with a1 a non-universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, the fit with Lmin = 16 yields ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='322(1) having a huge χ2/DoF (χ2/DoF ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0), which reduces to χ2/DoF ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7 with Lmin = 32 and ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='309(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For W = 1, 3 and 7, we obtain ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='310(1), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='295(3) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(1) as well as χ2/DoF ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 respectively, with Lmin = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Further, when ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 is fixed, we obtain l0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='08(3), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='866(3), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='01909(9) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0000119(3) as well as χ2/DoF ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3, with Lmin = 64, 48, 32 and 48, for W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Similar to that found for G, as W increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0, l0 decreases by several orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Meanwhile, the estimates of ˆq from χs are close to those from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' These observations confirm the scaling for- mula (8), which corresponds to a logarithmic scaling of two- point correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' From the FSS analyses of G and χs, we estimate the value of ˆq as ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We plot G and χsL−2 versus ln(L/l0) in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 2(a) and (b) respectively, where l0 takes above-mentioned values from preferred fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The mutually consistent scaling forms and critical exponents are hence il- lustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We analyze the helicity modulus Υ, which is defined through the fluctuations of winding number Wx of directed 3 2 4 6 8 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56 (c) Linear plot of ϒL vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' lnL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 3 5 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 (b) Log-log plot of χsL-2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' ln(L/l0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 3 5 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 (a) Log-log plot of G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' ln(L/l0) Plane-defect Villain model W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 W = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 W = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The two-point correlation G (a), the scaled susceptibility χsL−2 (b) and the scaled helicity modulus ΥL (c) for the extraordinary-log critical phase of the plane-defect Villain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In panels (a) and (b), the horizontal coordinates are written as ln(L/l0), where l0 are taken from preferred least-squares fits, and the plots are further made on log-log scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The slopes −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56 of dashed lines stand for −ˆq and α, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' flows along a periodic direction (say x direction), Υ = ⟨W2 x⟩/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (9) As indicated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' [36], such an estimator of Υ is effective in the analyses of ELU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For an extraordinary-log critical phase, Υ is expected to scale as ΥL ∝ lnL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' This behavior is roughly illustrated by the Monte Carlo data of Υ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Further, we perform least-squares fits to ΥL = α(lnL) + b + cL−ω, (10) where α is a universal parameter, whereas b and c are non- universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Unlike the scaling form ΥL ∼ 2α(lnL) [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (2)], which applies to the SCB that involves two open surfaces, the prefactor 2 is now removed due to the uniqueness of plane defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' ω denotes the exponent of leading finite-size correc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Accordingly, we estimate α by the least-squares fits to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We adopt ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='789, considering bulk irrelevant fields [45, 46], and compare the fits with those using ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We find that the inclusion of correction term stabilizes the fit- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7, we obtain α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='555(3), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='562(4), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='580(6) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='57(1) as well as χ2/DoF ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 respectively, with Lmin = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Comparing the fits, we estimate α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' There is a lack of a known critical universality class that the values of ˆq and α fall into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The scaling relation between ˆq and α is then investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We test the scaling relation (3) with n = 2, which was previously verified merely for SCB (Ta- ble I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Figure 3 displays the above-obtained fitting results of ˆq and α versus χ2/DoF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Next, from each of the estimates of ˆq and α, via the equation αˆq = 1/(2π), we obtain an estimate of α and ˆq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The estimates converted through the equation are also included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We note that the results of ˆq and α from direct fits and conversions are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For χ2/DoF ≈ 1, such consistency is apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Hence, we obtain strong evidence of the scaling relation αˆq = 1/(2π) for an ELU of plane-defect critical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 1 10 100 Plane-defect Villain model α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56(3) q^ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(2) χ2/DoF q^ = 1/(2πα) α = 1/(2πq^) Plane-defect Villain model q^ α Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Test of the scaling relation αˆq = 1/(2π) for the extraordinary-log critical phase of the plane-defect Villain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Circles denote the results from least-squares fits to FSS formulae, whereas squares denote the results converted from fitting results via αˆq = 1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The estimated critical exponent ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(2) and uni- versal parameter α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56(3) are denoted by the shadows centered at the red and black lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Phase Diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We have demonstrated that ELU exists in a strong-W regime with K = Kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' One also expects the occurrence of a transition at W = Kc and K = Kc in the 3D O(2) universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We proceed to investigate critical phenomena on the plane defect for K < Kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Figure 4(a) shows ΥL versus W at K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For W ⪆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7, ΥL tends to be independent of L in the L → ∞ limit and extrapolates to a W-dependent non- trivial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' These observations indicate a critical phase for the plane defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 4(b), at W ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='73, G 4 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='35 Kc (c) W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 ϒL K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='35 Kc (d) W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 GL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='16 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 WKT (a) K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 ϒL W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='75 WKT (b) K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2 GL1/4 W Plane-defect Villain model L = 4 L = 48 L = 8 L = 64 L = 16 L = 96 L = 32 L = 128 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Critical phenomena of the plane-defect Villain model with K < Kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The scaled helicity modulus ΥL (a) and the scaled two- point correlation GL1/4 (b) versus W for K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' ΥL (c) and GL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='16 (d) versus K for W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Dashed lines represent phase transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' scales as G ∝ L−η with η = 1/4, which is reminiscent of the anomalous dimension of the Kosterlitz-Thouless transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' It is natural to ask whether there is a critical phase that fea- tures quasi-long-range order with 0 < η < 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Figure 4(c) demonstrates, with W = 1, that ΥL is invariant for K < Kc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Meanwhile, the exponent η is nearly invariant (η ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='16) in the parameter regime [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 4(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Our results suggest that, as K → 0, the plane-defect criticality reduces drastically to an essential 2D behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The phase diagram, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 1, is then established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' It calls for a sophisticated renormalization- group study to reveal the distribution and properties of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Other O(2) Critical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The relevance of O(2) crit- icality to various systems of interest determines its signif- icance [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' One may ask whether the current findings hold true for a different O(2) critical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Here, we study a plane-defect critical XY model and investigate the extraordinary-log critical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The model Hamiltonian reads H = − � ⟨rr′⟩ Crr′ ⃗Sr · ⃗Sr′ , (11) where ⃗Sr = (Sa r , Sb r) represents 2D vectors of unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' As defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (5), Crr′ denotes the coupling strength for sites r and r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We set K = Kc, where Kc = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2018441 is the bulk critical point obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We simu- late the model with W = 1, 3 and 7 up to the linear size L = 128, using a cluster Monte Carlo algorithm [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We sample the helicity modulus Υ = 1 L3 (⟨E⟩ − ⟨T 2⟩) with E = � r Cr(r+ex)⃗Sr · ⃗Sr+ex and T = � r Cr(r+ex)(Sa r Sb r+ex − 2 7 12 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='9 2 3 4 5 1 4 7 10 G ln(L/l 0 ) Plane-defect XY model W=1 W=3 W=7 Log-log plot of G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' ln(L/l 0 ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 (a) (b) �L lnL Linear plot of �L vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' lnL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The two-point correlation G (a) and the scaled helicity modulus ΥL (b) for the extraordinary-log critical phase of the plane- defect XY model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In panel (a), the horizontal coordinate is written as ln(L/l0), where l0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='240 (W = 1), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0219 (W = 3) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0000146 (W = 7) are taken from preferred least-squares fits, and the plot is further made on a log-log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The slopes −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56 of dashed lines stand for −ˆq and α, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Sb rSa r+ex), as well as the two-point correlation G = ⟨⃗Sr · ⃗Sr′⟩ with r′ − r = (L/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' For G and Υ, we perform FSS anal- yses according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (7) and (10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Figure 5 illustrates the estimates of ˆq and α, which are compatible with those of the plane-defect Villain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' A specially designed χ2 test for scaling relation of ˆq and α as well as the details of fits are presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In the context of O(2) critical systems, we find that the ELU exists in plane defects of infinite-geometry lat- tices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The conclusion is based on extensive Monte Carlo sim- ulations of two models in a broad strong-coupling regime of plane defects, where the extraordinary-log critical phase fea- tures logarithmic scaling forms of two-point correlation and helicity modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The critical exponent ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(2) and the universal parameter α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56(3) quantitatively characterize the scaling forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Both ˆq and α are far from those of SCB, indicating the existence of a new critical universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Surprisingly, the scaling relation (3) is likely to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' As a result, the current findings strongly suggest that ELU may exist in a wider range of many-body systems than previously thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In comparison to SCB, the ˆq value of plane-defect criticality is relatively small, leaving more room for the dis- covery of ELU in O(n) models with n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Plane defects in O(2) critical bulks could serve as a platform for exotic critical phenomena, inspiring numerous follow-up studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Here, we limit ourselves to quantum ELU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Theoret- ically, a model of quantum ELU is precious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' On the experi- mental side, a realization of ELU is awaited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' A close classical- quantum mapping exists between the plane-defect Villain 5 model and the quantum rotor and Bose-Hubbard models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' On this basis, a possible pathway to realizing quantum ELU is the 2D quantum Hamiltonian ˆH = − � ⟨rr′⟩ trr′(ˆΦ† r ˆΦr′ + ˆΦr ˆΦ† r′) + U 2 � r ˆn2 r, where ˆΦ† r, ˆΦr and ˆnr are respectively the bosonic creation, annihilation and particle number opera- tors at site r, U represents onsite repulsion, and trr′ denotes the bond-dependent hopping strength realizing an edge de- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Recent advances in quantum simulations with ultra-cold bosons in optical lattices [48–50] enable the experimental re- alization of quantum ELU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' While we were finalizing this paper, an in- dependent work appeared on the ArXiv [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Based on a renormalization-group theory, this work predicts the existence of 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Ira (I) and Masha (M)], which obeys the detailed balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' A core part of the simulations relies on iteratively carrying out two update schemes, Algorithm 1 and Algorithm 2, which run in the whole simple-cubic lattice and the plane defect, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Once the source-sink points collide (I = M), a closed loop of directed flows is superposed on the directed-flow configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' After producing a permanent quantity of closed loop(s), the update process shifts between Algorithm 1 and Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Even if Algorithm 1 itself guarantees ergodicity, we include Algorithm 2 for the efficient simulations in plane defect as well as the sampling of some quantities that characterize plane-defect critical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Algorithm 1 Update in simple-cubic lattice 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' If I = M, randomly pick up a lattice site I′ in the simple-cubic lattice, with the probability 1/L3 for each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Let I = M = I′, sign(I) = 1, sign(M) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Interchange I ↔ M and sign(I) ↔ sign(M) with the probability 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Randomly pick up a neighbor In of I, with the probability 1/6 for each of the neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Propose to simultaneously move I → In and update the flow JIIn to J ′ IIn: J ′ IIn = JIIn + df(I → In)sign(I), where df(I → In) = ±1 is a fixed parameter, quantifying the direction of flow along bond-IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Accept the proposed change with the probability p = � � � min[1, e −(J ′2 IIn −J 2 IIn ) 2W ] I and In ∈ plane defect, min[1, e −(J ′2 IIn −J 2 IIn ) 2K ] I or In /∈ plane defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Algorithm 2 Update in plane defect 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' If I = M, randomly pick up a lattice site I′ in the plane defect, with the probability 1/L2 for each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Let I = M = I′, sign(I) = 1, sign(M) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The same as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Randomly pick up a neighbor In of I in the plane defect, with the probability 1/4 for each of the neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The same as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Accept the proposed change with the probability p = min[1, e −(J ′2 IIn −J 2 IIn ) 2W ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Appendix B: Extraordinary-log critical phase of the plane-defect Villain model We simulate the extraordinary-log critical phase of plane-defect Villain model using the worm algorithm, with the lattice sizes L = 4, 8, 16, 32, 48, 64, 96 and 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The number of generated closed loops ranges from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 × 108 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3 × 109 for L ≤ 32 and ranges from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 × 1010 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='9 × 1010 for 48 ≤ L ≤ 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The first one sixth of the generated closed loops are utilized for the thermalization in each Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Here, we analyze the FSS for the extraordinary-log critical phase with W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7 at K = Kc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='333 067 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' According to a standard criterion, we prefer the fits with χ2/DoF ≈ 1 and conclude by comparing the fits that are stable against gradually increasing Lmin, which is the size of the minimum system involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' To supplement the presentation in the main text, we provide more details about the FSS analyses of G, χs, and Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The data of these quantities are fitted to the scaling formulae G = a0[ln(L/l0)]−ˆq, (S1) χs = a1L2[ln(L/l0)]−ˆq, (S2) and ΥL = α(lnL) + b + cL−ω, (S3) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 3 5 10 20 Plane-defect Villain model Log-log plot of G′ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' ln(L/l0) W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 W = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 W = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Two-point correlation G′ versus ln(L/l0) for the plane-defect Villain model, where l0 are taken from preferred least-squares fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The plot is further made on a log-log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The slope −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 of dashed lines stands for −ˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' respectively, where ˆq and α are universal parameters, and ω is a correction exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' l0, a0, a1, b and c represent non-universal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The results of least-squares fits are presented in Tables S1, S2 and S3, for G, χs and Υ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' In addition to the quantities discussed in the main text, we take into consideration the two-point correlation G′ for the distances (δx, δy) = (L/4, 0) and (δx, δy) = (0, L/4), which is defined by G′ = [g(0, L/4) + g(L/4, 0)]/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (S4) For the extraordinary-log critical phase at W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7, we perform FSS analyses according to G′ = a0[ln(L/l0)]−ˆq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (S5) The results of fits are summarized in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Using the results of l0 from preferred fits, we plot G′ versus ln(L/l0) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The leading FSS behavior of G′ is similar to that of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Appendix C: Extraordinary-log critical phase of the plane-defect XY model We simulate the extraordinary-log critical phase of plane-defect XY model using the Wolff cluster algorithm, with the lattice sizes L = 8, 16, 32, 48, 64, 96 and 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The total number of Wolff steps ranges from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='6 × 108 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='4 × 108 for L ≤ 32 and ranges from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 × 108 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='6 × 108 for 48 ≤ L ≤ 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The first one sixth of the Wolff Monte Carlo steps are utilized for the thermalization in each Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We consider the coupling strengths W = 1, 3 and 7 of plane defect at K = Kc = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='2018441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The data for G and Υ are fitted according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' (S1) and (S3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Tables S5 and S6 provide a summary of the outcomes of least-squares fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Using the results of ˆq and α from least-squares fits, a test of the scaling relation αˆq = 1/(2π) is given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 3 Plane-defect XY model a =1/(2pa) a =1/(2p ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='1 q q q q c 2 /DoF a =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56(3) =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(2) q Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Test of the scaling relation αˆq = 1/(2π) for the extraordinary-log critical phase of the plane-defect XY model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Circles denote the results from least-squares fits to FSS formulae, whereas squares denote the results converted from fitting results via αˆq = 1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' The estimated critical exponent ˆq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29(2) and universal parameter α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='56(3) are denoted by the shadows centered at the red and black lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' 4 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Fits of the two-point correlation G to G = a0[ln(L/l0)]−ˆq for the extraordinary-log critical phase of plane-defect Villain model at W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' We investigate both cases where ˆq is free and fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' W Lmin Lmax χ2/DoF a0 l0 ˆq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 4 128 3119.' metadata={'source': 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Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Fits of the susceptibility χs to χs = a1L2[ln(L/l0)]−ˆq for the extraordinary-log critical phase of plane-defect Villain model at W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5, 1, 3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' W Lmin Lmax χ2/DoF a1 l0 ˆq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='5 4 128 11202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='68/5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} 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two-point correlation G to G = a0[ln(L/l0)]−ˆq for the extraordinary-log critical phase of plane-defect XY model at W = 1, 3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' W Lmin Lmax χ2/DoF a0 l0 ˆq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0 8 128 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='30/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='7730(7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='059(5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='09/1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='993(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='0000149(4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content='29 9 Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' Fits of the helicity modulus Υ to ΥL = α(lnL) + b + cL−ω for the extraordinary-log critical phase of plane-defect XY model at W = 1, 3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFKT4oBgHgl3EQfFi0Y/content/2301.11720v1.pdf'} +page_content=' W Lmin Lmax χ2/DoF α b c ω 1.' 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a/r9AzT4oBgHgl3EQfrf3T/content/tmp_files/2301.01646v1.pdf.txt b/r9AzT4oBgHgl3EQfrf3T/content/tmp_files/2301.01646v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..928343ce3929e33c47723b0c9e1d12db29b15fe4 --- /dev/null +++ b/r9AzT4oBgHgl3EQfrf3T/content/tmp_files/2301.01646v1.pdf.txt @@ -0,0 +1,491 @@ +arXiv:2301.01646v1 [math.NT] 4 Jan 2023 +An analogue of Mahler’s transference theorem for +multiplicative Diophantine approximation +Oleg N. German +Abstract +Khintchine’s and Dyson’s transference theorems can be very easily deduced from +Mahler’s transference theorem. In the multiplicative setting an obstacle appears, +which does not allow deducing the multiplicative transference theorem immediately +from Mahler’s theorem. Some extra considerations are required, for instance, induc- +tion by the dimension. In this paper we propose an analogue of Mahler’s theorem +which implies the multiplicative transference theorem immediately. +1 +Introduction +Consider a matrix +Θ = + + + +θ11 +· · · +θ1m +... +... +... +θn1 +· · · +θnm + + + , +θij ∈ R, +m + n ⩾ 3, +and a system of linear equations +Θx = y +with variables x = (x1, . . . , xm) ∈ Rm, y = (y1, . . . , yn) ∈ Rn. One of the main questions +in the theory od Diophantine approximation is how small the vector Θx − y can be as +x and y range independently through Zm\{0} and Zn respectively. There are several +classical ways of measuring the “size” of a vector. One can choose a norm, for instance, +the sup-norm, one can alter it turning it into a so called weighted norm, or one can +consider the product of the absolute values of a vector’s coordinates. In each of those +settings there exist transference theorems — statements reflecting the relation between +the approximation properties of Θ and those of Θ⊺. They are usually formulated in terms +of Diophantine exponents, which are probably the simplest quantities responsible for the +approximation properties. +Given a positive integer k and z = (z1, . . . , zk) ∈ Rk, we denote +|z| = max +1⩽i⩽k |zi|, +Π(z) = +� +1⩽i⩽k +|zi|1/k, +Π′(z) = +� +1⩽i⩽k +max +� +1, |zi| +�1/k. +Definition 1. Supremum of real numbers γ for which there exists t however large such +that the system of inequalities +|x| ⩽ t, +|Θx − y| ⩽ t−γ +(1) +admits a solution (x, y) ∈ Zm ⊕ Zn with nonzero x is called the Diophantine exponent of +Θ and is denoted by ω(Θ). +1 + +Definition 2. Supremum of real numbers γ for which there exists t however large such +that the system of inequalities +Π′(x) ⩽ t, +Π(Θx − y) ⩽ t−γ +(2) +admits a solution (x, y) ∈ Zm⊕Zn with nonzero x is called the multiplicative Diophantine +exponent of Θ and is denoted by ω×(Θ). +For each z ∈ Rk we have +Π(z) ⩽ |z|, +and for each z ∈ Zk we have +|z|1/k ⩽ Π′(z) ⩽ |z|. +Hence +m/n ⩽ ω(Θ) ⩽ ω×(Θ) ⩽ +� +mω(Θ) +for n = 1 ++∞ +for n ⩾ 2 +, +(3) +where the first inequality is a consequence of Minkowski’s convex body theorem. +The inequalities (3) can be called trivial. The transference theorems mentioned above +provide the following nontrivial relations: +ω(Θ⊺) ⩾ +nω(Θ) + n − 1 +(m − 1)ω(Θ) + m +(4) +and +ω×(Θ⊺) ⩾ +nω×(Θ) + n − 1 +(m − 1)ω×(Θ) + m . +(5) +The inequality (4) belongs to Dyson [1], the inequality (5) was proved by the author in [2]. +One can notice that the inequalities look identical, however, there is an essential difference +between their proofs. Dyson’s inequality follows almost immediately from Mahler’s trans- +ference theorem (see [3], [4], and also [5], [6]), whereas the inequality for the multiplicative +exponents, along with Mahler’s theorem, requires induction by n. Roughly speaking, the +reason is that the functionals Π(·) and Π′(·) are not the same. +The purpose of this paper is to find an analogue of Mahler’s transference theorem so +that it would imply (5) as immediately as the classical Mahler theorem implies (4). +The rest of the paper is organised as follows. In Section 2 we formulate Mahler’s +theorem and show how to derive Dyson’s inequality from it. In Section 3 we formulate +and prove the main result of this paper. In Section 4 we derive (5) from our result. +2 +Mahler’s theorem and Dyson’s inequality +Set +d = m + n. +In his original paper [7], Mahler formulated his famous theorem in terms of bilinear forms +with integer coefficients (see also [3] and [6])). In [6] Mahler’s theorem is interpreted in +terms of pseudocompound parallelepipeds and dual lattices. We deem this interpretation +more apt for applications. A pseudocompound parallelepiped is a concept proposed in +Schmidt’s book [4], it is a simplification of what Mahler calls in his papers [8], [9] the +(d − 1)-th compound body of a parallelepiped. +2 + +Definition 3. Let η1, . . . , ηd be positive real numbers. Consider the parallelepiped +P = +� +z = (z1, . . . , zd) ∈ Rd ��� |zi| ⩽ ηi, i = 1, . . . , d +� +. +(6) +The parallelepiped +P∗ = +� +z = (z1, . . . , zd) ∈ Rd ��� |zi| ⩽ 1 +ηi +d +� +j=1 +ηj, i = 1, . . . , d +� +is called the pseudocompound of P. +We remind that, given a full-rank lattice Λ in Rd, its dual lattice Λ∗ is defined as +Λ∗ = +� +z ∈ Rd �� ⟨z, z′⟩ ∈ Z for each z′ ∈ Λ +� +, +where ⟨ · , · ⟩ denotes the inner product. +The following version of Mahler’s transference theorem is proposed in [6]. +Theorem 1. Let Λ be a full-rank lattice in Rd with determinant equal to 1. Let P be a +parallelepiped centered at the origin with faces parallel to the coordinate planes. Then +P∗ ∩ Λ∗ ̸= {0} =⇒ cP ∩ Λ ̸= {0} +with c = +�√ +d +�1/(d−1). +Theorem 1 is actually a strengthening of the original Mahler’s theorem. Mahler for- +mulated his theorem with d − 1 instead of c. We note however that, from the point of +view of Diophantine exponents, any constant (depending on d only) will do. +Let us show how to derive (4) from Theorem 1. Recall that d = m + n. +Consider the lattices +Λ = Λ(Θ) = +� Im +−Θ +In +� +Zd, +Λ∗ = Λ∗(Θ) = +�Im +Θ⊺ +In +� +Zd. +(7) +Clearly, Λ∗ is the dual lattice of Λ. Furthermore, for each set of positive t, γ, s, δ, let us +define the parallelepipeds +P(t, γ) = +� +z = (z1, . . . , zd) ∈ Rd +����� +|zj| ⩽ t, +j = 1, . . . , m +|zm+i| ⩽ t−γ, +i = 1, . . . , n +� +, +(8) +Q(s, δ) = +� +z = (z1, . . . , zd) ∈ Rd +����� +|zj| ⩽ s−δ, +j = 1, . . . , m +|zm+i| ⩽ s, +i = 1, . . . , n +� +. +(9) +Then +ω(Θ) = sup +� +γ ⩾ m +n +���� ∀ t0 ∈ R ∃ t > t0 : P(t, γ) ∩ Λ ̸= {0} +� +, +ω(Θ⊺) = sup +� +δ ⩾ n +m +���� ∀ s0 ∈ R ∃ s > s0 : Q(s, δ) ∩ Λ∗ ̸= {0} +� +. +(10) +If t, γ, s, δ are related by +t = s((n−1)δ+n)/(d−1), +γ = mδ + m − 1 +(n − 1)δ + n, +(11) +3 + +then Q(s, δ) is the pseudocompound of P(t, γ), that is Q(s, δ) = P(t, γ)∗. By Theorem 1 +we get +Q(s, δ) ∩ Λ∗ ̸= {0} =⇒ cP(t, γ) ∩ Λ ̸= {0}. +Hence, in view of (10), +ω(Θ⊺) ⩾ δ =⇒ ω(Θ) ⩾ γ = mδ + m − 1 +(n − 1)δ + n . +Thus, +ω(Θ) ⩾ ω(mΘ⊺) + m − 1 +(n − 1)ω(Θ⊺) + n . +Swapping the triple (Θ, m, n) for (Θ⊺, n, m), we get (4). +3 +An analogue of Mahler’s theorem +For each tuple (λλλ,µµµ) = (λ1, . . . , λm, µ1, . . . , µn) ∈ Rd ++, we define the parallelepiped P(λλλ,µµµ) +as +P(λλλ,µµµ) = +� +z = (z1, . . . , zd) ∈ Rd +����� +|zj| ⩽ λj, +j = 1, . . . , m +|zm+i| ⩽ µi, +i = 1, . . . , n +� +. +(12) +Let us also set +λ∗ +j = λ−1 +j +m +� +k=1 +λk +n +� +k=1 +µk, +j = 1, . . . , m, +µ∗ +i = µ−1 +i +m +� +k=1 +λk +n +� +k=1 +µk, +i = 1, . . . , n. +(13) +Then, clearly, P(λλλ,µµµ)∗ = P(λλλ∗,µµµ∗). Finally, we define the tuple ˆλλλ = (ˆλ1, . . . , ˆλm) as +follows. Let us sort the elements of λλλ in ascending order: λj1 ⩽ . . . ⩽ λjm. If λj1 ⩾ 1, we +set ˆλλλ = λλλ. If λj1 < 1, we set p to be the greatest index such that λj1 · . . . · λjp < 1 and +define ˆλ1, . . . , ˆλm as +ˆλji = 1, +i = 1, . . . , p, +ˆλji = λji +� +λj1 · . . . · λjk +�1/(m−p), +i = p + 1, . . . , m. +(14) +The following theorem is the main result of the paper. +Theorem 2. Let Λ and Λ∗ be defined by (7). Consider arbitrary tuples λλλ = (λ1, . . . , λm) ∈ +Rm ++ and µµµ = (µ1, . . . , µn) ∈ Rn ++. Suppose +Π(λλλ) ⩾ 1. +(15) +Let λλλ∗, µµµ∗ be defined by (13), and let ˆλλλ be defined by (14). Then +min +1⩽j⩽m +ˆλj ⩾ 1, +Π′(ˆλλλ) = Π(ˆλλλ) = Π(λλλ), +(16) +and +P(λλλ∗,µµµ∗) ∩ Λ∗ ̸= {0} =⇒ c1P(ˆλλλ,µµµ) ∩ Λ ̸= {0} +(17) +with c1 = +�√n + 1 +�1/n. +4 + +Proof. Without loss of generality, we may assume that +λ1 ⩽ . . . ⩽ λm. +If λ1 ⩾ 1, then ˆλλλ = λλλ, (16) is obvious, and (17) is provided by Theorem 1, since c ⩽ c1. +Let us assume that λ1 < 1. Then p is correctly defined and p < m, since (15) holds. +Hence (16) follows immediately. +Let us consider the truncated tuples +λλλ↓ = (λp+1, . . . , λm), +λλλ∗ +↓ = (λ∗ +p+1, . . . , λ∗ +m), +ˆλλλ↓ = (ˆλp+1, . . . , ˆλm). +Then +P(ˆλλλ↓,µµµ)∗ = +� +(zp+1, . . . , zd) ∈ Rd−p +����� +|zj| ⩽ κλ∗ +j, +j = p + 1, . . . , m +|zm+i| ⩽ µ∗ +i , +i = 1, . . . , n +� +, +where κ = +� +λ1 · . . . · λp +�−1/(m−p). Since κ > 1, we have +P(λλλ∗ +↓,µµµ∗) ⊂ P(ˆλλλ↓,µµµ)∗. +(18) +Let us also consider the matrix +Θ↓ = + + + +θ1 p+1 +· · · +θ1m +... +... +... +θn p+1 +· · · +θnm + + + +obtained from Θ by deleting the first p columns, and the lattices +Λ↓ = +� +Im−p +−Θ↓ +In +� +Zd−p, +Λ∗ +↓ = +�Im−p +Θ⊺ +↓ +In +� +Zd−p. +We make the following two crucial observations: first, the set +� +(0, . . . , 0, zp+1, . . . , zd) ∈ Rd ��� (zp+1, . . . , zd) ∈ Λ↓ +� +is a sublattice of Λ; second, the set +� +(0, . . . , 0, zp+1, . . . , zd) ∈ Rd ��� (zp+1, . . . , zd) ∈ Λ∗ +↓ +� +is the orthogonal projection of Λ∗ onto the (zp+1, . . . , zd)–coordinate plane. Hence +P(λλλ∗,µµµ∗) ∩ Λ∗ ̸= {0} =⇒ P(λλλ∗ +↓,µµµ∗) ∩ Λ∗ +↓ ̸= {0}, +P(ˆλλλ↓,µµµ) ∩ Λ↓ ̸= {0} =⇒ P(ˆλλλ,µµµ) ∩ Λ ̸= {0}. +(19) +Finally, by Theorem 1 we have +P(ˆλλλ↓,µµµ)∗ ∩ Λ∗ +↓ ̸= {0} =⇒ c2P(ˆλλλ↓,µµµ) ∩ Λ↓ ̸= {0} +(20) +with c2 = +�√d − p +�1/(d−p−1). Gathering up together (18), (19), (20), and taking into +account that c2 ⩽ c1, we get the following chain of implications: +P(λλλ∗,µµµ∗) ∩ Λ∗ ̸= {0} =⇒ P(λλλ∗ +↓,µµµ∗) ∩ Λ∗ +↓ ̸= {0} =⇒ +=⇒ P(ˆλλλ↓,µµµ)∗ ∩ Λ∗ +↓ ̸= {0} =⇒ c2P(ˆλλλ↓,µµµ) ∩ Λ↓ ̸= {0} =⇒ +=⇒ c2P(ˆλλλ,µµµ) ∩ Λ ̸= {0} =⇒ c1P(ˆλλλ,µµµ) ∩ Λ ̸= {0}, +which proves (17). +5 + +4 +Proof of the multiplicative transference inequality +Let us show how to derive (5) from Theorem 2. For every positive t, γ, s, δ, let us define +the following two families of parallelepipeds: +F(t, γ) = +� +P(λλλ,µµµ) +��� Π(λλλ) = t, Π(µµµ) = t−γ, +min +1⩽j⩽m λj ⩾ 1 +� +, +G(s, δ) = +� +P(λλλ,µµµ) +��� Π(λλλ) = s−δ, Π(µµµ) = s, min +1⩽i⩽n µi ⩾ 1 +� +. +Each parallelepiped P(λλλ,µµµ) satisfying the conditions +Π′(λλλ) ⩽ t, +Π(µµµ) ⩽ t−γ +(21) +is contained in a parallelepiped from F(t, γ). Conversely, each parallelepiped P(λλλ,µµµ) from +F(t, γ) satisfies (21). Similarly, each parallelepiped P(λλλ,µµµ) satisfying the conditions +Π(λλλ) ⩽ s−δ, +Π′(µµµ) ⩽ s +(22) +is contained in a parallelepiped from G(s, δ). And conversely, each parallelepiped P(λλλ,µµµ) +from G(s, δ) satisfies (22). Thus, for multiplicative exponents, the following analogue of +(10) holds: +ω×(Θ) = sup +� +γ ⩾ m +n +���� ∀ t0 ∈ R ∃ t > t0 : ∃P ∈ F(t, γ) : P ∩ Λ ̸= {0} +� +, +ω×(Θ⊺) = sup +� +δ ⩾ n +m +���� ∀ s0 ∈ R ∃ s > s0 : ∃P ∈ G(s, δ) : P ∩ Λ∗ ̸= {0} +� +. +(23) +Let us assume again that t, γ, s, δ are related by (11). Consider an arbitrary parallelepiped +P(λλλ,µµµ) such that P(λλλ∗,µµµ∗) ∈ G(s, δ). Then +Π(λλλ) = t, +Π(µµµ) = t−γ. +We cannot guarantee that λλλ has no components strictly less that 1, so generally it is +not true that P(λλλ,µµµ) ∈ F(t, γ). Nevertheless, if t ⩾ 1, then by Theorem 2 we do have +P(ˆλλλ,µµµ) ∈ F(t, γ), and moreover, +P(λλλ∗,µµµ∗) ∩ Λ∗ ̸= {0} =⇒ c1P(ˆλλλ,µµµ) ∩ Λ ̸= {0}. +Hence, in view of (23), +ω×(Θ⊺) ⩾ δ =⇒ ω×(Θ) ⩾ γ = mδ + m − 1 +(n − 1)δ + n . +Thus, +ω×(Θ) ⩾ ω×(mΘ⊺) + m − 1 +(n − 1)ω×(Θ⊺) + n . +Swapping the triple (Θ, m, n) for (Θ⊺, n, m), we get (5). +Acknowledgements. +The author is a winner of the “Junior Leader” contest conducted +by Theoretical Physics and Mathematics Advancement Foundation “BASIS” and would +like to thank its sponsors and jury. +6 + +References +[1] F. J. Dyson On simultaneous Diophantine approximations. Proc. London Math. +Soc., (2) 49 (1947), 409–420. +[2] O. N. German Transference inequalities for multiplicative Diophantine exponents. +Proc. Steklov Inst. Math., 275 (2011), 216–228. +[3] J. W. S. Cassels An introduction to Diophantine approximation. Cambridge Uni- +versity Press (1957). +[4] W. M. Schmidt Diophantine Approximation. Lecture Notes in Math., +785, +Springer-Verlag (1980). +[5] O. N. German On Diophantine exponents and Khintchine’s transference principle. +Moscow J. Comb. Number Theory, 2:2 (2012), 22–51 +[6] O. N. German, K. G. Evdokimov A strengthening of Mahler’s transference theo- +rem. Izv. Math., 79:1 (2015), 60–73. +[7] K. Mahler Ein ¨Ubertragungsprinzip f¨ur lineare Ungleichungen. ˇCas. Peˇst. Mat. +Fys., 68 (1939), 85–92. +[8] K. Mahler On compound convex bodies, I. Proc. London Math. Soc. (3), 5 (1955), +358–379. +[9] K. Mahler On compound convex bodies, II. Proc. London Math. Soc. (3), 5 (1955), +380–384. +7 + diff --git a/r9AzT4oBgHgl3EQfrf3T/content/tmp_files/load_file.txt b/r9AzT4oBgHgl3EQfrf3T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e28297aab6d424ac58b477c87c08685dcff3207 --- /dev/null +++ b/r9AzT4oBgHgl3EQfrf3T/content/tmp_files/load_file.txt @@ -0,0 +1,311 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf,len=310 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content='01646v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content='NT] 4 Jan 2023 An analogue of Mahler’s transference theorem for multiplicative Diophantine approximation Oleg N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' German Abstract Khintchine’s and Dyson’s transference theorems can be very easily deduced from Mahler’s transference theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' In the multiplicative setting an obstacle appears, which does not allow deducing the multiplicative transference theorem immediately from Mahler’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Some extra considerations are required, for instance, induc- tion by the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' In this paper we propose an analogue of Mahler’s theorem which implies the multiplicative transference theorem immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' 1 Introduction Consider a matrix Θ = \uf8eb \uf8ec \uf8ed θ11 · · θ1m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' θn1 · · θnm \uf8f6 \uf8f7 \uf8f8 , θij ∈ R, m + n ⩾ 3, and a system of linear equations Θx = y with variables x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , xm) ∈ Rm, y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , yn) ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' One of the main questions in the theory od Diophantine approximation is how small the vector Θx − y can be as x and y range independently through Zm\\{0} and Zn respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' There are several classical ways of measuring the “size” of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' One can choose a norm, for instance, the sup-norm, one can alter it turning it into a so called weighted norm, or one can consider the product of the absolute values of a vector’s coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' In each of those settings there exist transference theorems — statements reflecting the relation between the approximation properties of Θ and those of Θ⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' They are usually formulated in terms of Diophantine exponents, which are probably the simplest quantities responsible for the approximation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Given a positive integer k and z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zk) ∈ Rk, we denote |z| = max 1⩽i⩽k |zi|, Π(z) = � 1⩽i⩽k |zi|1/k, Π′(z) = � 1⩽i⩽k max � 1, |zi| �1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Supremum of real numbers γ for which there exists t however large such that the system of inequalities |x| ⩽ t, |Θx − y| ⩽ t−γ (1) admits a solution (x, y) ∈ Zm ⊕ Zn with nonzero x is called the Diophantine exponent of Θ and is denoted by ω(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' 1 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Supremum of real numbers γ for which there exists t however large such that the system of inequalities Π′(x) ⩽ t, Π(Θx − y) ⩽ t−γ (2) admits a solution (x, y) ∈ Zm⊕Zn with nonzero x is called the multiplicative Diophantine exponent of Θ and is denoted by ω×(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' For each z ∈ Rk we have Π(z) ⩽ |z|, and for each z ∈ Zk we have |z|1/k ⩽ Π′(z) ⩽ |z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Hence m/n ⩽ ω(Θ) ⩽ ω×(Θ) ⩽ � mω(Θ) for n = 1 +∞ for n ⩾ 2 , (3) where the first inequality is a consequence of Minkowski’s convex body theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' The inequalities (3) can be called trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' The transference theorems mentioned above provide the following nontrivial relations: ω(Θ⊺) ⩾ nω(Θ) + n − 1 (m − 1)ω(Θ) + m (4) and ω×(Θ⊺) ⩾ nω×(Θ) + n − 1 (m − 1)ω×(Θ) + m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (5) The inequality (4) belongs to Dyson [1], the inequality (5) was proved by the author in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' One can notice that the inequalities look identical, however, there is an essential difference between their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Dyson’s inequality follows almost immediately from Mahler’s trans- ference theorem (see [3], [4], and also [5], [6]), whereas the inequality for the multiplicative exponents, along with Mahler’s theorem, requires induction by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Roughly speaking, the reason is that the functionals Π(·) and Π′(·) are not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' The purpose of this paper is to find an analogue of Mahler’s transference theorem so that it would imply (5) as immediately as the classical Mahler theorem implies (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' In Section 2 we formulate Mahler’s theorem and show how to derive Dyson’s inequality from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' In Section 3 we formulate and prove the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' In Section 4 we derive (5) from our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' 2 Mahler’s theorem and Dyson’s inequality Set d = m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' In his original paper [7], Mahler formulated his famous theorem in terms of bilinear forms with integer coefficients (see also [3] and [6])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' In [6] Mahler’s theorem is interpreted in terms of pseudocompound parallelepipeds and dual lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' We deem this interpretation more apt for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' A pseudocompound parallelepiped is a concept proposed in Schmidt’s book [4], it is a simplification of what Mahler calls in his papers [8], [9] the (d − 1)-th compound body of a parallelepiped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' 2 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Let η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , ηd be positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Consider the parallelepiped P = � z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Rd ��� |zi| ⩽ ηi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (6) The parallelepiped P∗ = � z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Rd ��� |zi| ⩽ 1 ηi d � j=1 ηj, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , d � is called the pseudocompound of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' We remind that, given a full-rank lattice Λ in Rd, its dual lattice Λ∗ is defined as Λ∗ = � z ∈ Rd �� ⟨z, z′⟩ ∈ Z for each z′ ∈ Λ � , where ⟨ · , · ⟩ denotes the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' The following version of Mahler’s transference theorem is proposed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Let Λ be a full-rank lattice in Rd with determinant equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Let P be a parallelepiped centered at the origin with faces parallel to the coordinate planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Then P∗ ∩ Λ∗ ̸= {0} =⇒ cP ∩ Λ ̸= {0} with c = �√ d �1/(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Theorem 1 is actually a strengthening of the original Mahler’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Mahler for- mulated his theorem with d − 1 instead of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' We note however that, from the point of view of Diophantine exponents, any constant (depending on d only) will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Let us show how to derive (4) from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Recall that d = m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Consider the lattices Λ = Λ(Θ) = � Im −Θ In � Zd, Λ∗ = Λ∗(Θ) = �Im Θ⊺ In � Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (7) Clearly, Λ∗ is the dual lattice of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Furthermore, for each set of positive t, γ, s, δ, let us define the parallelepipeds P(t, γ) = � z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Rd ����� |zj| ⩽ t, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , m |zm+i| ⩽ t−γ, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , n � , (8) Q(s, δ) = � z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Rd ����� |zj| ⩽ s−δ, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , m |zm+i| ⩽ s, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (9) Then ω(Θ) = sup � γ ⩾ m n ���� ∀ t0 ∈ R ∃ t > t0 : P(t, γ) ∩ Λ ̸= {0} � , ω(Θ⊺) = sup � δ ⩾ n m ���� ∀ s0 ∈ R ∃ s > s0 : Q(s, δ) ∩ Λ∗ ̸= {0} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (10) If t, γ, s, δ are related by t = s((n−1)δ+n)/(d−1), γ = mδ + m − 1 (n − 1)δ + n, (11) 3 then Q(s, δ) is the pseudocompound of P(t, γ), that is Q(s, δ) = P(t, γ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' By Theorem 1 we get Q(s, δ) ∩ Λ∗ ̸= {0} =⇒ cP(t, γ) ∩ Λ ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Hence, in view of (10), ω(Θ⊺) ⩾ δ =⇒ ω(Θ) ⩾ γ = mδ + m − 1 (n − 1)δ + n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Thus, ω(Θ) ⩾ ω(mΘ⊺) + m − 1 (n − 1)ω(Θ⊺) + n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Swapping the triple (Θ, m, n) for (Θ⊺, n, m), we get (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' 3 An analogue of Mahler’s theorem For each tuple (λλλ,µµµ) = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , λm, µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , µn) ∈ Rd +, we define the parallelepiped P(λλλ,µµµ) as P(λλλ,µµµ) = � z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Rd ����� |zj| ⩽ λj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , m |zm+i| ⩽ µi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (12) Let us also set λ∗ j = λ−1 j m � k=1 λk n � k=1 µk, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , m, µ∗ i = µ−1 i m � k=1 λk n � k=1 µk, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (13) Then, clearly, P(λλλ,µµµ)∗ = P(λλλ∗,µµµ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Finally, we define the tuple ˆλλλ = (ˆλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , ˆλm) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Let us sort the elements of λλλ in ascending order: λj1 ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' ⩽ λjm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' If λj1 ⩾ 1, we set ˆλλλ = λλλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' If λj1 < 1, we set p to be the greatest index such that λj1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' · λjp < 1 and define ˆλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , ˆλm as ˆλji = 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , p, ˆλji = λji � λj1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' · λjk �1/(m−p), i = p + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (14) The following theorem is the main result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Let Λ and Λ∗ be defined by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Consider arbitrary tuples λλλ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , λm) ∈ Rm + and µµµ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , µn) ∈ Rn +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Suppose Π(λλλ) ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (15) Let λλλ∗, µµµ∗ be defined by (13), and let ˆλλλ be defined by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Then min 1⩽j⩽m ˆλj ⩾ 1, Π′(ˆλλλ) = Π(ˆλλλ) = Π(λλλ), (16) and P(λλλ∗,µµµ∗) ∩ Λ∗ ̸= {0} =⇒ c1P(ˆλλλ,µµµ) ∩ Λ ̸= {0} (17) with c1 = �√n + 1 �1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Without loss of generality, we may assume that λ1 ⩽ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' ⩽ λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' If λ1 ⩾ 1, then ˆλλλ = λλλ, (16) is obvious, and (17) is provided by Theorem 1, since c ⩽ c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Let us assume that λ1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Then p is correctly defined and p < m, since (15) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Hence (16) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Let us consider the truncated tuples λλλ↓ = (λp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , λm), λλλ∗ ↓ = (λ∗ p+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , λ∗ m), ˆλλλ↓ = (ˆλp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , ˆλm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Then P(ˆλλλ↓,µµµ)∗ = � (zp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Rd−p ����� |zj| ⩽ κλ∗ j, j = p + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , m |zm+i| ⩽ µ∗ i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , n � , where κ = � λ1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' · λp �−1/(m−p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Since κ > 1, we have P(λλλ∗ ↓,µµµ∗) ⊂ P(ˆλλλ↓,µµµ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (18) Let us also consider the matrix Θ↓ = \uf8eb \uf8ec \uf8ed θ1 p+1 · · θ1m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' θn p+1 · · θnm \uf8f6 \uf8f7 \uf8f8 obtained from Θ by deleting the first p columns, and the lattices Λ↓ = � Im−p −Θ↓ In � Zd−p, Λ∗ ↓ = �Im−p Θ⊺ ↓ In � Zd−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' We make the following two crucial observations: first, the set � (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , 0, zp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Rd ��� (zp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Λ↓ � is a sublattice of Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' second, the set � (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , 0, zp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Rd ��� (zp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd) ∈ Λ∗ ↓ � is the orthogonal projection of Λ∗ onto the (zp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' , zd)–coordinate plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Hence P(λλλ∗,µµµ∗) ∩ Λ∗ ̸= {0} =⇒ P(λλλ∗ ↓,µµµ∗) ∩ Λ∗ ↓ ̸= {0}, P(ˆλλλ↓,µµµ) ∩ Λ↓ ̸= {0} =⇒ P(ˆλλλ,µµµ) ∩ Λ ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (19) Finally, by Theorem 1 we have P(ˆλλλ↓,µµµ)∗ ∩ Λ∗ ↓ ̸= {0} =⇒ c2P(ˆλλλ↓,µµµ) ∩ Λ↓ ̸= {0} (20) with c2 = �√d − p �1/(d−p−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Gathering up together (18), (19), (20), and taking into account that c2 ⩽ c1, we get the following chain of implications: P(λλλ∗,µµµ∗) ∩ Λ∗ ̸= {0} =⇒ P(λλλ∗ ↓,µµµ∗) ∩ Λ∗ ↓ ̸= {0} =⇒ =⇒ P(ˆλλλ↓,µµµ)∗ ∩ Λ∗ ↓ ̸= {0} =⇒ c2P(ˆλλλ↓,µµµ) ∩ Λ↓ ̸= {0} =⇒ =⇒ c2P(ˆλλλ,µµµ) ∩ Λ ̸= {0} =⇒ c1P(ˆλλλ,µµµ) ∩ Λ ̸= {0}, which proves (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' 5 4 Proof of the multiplicative transference inequality Let us show how to derive (5) from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' For every positive t, γ, s, δ, let us define the following two families of parallelepipeds: F(t, γ) = � P(λλλ,µµµ) ��� Π(λλλ) = t, Π(µµµ) = t−γ, min 1⩽j⩽m λj ⩾ 1 � , G(s, δ) = � P(λλλ,µµµ) ��� Π(λλλ) = s−δ, Π(µµµ) = s, min 1⩽i⩽n µi ⩾ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Each parallelepiped P(λλλ,µµµ) satisfying the conditions Π′(λλλ) ⩽ t, Π(µµµ) ⩽ t−γ (21) is contained in a parallelepiped from F(t, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Conversely, each parallelepiped P(λλλ,µµµ) from F(t, γ) satisfies (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Similarly, each parallelepiped P(λλλ,µµµ) satisfying the conditions Π(λλλ) ⩽ s−δ, Π′(µµµ) ⩽ s (22) is contained in a parallelepiped from G(s, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' And conversely, each parallelepiped P(λλλ,µµµ) from G(s, δ) satisfies (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Thus, for multiplicative exponents, the following analogue of (10) holds: ω×(Θ) = sup � γ ⩾ m n ���� ∀ t0 ∈ R ∃ t > t0 : ∃P ∈ F(t, γ) : P ∩ Λ ̸= {0} � , ω×(Θ⊺) = sup � δ ⩾ n m ���� ∀ s0 ∈ R ∃ s > s0 : ∃P ∈ G(s, δ) : P ∩ Λ∗ ̸= {0} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' (23) Let us assume again that t, γ, s, δ are related by (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Consider an arbitrary parallelepiped P(λλλ,µµµ) such that P(λλλ∗,µµµ∗) ∈ G(s, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Then Π(λλλ) = t, Π(µµµ) = t−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' We cannot guarantee that λλλ has no components strictly less that 1, so generally it is not true that P(λλλ,µµµ) ∈ F(t, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Nevertheless, if t ⩾ 1, then by Theorem 2 we do have P(ˆλλλ,µµµ) ∈ F(t, γ), and moreover, P(λλλ∗,µµµ∗) ∩ Λ∗ ̸= {0} =⇒ c1P(ˆλλλ,µµµ) ∩ Λ ̸= {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Hence, in view of (23), ω×(Θ⊺) ⩾ δ =⇒ ω×(Θ) ⩾ γ = mδ + m − 1 (n − 1)δ + n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Thus, ω×(Θ) ⩾ ω×(mΘ⊺) + m − 1 (n − 1)ω×(Θ⊺) + n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Swapping the triple (Θ, m, n) for (Θ⊺, n, m), we get (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} +page_content=' The author is a winner of the “Junior Leader” contest conducted by Theoretical Physics and Mathematics Advancement Foundation “BASIS” and would like 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+page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfrf3T/content/2301.01646v1.pdf'} diff --git a/r9E1T4oBgHgl3EQfQAMJ/content/tmp_files/2301.03033v1.pdf.txt b/r9E1T4oBgHgl3EQfQAMJ/content/tmp_files/2301.03033v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f73aa296927c7a2b3ac14b02d4f100dbcad2a02d --- /dev/null +++ b/r9E1T4oBgHgl3EQfQAMJ/content/tmp_files/2301.03033v1.pdf.txt @@ -0,0 +1,842 @@ +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +1 +RGB-T Multi-Modal Crowd Counting +Based on Transformer +Zhengyi Liu +liuzywen@ahu.edu.cn +Wei Wu +2640947588@qq.com +Yacheng Tan +1084043983@qq.com +Guanghui Zhang +2532950974@qq.com +School of Computer Science and +Technology +Anhui University +Hefei, China +Abstract +Crowd counting aims to estimate the number of persons in a scene. Most state-of-the- +art crowd counting methods based on color images can’t work well in poor illumination +conditions due to invisible objects. With the widespread use of infrared cameras, crowd +counting based on color and thermal images is studied. Existing methods only achieve +multi-modal fusion without count objective constraint. To better excavate multi-modal +information, we use count-guided multi-modal fusion and modal-guided count enhance- +ment to achieve the impressive performance. The proposed count-guided multi-modal +fusion module utilizes a multi-scale token transformer to interact two-modal information +under the guidance of count information and perceive different scales from the token per- +spective. The proposed modal-guided count enhancement module employs multi-scale +deformable transformer decoder structure to enhance one modality feature and count in- +formation by the other modality. Experiment in public RGBT-CC dataset shows that our +method refreshes the state-of-the-art results. https://github.com/liuzywen/RGBTCC +1 +Introduction +Crowd counting can predict the distribution of crowd and estimate the number of persons +in unconstraint scenes. It is widely studied by the academia and industrial communities +since the number of persons is an important indicator of incident monitoring[31], traffic +control[19], and infectious disease prevention[32]. The existing crowd counting methods +have achieved tremendous improvement due to the introduce of convolutional neural net- +works [7, 8] and transformer[28, 40]. +However, when light is insufficient, the performance of crowd counting is unsatisfying, +as shown in the first line of Fig.1. The thermal image can percept the temperature of objects +to recognize the persons. Therefore, RGB-Thermal (RGB-T) crowd counting by introducing +the thermal modality has attracted a lot of attentions. +© 2022. The copyright of this document resides with its authors. +It may be distributed unchanged freely in print or electronic forms. +arXiv:2301.03033v1 [cs.CV] 8 Jan 2023 + +2 +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +Figure 1: Three examples to show the performances of different methods in poor light condi- +tion, thermal disturbance, and large-scale variation, respectively. “Estimate" means predicted +counts. “Difference" means the counting difference from the ground truth. (a) RGB image +(b) the paired thermal image (c) MAN[18] result based on RGB image (d) MAN[18] result +based on RGB-T image (e) CMCRL[20] result (f) our result (g) ground truth. +In RGB-T crowd counting task, a most important challenge is the multi-modal fusion +problem. The color modality is good at perceiving the shape and texture of persons, but it +is also interfered by the cluttered background. The thermal modality is skilled in recogniz- +ing persons which have temperature from scattered environments, but it also highlights the +other heating objects, as shown in the second line of Fig.1 where the cars in the right are high- +lighted. Existing RGB-T crowd counting methods fuse complementary multi-modal features +by Information Aggregation and Distribution Module (IADM) [20], Information Improve- +ment Module (IIM) [29], and Mutual Attention Transformer (MAT) [41]. However, these +multi-modal interactions are lack of a constraint. If we add a counting constraint on multi- +modal fusion process, two-modal fusion has a clear goal. Therefore, we use a transformer +structure to fuse two-modal information and design a learnable count token to participant the +two-modal fusion. It makes the color and thermal modality interact under the guidance of a +common count token. +In RGB-T crowd counting task, the other challenge is large-scale variation which is also +the common issue in crowd counting, as shown in the third line of Fig.1 where persons +that are far from the camera appear much smaller than those close to it. Existing methods +use multi-column structure [1, 21, 46, 48], dilated convolution [2, 6, 15], high-resolution +representation [24], and attention mechanism [18] to enlarge the receptive fields. Under +the transformer framework, we propose a multi-scale token transformer to perceive persons +with different scales. The tokens are merged to form token sequences with different lengths +and then fed into some parallel transformers. After the enhancement of transformers, the +receptive fields of features will be diversified. +To further improve the accuracy of crowd counting, we use a modality to guide the +learning of the other modality and count token. A multi-scale deformable transformer is +adopted to decode a modality and count token by the other modality. As a result, the count +ability of the feature is enhanced. + +Estimate: 19. +Estimate +Estimate: +GT:25 +istimate: +160. +8 +Difference: 5.3 +Difference: 2.8 +Difference: 2.3 +Estimate: 42 +Estimate: 31 +Estimate: 20.9 +Estimate: 37.3 +GT: 38 +Difference: 4.5 +Difference: 6.3 +Difference: 17.1 +Difference: 0.7 +Estimate: 40.6 +Estimate: 39.1 +Estimate: 37.5 +Estimate: 27.4 +GT: 32 +i.: +Difference: 8.6 +Difference: 7.1 +Difference: 5.5 +Difference: 4.6 +(a) RGB +(b) Thermal +(c) MAN (RGB) (d) MAN (RGBT) +(e) CMCRL +(f) Ours +(g) GTLIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +3 +In summary, the main contributions are summarized as follows: +• An RGB-T multi-modal crowd counting model is proposed based on the transformer. +Multi-head self-attention is used to achieve the count-guided multi-modal fusion. Multi- +head cross-attention is adopted to achieve the modal-guided count enhancement. +• A count-guided multi-modal fusion transformer is proposed to solve the fusion prob- +lem. Under the guidance of count global information, color and thermal modalities +are well combined and aligned. +• A multi-scale token transformer is proposed to solve the large-scale variation problem. +Three-scale token sequences are parallel handled to achieve multi-scale concept. +• The ablation experiments verify the effectiveness of modules, multi-scale design, and +count guidance. The comparison experiments show the significant improvement over +existing RGB-T crowd counting methods. +2 +Related work +2.1 +Crowd counting +Crowd counting can be achieved by detection [12, 13, 16, 27] or density map estimation +[3, 14, 36, 38]. Since the latter can solve high overlap and occlusion problem, it shows better +performance than the former. +The large-scale variation generated by the wide viewing angle of cameras and 2D per- +spective projection is a major challenge in crowd counting. The persons which are close to +the camera are large, while the persons which are far from the camera are small. Multi-scale +architecture[1, 2, 6, 15, 47, 48] and perspective information[9, 25, 42, 44, 45] are two main +solutions. Recently, to deal with the scale changes, some attention based methods are pro- +posed. MAN[18] improves global attention in the transformer by adding region attention. +HANet[35] introduces scale context in the parallel spatial attention and channel attention. +In the paper, we solve the large-scale variation problem by multi-scale transformer based +on tokens. The original token sequence is merged into a middle-scale token sequence and a +large-scale token sequence, respectively. Then the three are parallel handled by three multi- +head self-attention structures. Finally, three branches are concatenated and combined. The +multi-scale concept ensures abundant receptive fields which benefits the crowd counting task. +2.2 +Transformer based crowd counting +Previous works utilize the convolution neural network as the backbone and regress density +map to predict the crowd count. The advent of transformer has pushed the crowd count- +ing model forward. BCCTrans [28] introduces a global context learnable token to guide the +counting. SAANet [40] designs a deformer backbone to extract the features, aggregates +multi-level features by a deformable transformer encoder, and introduces a count query +in a transformer decoder to re-calibrates the multi-level feature maps. DCSwinTrans[10] +enhances the large-range contextual information by a dilated Swin Transformer backbone, +and equips with a feature pyramid networks decoder to achieve crowd instant localization. +CrowdFormer [43] models the human top-down visual perception mechanism by an overlap + +4 +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +patching transformer block. CCTrans [30] adopts a pyramid transformer and a multi-scale re- +gression head to achieve both fully-supervised and weakly-supervised crowd counting task. +In addition, in weakly-supervised crowd counting, there are some other transformer based +methods. TransCrowd [17] uses a learnable counting token or global average pooling on +high-layer semantic tokens to represent the crowd numbers. It constructs a weakly super- +vised model from sequence-to-count perspective. SFSL [5] introduces a learnable unbiased +feature estimation of persons and utilizes the feature similarity for the regression of crowd +numbers to solve the lack of local supervision. CrowdMLP [37] proposes a multi-granularity +multilayer perceptron (MLP) regressor to enlarge receptive fields and a split-counting to de- +couple spatial constraints. JCTNet [34] introduces transformer structure upon the high-layer +feature of convolutional neural network and regresses the count. +In the paper, we use transformer encoder structure to achieve count-guided multi-modal +fusion, and use transformer decoder structure to perform modal-guided count enhancement. +2.3 +RGB-T crowd counting +Although the crowd counting methods have achieved many significant improvements, they +rely on optical information and often perform poorly when the light is insufficient. To solve +this problem, RGB-T crowd counting has been getting a lot of attentions. On one hand, +thermal image can recognize pedestrians in poor illumination conditions. On the other hand, +thermal image can reduce wrong recognition about some human-shaped objects. Mean- +while, RGB image can suppress interference in thermal images. For example, heating walls +and lamps that are highlighted in thermal images can be filtered from color perspective. +Therefore, RGB and thermal images need to be simultaneously explored. +CMCRL [20] introduces a two-stream framework that first aggregates two features and +second propagates the common information to further refine each feature. TAFNet [29] uses +a three-stream network to learn the RGB feature, the thermal feature, and the concatenated +RGB-T feature for crowd counting. The proposed Information Improvement Module (IIM) +is used to fuse the modal-specific and combination features. Mutual Attention Transformer +(MAT) [41] uses cross-modal mutual attention to build long-range dependencies and enhance +semantic features in crowd counting task. DEFNet [49] uses multi-modal fusion, receptive +field enhancement, and multi-layer fusion to highlight the crowd position and suppress the +background noise. +In these works, the fusion of the RGB and thermal images are short of count objective +constraint. We design a learnable count token to guide multi-modal fusion. +3 +Proposed Method +We propose an RGB-T multi-modal crowd counting method which includes a count-guide +multi-modal fusion, a modal-guide count enhancement, and a regression head, as shown in +Fig.2. To solve multi-modal fusion problem, we introduce a count guidance. Moreover, +to perceive the large-scale variation, we propose a multi-scale token concept. Combining +both, multi-modal features are well fused towards a global objective. Furthermore, counting +information is further enhanced from one modality under the guidance of the other modality. + +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +5 +Figure 2: Our proposed RGB-T multi-modal crowd counting model based on transformer. +3.1 +Count guided multi-modal fusion +Given a paired RGB-T image I = {Ir,It}, we use two PVT encoders [39] as the feature +extractors to capture hierarchical features. +Fr = EPVT(Ir) +Ft = EPVT(It) +(1) +where EPVT denotes a PVT encoder, Fr = {Fi +r }4 +i=1 and Ft = {Fi +t }4 +i=1 represent color features +and thermal features, respectively, i is the feature layer number. +The high-layer features contain more semantic information, which are suitable to obtain +the global counting cues. Besides, color feature and thermal feature have each advantage +in representing the crowd. Therefore, we use the high-level tokens from color modality +and thermal modality to excavate the number of crowd. To fully align two-modal data and +generate a consistent result, a learnable count token is designed to guide the two-modal +fusion. Specifically, as is illustrated in Fig.2, high-layer semantic features F4 +r and F4 +t are +generated from color encoder and thermal encoder, respectively. They represent unaligned +multi-modal semantic concept. We design a learnable count token Fcount which implies the +coarse number of crowd. The three are concatenated along the token direction, and then fed +into a Multi-Scale Token Transformer (MSTTrans) which spreads information among color, +thermal, and crowd count by the multi-head self-attention. +MSTTrans is proposed to solve large-scale variations. Inspired by multi-scale design +in atrous spatial pyramid pooling (ASPP) [4], MSTTrans achieves multi-scale transformer +based on tokens. At first, we concatenate high-layer color feature, high-layer thermal feature, +and the learnable count token to form an initial token sequence. Then, we merge the initial + +Density Map +Count +① Modal-Guided Count Enhancement +66.61 +② Count-Guided Multi-Modal Fusion +MLP +MLP +MLP +Multi-Scale Deformable +1 +Transform er (MSDTrans) +FC+Reshape +FC+Reshape +MHSA +MHSA +MHSA +Multi-Scale Token Transformer (MSTTrans) +Reshape+FC +Reshape+FC +Count Token' +PVT +PVT +RGB +Thermal +Count +Patch Embedding +Patch Embedding +MHSA +: Multi-Head Self-Attention +MLP +: Multilayer Perceptron +FC +: Fully Connected Layer +: Concatenation +RGB +Thermal6 +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +token sequence to generate a middle-scale token sequence. The middle-scale token sequence +has the larger receptive fields than original token sequence. Besides, we merge the initial +token sequence to generate a large-scale token sequence, where a modality is represented +by a token. According to above merge strategy, three parallel branches which all include +color modality, thermal modality, and count token are constructed. They are fed into three +multi-head self-attention modules for in-depth fusion. +Specifically, as is illustrated in the right of Fig. 2, suppose the high-layer semantic feature +F4 +r ∈ RN2×C and F4 +t ∈ RN2×C, where N2 and C represent the number of tokens and channels, +respectively. The two-modal features and the learnable count token are concatenated to +generate the initial token sequence f1 ∈ R(2N2+1)×C. +f1 = [F4 +r ,F4 +t ,Fcount] +(2) +where [·] denotes concatenation operation along token direction. +Then, the two-modal features are merged to N groups and each group generates N +middle-scale tokens. All the middle-scale tokens and the learnable count token are con- +catenated to generate the middle-scale token sequence f2 ∈ R(2N+1)×C. +f2 = [mergeN2→N(F4 +r ),mergeN2→N(F4 +t ),Fcount] +(3) +where mergea→b denotes the aggregation operation from a tokens to b tokens which applies +a reshape operation and a fully connected layer. +Meanwhile, the two-modal features are merged to two groups and each group generates +a large-scale token. The large-scale tokens and the learnable count token are concatenated to +generate the large-scale token sequence f3 ∈ R(2+1)×C. There are a color token, a thermal to- +ken, and a learnable count token. It ensures two-modal whole alignment under the guidance +of count. +f3 = [mergeN2→1(F4 +r ),mergeN2→1(F4 +t ),Fcount] +(4) +Three token sequences with different scales are fed into three multi-head self-attention +modules for multi-modal interaction. +f ′ +i = MHSA( fi) +(5) +where MHSA represents two multi-head self-attention layers. +Since the lengths of middle-scale and large-scale token sequences are different from +initial token sequences, we apply fully connection layer and reshape operation to restore +token sequence length. +gi = Reshape(FC( f ′ +i )) +(6) +where i = 2,3 because only middle-scale and large-scale token sequences should be restored, +FC is a fully-connected layer, and Reshape is a reshape operation to restore token length. +Further, to retain the original features in the middle-scale and large-scale branches, the +concatenation and MLP operations are successively conducted. +g′ +i = MLP(Concat(gi, f1)) +(7) +where i = 2,3, Concat is concatenation operation along channel direction, and MLP is a +two-layer perceptron. + +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +7 +Last, three features are concatenated and shrunk in channels. +G = [Gr,Gt,Gcount] = MLP(Concat( f ′ +1,g′ +2,g′ +3)) +(8) +where G has the same size as the input f1 of MSTTrans module, and consists of optimized +color feature Gr, thermal feature Gt, and count feature Gcount. +In MSTTrans module, the count token is responsible for incorporating the global infor- +mation and perceiving the number of persons. Besides, it is used to guide the fusion of color +feature and thermal feature. Under the guidance of count token, color feature and thermal +feature are in-depth interacted. Moreover, multi-scale token concept ensures the abundant +receptive fields adaptive to recognizing the persons with different sizes. +3.2 +Modal-guided counting enhancement +The researches pointed out that the thermal image can provide strong support on density map +estimation, especially in the dark background[29]. In the paper, we use the thermal modality +to predict the density map and count, and further use color modality to refine the prediction. +Therefore, after the previous count-guided multi-modal fusion, we design a modal-guided +counting enhancement module which is responsible for generating the density map and final +count from one modality under the guidance of the other modality. A multi-scale deformable +transformer (MSDTrans) is employed to achieve the above objective. +Specifically, the thermal feature Gt and the learnable count token Gcount are concatenated +as query (Q), and the enhanced color feature Gr and the encoded low-layer features Fi +r (i = +1,2,3) compose multi-scale color features which are regarded as key (K) and value (V). We +use multi-scale deformable attention [50] to enhance Q by K and V. Last, it will output +modal-guided enhanced feature Ot and count token Ocount. +[Ot,Ocount] = DeformAttn([Gt,Gcount],{Gr,F3 +r ,F2 +r ,F1 +r }) +(9) +where DeformAttn(a,b) is the multi-scale deformable attention [50], a represents content +feature, b is multi-scale features. +3.3 +Regression head and loss function +To obtain the density map, we use a simple regression head which consists of two 3×3 +convolution layers and one 1×1 convolution layer. +D = RH(Ot) +(10) +where RH is the regression head. +The loss includes a loss about the density map and a loss about the learnable count token. +L = LD(D,D⋆)+LC(Ocount,C⋆) +(11) +where LD adopts distribution matching loss proposed in[33], which supervises the density +map regression and count estimation, LC adopts L1 norm (∥·∥1) to supervise the count token. +D⋆ and C⋆ represent the ground truth of density map and count, respectively. + +8 +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +4 +Experiments +4.1 +Datasets and evaluation metrics +Dataset. The public RGBT-CC[20] dataset is adopted to evaluate our method. RGBT-CC +consists of 1,030 training samples, 200 validation samples, and 800 testing ones. +Evaluation Metrics. The widely used Grid Average Mean Absolute Error (GAME)[11] +and Root Mean Square Error (RMSE) are used as evaluation metrics[20, 29]. +GAME(l) = 1 +N +N +∑ +i=1 +4l +∑ +j=1 +| ˆP j +i −P j +i | +(12) +where ˆP j +i represents the predicted value of the jth region of the ith image, Pj +i indicates the +ground truth corresponding to ˆP j +i , 4l means the number of the divided non-overlapping re- +gions of the image, and N is the number of paired images in testing dataset. GAME sums the +counting errors in all the regions. +RMSE = +� +1 +N +N +∑ +i=1 +( ˆPi −Pi)2 +(13) +where ˆPi represents the predicted value of the ith image, Pi indicates the ground truth corre- +sponding to ˆPi. For both RMSE and GAME, lower value means the better performance. +4.2 +Implementation details +The implementation setting includes: (1) GPU (NVIDIA RTX 3090); (2) input image size +(224×224); (3) train time (17 hours); (4) learning rate (1e−5); (5) weight decay (1e−4). +4.3 +Comparison with state-of-the-art methods +To make quantitative comparisons, our method is compared with recent prominent approaches, +including CSRNet[15], BL[22], DM-Count[33], P2PNet[26], MARUNet[23], MAN[18], +CMCRL[20], TAFNet [29], MAT[41], and DEFNet[49] which are shown in Table 1. The +top of the table shows six single-modal crowd counting models which are retrained by the +input fusion of RGB and thermal images. The bottom of the table shows four RGB-T crowd +counting models and ours. From the observation, we can conclude our method performs the +best among all the methods. It achieves about 8.4%, 7.8%, 5.7%, 4.1%, 10.9% improvement +over the second best result in GAME(0), GAME(1), GAME(2), GAME(3) and RMSE, re- +spectively. The great improvement profits from the multi-modal fusion under the guidance of +count token and count enhancement of a modality under the guidance of the other modality. +4.4 +Ablation studies +4.4.1 +Effectiveness analysis of the proposed modules +To verify the effectiveness of the proposed modules, we conduct the ablation studies. Table +2 show the result. At first, we construct a baseline model. It concatenates high-layer features +of two PVT encoders and applies regression head to predict the density map and sum up. + +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +9 +Table 1: Comparison results of different methods on RGBT-CC benchmark dataset. The top +part: +some RGB crowd counting models are retrained by input fusion of color modality +and thermal modality. The bottom part: some RGB-T crowd counting models. The best +result is in bold. +Methods +Source +GAME(0)↓ +GAME(1)↓ +GAME(2)↓ +GAME(3)↓ +RMSE↓ +CSRNet[15] +CVPR2018 +20.40 +23.58 +28.03 +35.51 +35.26 +BL[22] +ICCV2019 +18.70 +22.55 +26.83 +34.62 +32.67 +DM-Count[33] +NeurIPS2020 +16.54 +20.73 +25.23 +32.23 +27.22 +P2PNet[26] +ICCV2021 +16.24 +19.42 +23.48 +30.27 +29.94 +MARUNet[23] +WACV2021 +17.39 +20.54 +23.69 +27.36 +30.84 +MAN[18] +CVPR2022 +17.16 +21.78 +28.74 +41.59 +33.84 +CMCRL[20] +CVPR2021 +15.61 +19.95 +24.69 +32.89 +28.18 +TAFNet[29] +ISCAS2022 +12.38 +16.98 +21.86 +30.19 +22.45 +MAT[41] +ICME2022 +12.35 +16.29 +20.81 +29.09 +22.53 +DEFNet[49] +TITS2022 +11.90 +16.08 +20.19 +27.27 +21.09 +Ours +BMVC2022 +10.90 +14.81 +19.02 +26.14 +18.79 +The baseline result is shown in the first line. Then, we add count-guided multi-modal fusion +module and modal-guided count enhancement module based on the baseline, respectively. +The result is shown in the second and the third lines, respectively. Finally, we add all the +modules. The result is shown in the fourth line. By the observation, MSTTrans improves +the performance from GAME0 (11.62) to GAME0 (10.91). It benefits from the better fusion +which has a global common objective and multi-scale concept. MSDTrans improves the +performance from GAME0 (11.62) to GAME0 (11.17). It indicates the supplementary effect +of a modality on the other modality. Last, the whole model achieves a best GAME0 (10.90), +which shows the effectiveness of both modules. However, we also find that RMSE value in +the second line achieves the best result. It suggests our future work to improve the model. +Table 2: Ablation study about modules. The best result is in bold. +Variant +Candidate +GAME(0) +GAME(1) +GAME(2) +GAME(3) +RMSE +Baseline +MSTTrans +MSDTrans +No.1 +✓ +11.62 +16.25 +20.38 +27.17 +19.88 +No.2 +✓ +✓ +10.91 +15.26 +19.88 +26.99 +18.32 +No.3 +✓ +✓ +11.22 +15.20 +19.42 +26.30 +19.75 +No.4 +✓ +✓ +✓ +10.90 +14.81 +19.02 +26.14 +18.79 +4.4.2 +Effectiveness analysis of the count-guided multi-modal fusion design +To verify our contributions, we conduct the ablation studies about the count-guided multi- +modal fusion design. There are two essential design conceptions in the module. One is the +guidance of the learnable count token. The other is multi-scale strategy. Table 3 show the +result. At first, we show our result in the first line. Then, we remove the learnable count +token from the whole model. Finally, we replace the multi-scale token transformer with +vanilla multi-head self-attention. By the observation, we find that the performance declines +obviously when removing the count token. It just verifies the effectiveness of the count +token. Furthermore, multi-scale concept is also effective because the performance is worse +when replacing our proposed multi-scale token transformer with multi-head self-attention. +Compared with both, multi-scale concept plays a more important role than the learnable + +10 +LIU ET AL.: RGB-T MULTI-MODAL CROWD COUNTING +count token. It also verifies our most important contribution which introduces a token level +multi-scale transformer. +Table 3: Ablation study about count guidance and multi-scale concept in count-guided multi- +modal fusion module. The best result is in bold. “Ours/count" represents our model remov- +ing the learnable count token. “Ours/multi-scale" represents our model with vanilla multi- +head self-attention instead of the multi-scale token transformer. +Variant +Candidate +GAME(0) +GAME(1) +GAME(2) +GAME(3) +RMSE +Ours +Ours/count +Ours/multi-scale +No.1 +✓ +10.90 +14.81 +19.02 +26.14 +18.79 +No.2 +✓ +11.82 +15.91 +20.10 +27.13 +20.54 +No.3 +✓ +11.82 +16.39 +20.89 +28.37 +21.73 +5 +Conclusions +In the paper, we propose an RGB-T multi-modal crowd counting method based on Trans- +former. Two-modal features are fused under the guidance of a learnable count token. Then +crowd density map is predicted by a modality and guided by the other modality. To solve +the large-scale variation problem, a multi-scale token transformer is proposed to diversify +the receptive fields. The experimental results demonstrate a significant improvement over +existing RGB-T crowd counting methods and verify the effectiveness of all the designs. +6 +Acknowledgment +This work is supported by Natural Science Foundation of Anhui Province (1908085MF182) +and Science Research Project for Graduate Student of Anhui Provincial Education Depart- +ment (YJS20210047). +References +[1] Deepak Babu Sam, Shiv Surya, and R Venkatesh Babu. Switching Convolutional Neu- +ral Network for Crowd Counting. In Proceedings of the IEEE Conference on Computer +Vision and Pattern Recognition, pages 5744–5752, 2017. +[2] Shuai Bai, Zhiqun He, Yu Qiao, Hanzhe Hu, Wei Wu, and Junjie Yan. +Adaptive +Dilated Network with Self-Correction Supervision for Counting. In Proceedings of +the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4594– +4603, 2020. +[3] Jiwei Chen, Kewei Wang, Wen Su, and Zengfu Wang. 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In International +Conference on Learning Representations, pages 1–12, 2020. + diff --git a/r9E1T4oBgHgl3EQfQAMJ/content/tmp_files/load_file.txt b/r9E1T4oBgHgl3EQfQAMJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c50c168ec5437a93feb49807ad2028f03b293df --- /dev/null +++ b/r9E1T4oBgHgl3EQfQAMJ/content/tmp_files/load_file.txt @@ -0,0 +1,543 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf,len=542 +page_content='LIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING 1 RGB-T Multi-Modal Crowd Counting Based on Transformer Zhengyi Liu liuzywen@ahu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='cn Wei Wu 2640947588@qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='com Yacheng Tan 1084043983@qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='com Guanghui Zhang 2532950974@qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='com School of Computer Science and Technology Anhui University Hefei, China Abstract Crowd counting aims to estimate the number of persons in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Most state-of-the- art crowd counting methods based on color images can’t work well in poor illumination conditions due to invisible objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' With the widespread use of infrared cameras, crowd counting based on color and thermal images is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Existing methods only achieve multi-modal fusion without count objective constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' To better excavate multi-modal information, we use count-guided multi-modal fusion and modal-guided count enhance- ment to achieve the impressive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The proposed count-guided multi-modal fusion module utilizes a multi-scale token transformer to interact two-modal information under the guidance of count information and perceive different scales from the token per- spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The proposed modal-guided count enhancement module employs multi-scale deformable transformer decoder structure to enhance one modality feature and count in- formation by the other modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Experiment in public RGBT-CC dataset shows that our method refreshes the state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='com/liuzywen/RGBTCC 1 Introduction Crowd counting can predict the distribution of crowd and estimate the number of persons in unconstraint scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It is widely studied by the academia and industrial communities since the number of persons is an important indicator of incident monitoring[31], traffic control[19], and infectious disease prevention[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The existing crowd counting methods have achieved tremendous improvement due to the introduce of convolutional neural net- works [7, 8] and transformer[28, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' However, when light is insufficient, the performance of crowd counting is unsatisfying, as shown in the first line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The thermal image can percept the temperature of objects to recognize the persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Therefore, RGB-Thermal (RGB-T) crowd counting by introducing the thermal modality has attracted a lot of attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' © 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The copyright of this document resides with its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It may be distributed unchanged freely in print or electronic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='03033v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='CV] 8 Jan 2023 2 LIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING Figure 1: Three examples to show the performances of different methods in poor light condi- tion, thermal disturbance, and large-scale variation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' “Estimate" means predicted counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' “Difference" means the counting difference from the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' (a) RGB image (b) the paired thermal image (c) MAN[18] result based on RGB image (d) MAN[18] result based on RGB-T image (e) CMCRL[20] result (f) our result (g) ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In RGB-T crowd counting task, a most important challenge is the multi-modal fusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The color modality is good at perceiving the shape and texture of persons, but it is also interfered by the cluttered background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The thermal modality is skilled in recogniz- ing persons which have temperature from scattered environments, but it also highlights the other heating objects, as shown in the second line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 where the cars in the right are high- lighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Existing RGB-T crowd counting methods fuse complementary multi-modal features by Information Aggregation and Distribution Module (IADM) [20], Information Improve- ment Module (IIM) [29], and Mutual Attention Transformer (MAT) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' However, these multi-modal interactions are lack of a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' If we add a counting constraint on multi- modal fusion process, two-modal fusion has a clear goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Therefore, we use a transformer structure to fuse two-modal information and design a learnable count token to participant the two-modal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It makes the color and thermal modality interact under the guidance of a common count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In RGB-T crowd counting task, the other challenge is large-scale variation which is also the common issue in crowd counting, as shown in the third line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 where persons that are far from the camera appear much smaller than those close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Existing methods use multi-column structure [1, 21, 46, 48], dilated convolution [2, 6, 15], high-resolution representation [24], and attention mechanism [18] to enlarge the receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Under the transformer framework, we propose a multi-scale token transformer to perceive persons with different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The tokens are merged to form token sequences with different lengths and then fed into some parallel transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' After the enhancement of transformers, the receptive fields of features will be diversified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' To further improve the accuracy of crowd counting, we use a modality to guide the learning of the other modality and count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' A multi-scale deformable transformer is adopted to decode a modality and count token by the other modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' As a result, the count ability of the feature is enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Estimate: 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Estimate Estimate: GT:25 istimate: 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 8 Difference: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 Difference: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='8 Difference: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 Estimate: 42 Estimate: 31 Estimate: 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='9 Estimate: 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 GT: 38 Difference: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='5 Difference: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 Difference: 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 Difference: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='7 Estimate: 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='6 Estimate: 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 Estimate: 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='5 Estimate: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='4 GT: 32 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=': Difference: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='6 Difference: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 Difference: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='5 Difference: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='6 (a) RGB (b) Thermal (c) MAN (RGB) (d) MAN (RGBT) (e) CMCRL (f) Ours (g) GTLIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING 3 In summary, the main contributions are summarized as follows: An RGB-T multi-modal crowd counting model is proposed based on the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Multi-head self-attention is used to achieve the count-guided multi-modal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Multi- head cross-attention is adopted to achieve the modal-guided count enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' A count-guided multi-modal fusion transformer is proposed to solve the fusion prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Under the guidance of count global information, color and thermal modalities are well combined and aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' A multi-scale token transformer is proposed to solve the large-scale variation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Three-scale token sequences are parallel handled to achieve multi-scale concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The ablation experiments verify the effectiveness of modules, multi-scale design, and count guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The comparison experiments show the significant improvement over existing RGB-T crowd counting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 2 Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 Crowd counting Crowd counting can be achieved by detection [12, 13, 16, 27] or density map estimation [3, 14, 36, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Since the latter can solve high overlap and occlusion problem, it shows better performance than the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The large-scale variation generated by the wide viewing angle of cameras and 2D per- spective projection is a major challenge in crowd counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The persons which are close to the camera are large, while the persons which are far from the camera are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Multi-scale architecture[1, 2, 6, 15, 47, 48] and perspective information[9, 25, 42, 44, 45] are two main solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Recently, to deal with the scale changes, some attention based methods are pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' MAN[18] improves global attention in the transformer by adding region attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' HANet[35] introduces scale context in the parallel spatial attention and channel attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In the paper, we solve the large-scale variation problem by multi-scale transformer based on tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The original token sequence is merged into a middle-scale token sequence and a large-scale token sequence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Then the three are parallel handled by three multi- head self-attention structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Finally, three branches are concatenated and combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The multi-scale concept ensures abundant receptive fields which benefits the crowd counting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='2 Transformer based crowd counting Previous works utilize the convolution neural network as the backbone and regress density map to predict the crowd count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The advent of transformer has pushed the crowd count- ing model forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' BCCTrans [28] introduces a global context learnable token to guide the counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' SAANet [40] designs a deformer backbone to extract the features, aggregates multi-level features by a deformable transformer encoder, and introduces a count query in a transformer decoder to re-calibrates the multi-level feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' DCSwinTrans[10] enhances the large-range contextual information by a dilated Swin Transformer backbone, and equips with a feature pyramid networks decoder to achieve crowd instant localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' CrowdFormer [43] models the human top-down visual perception mechanism by an overlap 4 LIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING patching transformer block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' CCTrans [30] adopts a pyramid transformer and a multi-scale re- gression head to achieve both fully-supervised and weakly-supervised crowd counting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In addition, in weakly-supervised crowd counting, there are some other transformer based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' TransCrowd [17] uses a learnable counting token or global average pooling on high-layer semantic tokens to represent the crowd numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It constructs a weakly super- vised model from sequence-to-count perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' SFSL [5] introduces a learnable unbiased feature estimation of persons and utilizes the feature similarity for the regression of crowd numbers to solve the lack of local supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' CrowdMLP [37] proposes a multi-granularity multilayer perceptron (MLP) regressor to enlarge receptive fields and a split-counting to de- couple spatial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' JCTNet [34] introduces transformer structure upon the high-layer feature of convolutional neural network and regresses the count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In the paper, we use transformer encoder structure to achieve count-guided multi-modal fusion, and use transformer decoder structure to perform modal-guided count enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 RGB-T crowd counting Although the crowd counting methods have achieved many significant improvements, they rely on optical information and often perform poorly when the light is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' To solve this problem, RGB-T crowd counting has been getting a lot of attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' On one hand, thermal image can recognize pedestrians in poor illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' On the other hand, thermal image can reduce wrong recognition about some human-shaped objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Mean- while, RGB image can suppress interference in thermal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' For example, heating walls and lamps that are highlighted in thermal images can be filtered from color perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Therefore, RGB and thermal images need to be simultaneously explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' CMCRL [20] introduces a two-stream framework that first aggregates two features and second propagates the common information to further refine each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' TAFNet [29] uses a three-stream network to learn the RGB feature, the thermal feature, and the concatenated RGB-T feature for crowd counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The proposed Information Improvement Module (IIM) is used to fuse the modal-specific and combination features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Mutual Attention Transformer (MAT) [41] uses cross-modal mutual attention to build long-range dependencies and enhance semantic features in crowd counting task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' DEFNet [49] uses multi-modal fusion, receptive field enhancement, and multi-layer fusion to highlight the crowd position and suppress the background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In these works, the fusion of the RGB and thermal images are short of count objective constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' We design a learnable count token to guide multi-modal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 3 Proposed Method We propose an RGB-T multi-modal crowd counting method which includes a count-guide multi-modal fusion, a modal-guide count enhancement, and a regression head, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' To solve multi-modal fusion problem, we introduce a count guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Moreover, to perceive the large-scale variation, we propose a multi-scale token concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Combining both, multi-modal features are well fused towards a global objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Furthermore, counting information is further enhanced from one modality under the guidance of the other modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' LIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING 5 Figure 2: Our proposed RGB-T multi-modal crowd counting model based on transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 Count guided multi-modal fusion Given a paired RGB-T image I = {Ir,It}, we use two PVT encoders [39] as the feature extractors to capture hierarchical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Fr = EPVT(Ir) Ft = EPVT(It) (1) where EPVT denotes a PVT encoder, Fr = {Fi r }4 i=1 and Ft = {Fi t }4 i=1 represent color features and thermal features, respectively, i is the feature layer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The high-layer features contain more semantic information, which are suitable to obtain the global counting cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Besides, color feature and thermal feature have each advantage in representing the crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Therefore, we use the high-level tokens from color modality and thermal modality to excavate the number of crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' To fully align two-modal data and generate a consistent result, a learnable count token is designed to guide the two-modal fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Specifically, as is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='2, high-layer semantic features F4 r and F4 t are generated from color encoder and thermal encoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' They represent unaligned multi-modal semantic concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' We design a learnable count token Fcount which implies the coarse number of crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The three are concatenated along the token direction, and then fed into a Multi-Scale Token Transformer (MSTTrans) which spreads information among color, thermal, and crowd count by the multi-head self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' MSTTrans is proposed to solve large-scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Inspired by multi-scale design in atrous spatial pyramid pooling (ASPP) [4], MSTTrans achieves multi-scale transformer based on tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' At first, we concatenate high-layer color feature, high-layer thermal feature, and the learnable count token to form an initial token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Then, we merge the initial Density Map Count ① Modal-Guided Count Enhancement 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content="61 ② Count-Guided Multi-Modal Fusion MLP MLP MLP Multi-Scale Deformable 1 Transform er (MSDTrans) FC+Reshape FC+Reshape MHSA MHSA MHSA Multi-Scale Token Transformer (MSTTrans) Reshape+FC Reshape+FC Count Token' PVT PVT RGB Thermal Count Patch Embedding Patch Embedding MHSA : Multi-Head Self-Attention MLP : Multilayer Perceptron FC : Fully Connected Layer : Concatenation RGB Thermal6 LIU ET AL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING token sequence to generate a middle-scale token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The middle-scale token sequence has the larger receptive fields than original token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Besides, we merge the initial token sequence to generate a large-scale token sequence, where a modality is represented by a token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' According to above merge strategy, three parallel branches which all include color modality, thermal modality, and count token are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' They are fed into three multi-head self-attention modules for in-depth fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Specifically, as is illustrated in the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 2, suppose the high-layer semantic feature F4 r ∈ RN2×C and F4 t ∈ RN2×C, where N2 and C represent the number of tokens and channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The two-modal features and the learnable count token are concatenated to generate the initial token sequence f1 ∈ R(2N2+1)×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' f1 = [F4 r ,F4 t ,Fcount] (2) where [·] denotes concatenation operation along token direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Then, the two-modal features are merged to N groups and each group generates N middle-scale tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' All the middle-scale tokens and the learnable count token are con- catenated to generate the middle-scale token sequence f2 ∈ R(2N+1)×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' f2 = [mergeN2→N(F4 r ),mergeN2→N(F4 t ),Fcount] (3) where mergea→b denotes the aggregation operation from a tokens to b tokens which applies a reshape operation and a fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Meanwhile, the two-modal features are merged to two groups and each group generates a large-scale token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The large-scale tokens and the learnable count token are concatenated to generate the large-scale token sequence f3 ∈ R(2+1)×C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' There are a color token, a thermal to- ken, and a learnable count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It ensures two-modal whole alignment under the guidance of count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' f3 = [mergeN2→1(F4 r ),mergeN2→1(F4 t ),Fcount] (4) Three token sequences with different scales are fed into three multi-head self-attention modules for multi-modal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' f ′ i = MHSA( fi) (5) where MHSA represents two multi-head self-attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Since the lengths of middle-scale and large-scale token sequences are different from initial token sequences, we apply fully connection layer and reshape operation to restore token sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' gi = Reshape(FC( f ′ i )) (6) where i = 2,3 because only middle-scale and large-scale token sequences should be restored, FC is a fully-connected layer, and Reshape is a reshape operation to restore token length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Further, to retain the original features in the middle-scale and large-scale branches, the concatenation and MLP operations are successively conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' g′ i = MLP(Concat(gi, f1)) (7) where i = 2,3, Concat is concatenation operation along channel direction, and MLP is a two-layer perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' LIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING 7 Last, three features are concatenated and shrunk in channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' G = [Gr,Gt,Gcount] = MLP(Concat( f ′ 1,g′ 2,g′ 3)) (8) where G has the same size as the input f1 of MSTTrans module, and consists of optimized color feature Gr, thermal feature Gt, and count feature Gcount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In MSTTrans module, the count token is responsible for incorporating the global infor- mation and perceiving the number of persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Besides, it is used to guide the fusion of color feature and thermal feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Under the guidance of count token, color feature and thermal feature are in-depth interacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Moreover, multi-scale token concept ensures the abundant receptive fields adaptive to recognizing the persons with different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='2 Modal-guided counting enhancement The researches pointed out that the thermal image can provide strong support on density map estimation, especially in the dark background[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In the paper, we use the thermal modality to predict the density map and count, and further use color modality to refine the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Therefore, after the previous count-guided multi-modal fusion, we design a modal-guided counting enhancement module which is responsible for generating the density map and final count from one modality under the guidance of the other modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' A multi-scale deformable transformer (MSDTrans) is employed to achieve the above objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Specifically, the thermal feature Gt and the learnable count token Gcount are concatenated as query (Q), and the enhanced color feature Gr and the encoded low-layer features Fi r (i = 1,2,3) compose multi-scale color features which are regarded as key (K) and value (V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' We use multi-scale deformable attention [50] to enhance Q by K and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Last, it will output modal-guided enhanced feature Ot and count token Ocount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' [Ot,Ocount] = DeformAttn([Gt,Gcount],{Gr,F3 r ,F2 r ,F1 r }) (9) where DeformAttn(a,b) is the multi-scale deformable attention [50], a represents content feature, b is multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 Regression head and loss function To obtain the density map, we use a simple regression head which consists of two 3×3 convolution layers and one 1×1 convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' D = RH(Ot) (10) where RH is the regression head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The loss includes a loss about the density map and a loss about the learnable count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' L = LD(D,D⋆)+LC(Ocount,C⋆) (11) where LD adopts distribution matching loss proposed in[33], which supervises the density map regression and count estimation, LC adopts L1 norm (∥·∥1) to supervise the count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' D⋆ and C⋆ represent the ground truth of density map and count, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 8 LIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 Datasets and evaluation metrics Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The public RGBT-CC[20] dataset is adopted to evaluate our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' RGBT-CC consists of 1,030 training samples, 200 validation samples, and 800 testing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The widely used Grid Average Mean Absolute Error (GAME)[11] and Root Mean Square Error (RMSE) are used as evaluation metrics[20, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' GAME(l) = 1 N N ∑ i=1 4l ∑ j=1 | ˆP j i −P j i | (12) where ˆP j i represents the predicted value of the jth region of the ith image, Pj i indicates the ground truth corresponding to ˆP j i , 4l means the number of the divided non-overlapping re- gions of the image, and N is the number of paired images in testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' GAME sums the counting errors in all the regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' RMSE = � 1 N N ∑ i=1 ( ˆPi −Pi)2 (13) where ˆPi represents the predicted value of the ith image, Pi indicates the ground truth corre- sponding to ˆPi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' For both RMSE and GAME, lower value means the better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='2 Implementation details The implementation setting includes: (1) GPU (NVIDIA RTX 3090);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' (2) input image size (224×224);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' (3) train time (17 hours);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' (4) learning rate (1e−5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' (5) weight decay (1e−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 Comparison with state-of-the-art methods To make quantitative comparisons, our method is compared with recent prominent approaches, including CSRNet[15], BL[22], DM-Count[33], P2PNet[26], MARUNet[23], MAN[18], CMCRL[20], TAFNet [29], MAT[41], and DEFNet[49] which are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The top of the table shows six single-modal crowd counting models which are retrained by the input fusion of RGB and thermal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The bottom of the table shows four RGB-T crowd counting models and ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' From the observation, we can conclude our method performs the best among all the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It achieves about 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='4%, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='8%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='7%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1%, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='9% improvement over the second best result in GAME(0), GAME(1), GAME(2), GAME(3) and RMSE, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The great improvement profits from the multi-modal fusion under the guidance of count token and count enhancement of a modality under the guidance of the other modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='4 Ablation studies 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 Effectiveness analysis of the proposed modules To verify the effectiveness of the proposed modules, we conduct the ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Table 2 show the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' At first, we construct a baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It concatenates high-layer features of two PVT encoders and applies regression head to predict the density map and sum up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' LIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING 9 Table 1: Comparison results of different methods on RGBT-CC benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The top part: some RGB crowd counting models are retrained by input fusion of color modality and thermal modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The bottom part: some RGB-T crowd counting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The best result is in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Methods Source GAME(0)↓ GAME(1)↓ GAME(2)↓ GAME(3)↓ RMSE↓ CSRNet[15] CVPR2018 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='40 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='58 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='03 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='51 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='26 BL[22] ICCV2019 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='70 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='55 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='83 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='62 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='67 DM-Count[33] NeurIPS2020 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='54 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='73 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='23 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='23 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='22 P2PNet[26] ICCV2021 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='24 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='42 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='48 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='27 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='94 MARUNet[23] WACV2021 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='39 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='54 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='69 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='36 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='84 MAN[18] CVPR2022 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='16 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='78 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='74 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='59 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='84 CMCRL[20] CVPR2021 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='61 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='95 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='69 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='89 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='18 TAFNet[29] ISCAS2022 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='38 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='98 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='86 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='19 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='45 MAT[41] ICME2022 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='35 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='29 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='81 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='09 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='53 DEFNet[49] TITS2022 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='90 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='08 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='19 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='27 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='09 Ours BMVC2022 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='90 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='81 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='02 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='14 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='79 The baseline result is shown in the first line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Then, we add count-guided multi-modal fusion module and modal-guided count enhancement module based on the baseline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The result is shown in the second and the third lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Finally, we add all the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The result is shown in the fourth line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' By the observation, MSTTrans improves the performance from GAME0 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='62) to GAME0 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It benefits from the better fusion which has a global common objective and multi-scale concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' MSDTrans improves the performance from GAME0 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='62) to GAME0 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It indicates the supplementary effect of a modality on the other modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Last, the whole model achieves a best GAME0 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='90), which shows the effectiveness of both modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' However, we also find that RMSE value in the second line achieves the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It suggests our future work to improve the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Table 2: Ablation study about modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The best result is in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Variant Candidate GAME(0) GAME(1) GAME(2) GAME(3) RMSE Baseline MSTTrans MSDTrans No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 ✓ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='62 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='25 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='38 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='88 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='2 ✓ ✓ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='91 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='26 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='88 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='99 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='32 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 ✓ ✓ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='22 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='20 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='42 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='30 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='75 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='4 ✓ ✓ ✓ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='90 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='81 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='02 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='14 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='79 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='2 Effectiveness analysis of the count-guided multi-modal fusion design To verify our contributions, we conduct the ablation studies about the count-guided multi- modal fusion design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' There are two essential design conceptions in the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' One is the guidance of the learnable count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The other is multi-scale strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Table 3 show the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' At first, we show our result in the first line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Then, we remove the learnable count token from the whole model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Finally, we replace the multi-scale token transformer with vanilla multi-head self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' By the observation, we find that the performance declines obviously when removing the count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It just verifies the effectiveness of the count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Furthermore, multi-scale concept is also effective because the performance is worse when replacing our proposed multi-scale token transformer with multi-head self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Compared with both, multi-scale concept plays a more important role than the learnable 10 LIU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' : RGB-T MULTI-MODAL CROWD COUNTING count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' It also verifies our most important contribution which introduces a token level multi-scale transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Table 3: Ablation study about count guidance and multi-scale concept in count-guided multi- modal fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The best result is in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' “Ours/count" represents our model remov- ing the learnable count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' “Ours/multi-scale" represents our model with vanilla multi- head self-attention instead of the multi-scale token transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Variant Candidate GAME(0) GAME(1) GAME(2) GAME(3) RMSE Ours Ours/count Ours/multi-scale No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='1 ✓ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='90 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='81 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='02 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='14 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='79 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='2 ✓ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='82 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='91 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='10 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='13 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='54 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='3 ✓ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='82 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='39 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='89 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='37 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content='73 5 Conclusions In the paper, we propose an RGB-T multi-modal crowd counting method based on Trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Two-modal features are fused under the guidance of a learnable count token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Then crowd density map is predicted by a modality and guided by the other modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' To solve the large-scale variation problem, a multi-scale token transformer is proposed to diversify the receptive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' The experimental results demonstrate a significant improvement over existing RGB-T crowd counting methods and verify the effectiveness of all the designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' 6 Acknowledgment This work is supported by Natural Science Foundation of Anhui Province (1908085MF182) and Science Research Project for Graduate Student of Anhui Provincial Education Depart- ment (YJS20210047).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' References [1] Deepak Babu Sam, Shiv Surya, and R Venkatesh Babu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' Switching Convolutional Neu- ral Network for Crowd Counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5744–5752, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E1T4oBgHgl3EQfQAMJ/content/2301.03033v1.pdf'} +page_content=' [2] Shuai Bai, Zhiqun He, Yu Qiao, Hanzhe Hu, Wei Wu, and Junjie Yan.' metadata={'source': 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Yang1*, Yigui Zhong2, Sougata Mardanya3, Tyler A. Cochran1, Ramakanta Chapai4, +Akifumi Mine2, Junyi Zhang5*, Jaime Sánchez-Barriga6, 7, Zi-Jia Cheng1, Oliver J. Clark6, Jia- +Xin Yin8, Joanna Blawat4, 9, Guangming Cheng10, Ilya Belopolski1, Tsubaki Nagashima2, +Najafzadeh Sahand2, Shiyuan Gao5, Nan Yao10, Arun Bansil11, Rongying Jin4, 9, Tay-Rong +Chang3, 12, 13, Shik Shin2, 14, 15, Kozo Okazaki2, 15, 16, M. Zahid Hasan1, 10, 17† + +1Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department +of Physics, Princeton University, Princeton, NJ 08544, USA. +2Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan. +3Department of Physics, National Cheng Kung University, Tainan 701, Taiwan. +4Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA +70803, USA. +5Institute for Quantum Matter and Department of Physics and Astronomy, Johns Hopkins +University, Baltimore, MD 21218, USA. +6Helmholtz-Zentrum Berlin für Materialien und Energie, Elektronenspeicherring BESSY II, +Albert-Einstein Strasse 15, Berlin 12489, Germany. +7IMDEA Nanoscience, C/ Faraday 9, Campus de Cantoblanco, Madrid 28049, Spain. +8Department of Physics, Southern University of Science and Technology, Shenzhen, +Guangdong 518055, China. +9Center for Experimental Nanoscale Physics, Department of Physics and Astronomy, +University of South Carolina, Columbia, SC 29208, USA. +10Princeton Institute for Science and Technology of Materials, Princeton University, +Princeton, NJ 08544, USA. +11Department of Physics, Northeastern University, Boston, MA 02115, USA. +12Center for Quantum Frontiers of Research and Technology (QFort), Tainan 701, Taiwan. +13Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan. +14Office of University Professor, University of Tokyo, Kashiwa, Chiba 277-8581, Japan. +15Material Innovation Research Center, University of Tokyo, Kashiwa, Chiba 277-8581, +Japan. +16Trans-scale Quantum Science Institute, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, +Japan. +17Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. + + +*xiany@princeton.edu +*jzhan312@jhu.edu +†mzhasan@princeton.edu + + + + + + + + + + +2 + +The interplay of nontrivial topology and superconductivity in condensed matter physics +gives rise to exotic phenomena. However, materials are extremely rare where it is possible to +explore the full details of the superconducting pairing. Here, we investigate the momentum +dependence of the superconducting gap distribution in a novel Dirac material PdTe. Using +high resolution, low temperature photoemission spectroscopy, we establish it as a spin-orbit +coupled Dirac semimetal with the topological Fermi arc crossing the Fermi level on the (010) +surface. This spin-textured surface state exhibits a fully gapped superconducting Cooper +pairing structure below 𝐓𝐜 ~ 4.5 K. Moreover, we find a node in the bulk near the Brillouin +zone boundary, away from the topological Fermi arc. These observations not only +demonstrate the band resolved electronic correlation between topological Fermi arc states +and the way it induces Cooper pairing in PdTe, but also provide a rare case where surface +and bulk states host a coexistence of nodeless and nodal gap structures enforced by spin- +orbit coupling. + + +Superconducting topological materials are emerging as the new frontier of condensed matter +physics. Despite increasing interest in this field, few candidates have been identified because it is +very challenging to confirm topological superconductivity experimentally [1-22]. While the +topological surface states of three-dimensional (3D) topological insulators (TIs) can host +topological superconductivity below the critical temperature (T"), spin-momentum locked Fermi +arcs in 3D Dirac semimetals could also induce topological superconductivity. Such a +superconducting pairing between Fermi arcs with opposite spins and momenta has been proposed +to reveal the nonlocal electronic correlations in topological semimetals [16]. However, the +superconducting state of Fermi arcs has rarely been explored experimentally. Therefore, +understanding the Cooper pairings of the Fermi arc states offers a new perspective for topological +superconductors (TSCs) [23-27]. Moreover, recent theoretical work predicts the existence of nodes +from the bulk bands in superconducting semimetals [28-30]. Thus, in superconducting semimetals +that are also topological, surface Fermi arcs and nodal gap distribution of the bulk states can coexist, +which in turn provide insight into the mechanism of unconventional and topological +superconductivity. + +Angle-resolved photoemission spectroscopy (ARPES) is arguably the most powerful technique for +probing the superconducting gap distribution in the momentum space. Here, we use ARPES to +demonstrate spin-orbit coupling (SOC) enforced surface-nodeless plus bulk-nodal pairings in a +topological material PdTe. We first establish it as a 3D Dirac semimetal with a bulk Dirac point +just below the Fermi level in its normal state with Fermi arc surface states on the (010) surface. +Furthermore, our spin-resolved photoemission measurements indicate a polarized spin texture of +these surface states in PdTe. Using high resolution ARPES, we observe a fully gapped Fermi arc +below T", thus could potentially give rise to topological superconductivity. Moreover, we discover +a node in the bulk bands near the Brillouin zone boundary below T". Therefore, PdTe displays a +nodal gap in the bulk but nodeless gap distribution from the nontrivial surface states. Our study +demonstrates a rare and exotic unconventional superconductor to study the interplay between +nontrivial topology and superconductivity. + +PdTe crystalizes in the NiAs type hexagonal structure with space group P6#/mmc [Fig. 1(a)]. Fig. +1(b) displays the Brillouin zone (BZ) where high symmetry points are indicated. For ARPES study, + + + + +3 + +we focus on the (010) surface [k$-k% surface in Fig. 1(b)]. Millimeter-sized PdTe single crystals +were successfully synthesized. Powder X-ray diffraction (XRD) patterns [Fig. S1(a)] confirm that +the sample has only a single phase with the hexagonal P6#/mmc (194) structure [31]. The XRD +pattern from the largest surface of a single crystal shows only (00n) peaks without any impurity +peak, indicating the (001) direction [Fig. S1(b)]. The high quality of our single crystals is further +confirmed by core level photoemission measurements with clear Pd d core levels [Fig. S1(c)]. +Moreover, we identify a sharp superconducting transition in the electrical resistivity of PdTe with +the critical temperature T" ~ 4.5 K [Fig. 1(c)], in agreement with previous reports [32, 33]. + +Like the well-established Dirac semimetal Na#Bi [34], there is a type I bulk band crossing near +the Fermi level along the Γ-A direction [Fig. 1(d)] in PdTe. Given the threefold rotational +symmetry around the c axis [Fig. 1(b)], this band crossing remains gapless when spin-orbit +coupling is included [34]. Since the two bulk Dirac crossings along the Γ-A and A-Γ will project +onto the same position on the (001) surface in momentum space, it is impossible to experimentally +observe the Fermi arc states on this surface. However, on the (010) surface, the two Dirac crossings +are well separated in momentum, and the double Fermi arc surface states connecting the bulk Dirac +points can be observed. Therefore, here, we focus on the (010) surface [Fig. 1(e)]. Indeed, the bulk +band constant energy contour at the energy of the Dirac point [Fig. S2] clearly demonstrates the +location of the Dirac crossings on (010). To help understand the connection of the surface Fermi +arc states with the Dirac point, we calculate the surface spectrum of a semi-infinite Green’s +function projected on the (010), which displays the bulk Dirac cone as well as the Fermi arc surface +states connecting to the Dirac crossing [Fig. 1(f)]. The connection of the surface Fermi arcs to the +bulk Dirac points confirms PdTe as a Dirac semimetal. + +Having identified the expected topology of PdTe, we systematically study its electronic structure +and spin polarization at the (010) surface in its normal state. We demonstrate the topological Dirac +fermion along the Γ-A direction and the corresponding Fermi arc states associated with the Dirac +point. Fig. 2(a) shows the experimentally measured constant energy contour of PdTe at 100 meV +binding energy corresponding to the binding energy of the Dirac point. The constant energy +contour displays two bulk Dirac points and two Fermi arcs connecting the two Dirac points in the +Brillouin zone (BZ). This is consistent with DFT calculations [Fig. 2(b)]. Overall, our ARPES data +agree well with theoretical calculations [Fig. S3]. To better demonstrate the topology, we show +the energy dispersion along the k$ direction in cut 1 [Fig. 2(c)]. Several dispersive bands cross the +Fermi level in Fig. 2(c)i. A comparison between (ii) and (iii) suggests that the inner dispersive +band is a surface state. As shown in Fig. 2(a), cut 1 should pass through the Fermi arc states. +Therefore, we conclude that the inner band is the Fermi arc surface state. Photon energy dependent +ARPES data also verify the surface nature of this Fermi arc state [Fig. S4]. For further confirmation, +we focus on cut 2 [Fig. 2(d)]. Since cut 2 passes through the bulk Dirac point as well as Fermi arc +states, it can display the bulk-boundary correspondence. Indeed, (ii) and (iii) indicate the Dirac +crossing below the Fermi level is a bulk state. Moreover, the bands emerging from the Dirac states +have a surface origin, as they only appear in surface calculations. Our ARPES data in Fig. 2(d)i +clearly show the surface states from the Dirac point. To observe the bulk Dirac crossing, we choose +another direction, cut 3. Here, we can see the type I Dirac point through our data [Fig. 2(e)i], which +agrees well with DFT calculations [Figs. 2(c)ii-iii]. By comparing ARPES and DFT in three cuts, +we confirm the bulk Dirac nodes and the connecting Fermi arc surface states in PdTe. Apart from + + + + +4 + +the Fermi arc surface states, there are also many topologically nontrivial surface states near the +Fermi level [Fig. S5]. + +Next, we perform spin-resolved photoemission measurements to probe the spin polarization of the +Fermi arc surface states. We measure two energy distribution curves (EDCs) indicated in Fig. 3(a). +Since the two EDCs cut through the Fermi arc at the opposite side of Γ, they are expected to show +reversed in-plane spin polarizations (along the k% direction). The spin-resolved EDCs and spin +polarization curves [Figs. 3(b-c)] indeed display reversed spin polarizations, which confirm that +the Fermi arc surface states are spin polarized in PdTe. The Fermi arcs might mix with bulk states, +thus reducing the value of the experimentally measured spin polarization. As a comparison, the +other two spin components along the k$ and k& directions show negligible spin polarizations [Fig. +S6]. + +To precisely determine the fermiology of the Fermi arcs across the transition temperature, we +utilize low-temperature, high-resolution laser ARPES [35]. Our experimental data clearly reveal +the curved Fermi arc surface state around the center of the BZ, in excellent agreement with +theoretical calculations, as shown in Fig. 4(a). The shape of the Fermi arc states is slightly different +from vacuum ultraviolet (VUV) ARPES results in Fig. 2 since laser ARPES measures a distinct +surface termination [Fig. S7], but the nontrivial topology remains the same for both terminations +[Fig. S8]. Having established the fermiology with laser ARPES, we show the superconducting gap +in the momentum space for both the Fermi arc surface states and the bulk bands in PdTe. The +positions of the correspondingly extracted EDCs are defined by an angle ∅ in Fig. 4(a). For all the +angles, the EDCs with the base temperature at around 2K show a small yet finite leading-edge shift +compared to EDCs above the transition temperature [Fig. S9]. This finite edge shift across +T" suggests a vanishing density of states at the Fermi level and thus displays the signature of a +superconducting gap opening for both bulk and surface bands. For a better visualization, we +symmetrize these EDCs with respect to the Fermi level in Figs. 4(b-c), and they all display a clear +gap opening. Therefore, our low-temperature data visualize the formation of Cooper pairs inside a +superconducting Fermi arc state in PdTe. This fully gapped Fermi arc may possibly lead to a +topological superconducting phase in this 3D Dirac semimetal like the topological surface states +in some iron-based superconductors do [5, 7]. Further theoretical works should address whether +topological edge states can form in the potentially topological superconducting state in PdTe. + +More intriguingly, despite the fully gapped bulk bands near the Fermi arc [Fig. 4(c)], we find that +the bulk states near Z. have a node in the superconducting state by exploring the bands close to the +Brillouin zone boundary, away from the Fermi arc [Fig. 4(d)]. As shown in Fig. 4(e), the +corresponding EDC displays no leading-edge shift across the critical temperature. Additionally, +the symmetrized EDCs don’t show any indication of a valley structure [Fig. 4(f)]. In other words, +in contrast with the clear superconducting gap structures in Figs. 4(b-c), there is no sign of +conventional superconductivity such as coherence peaks or a dip at the Fermi level in Fig. 4(f). +Therefore, our measurements have uncovered a gapless node in the bulk bands, below the critical +temperature. + +Our results highlight a novel platform to study the interplay between superconductivity and +topology. The spin-resolved ARPES data clearly demonstrate the spin-momentum locking of the +Fermi arcs, which are fully gapped below T" as revealed by laser ARPES measurements. + + + + +5 + +Furthermore, the bulk bands have a node below T". So far, there are just a few photoemission +studies on superconducting Dirac semimetal and none of them probes the superconducting Fermi +arc states below T" [18-22]. Challenges are due to either the Dirac crossing far below the Fermi +level [19, 21] or difficulty in measuring the side surface such as the (010) surface to directly +observe the Fermi arc states by ARPES. Therefore, it is essential to search for an ideal 3D Dirac +material system that (i) contains Dirac points near the Fermi level, (ii) has an accessible T" and (iii) +can be cleaved for measuring superconducting Fermi arc states. PdTe is such a candidate satisfying +all the above requirements. Moreover, it provides a rare case to study not only superconducting +surface states already hard to access in other materials, but also is an uncommon demonstration of +a gapless bulk node in a superconducting Dirac semimetal. The existence of a bulk node suggests +the possibility of unconventional superconductivity in PdTe, thus distinguishing it from related +superconducting semimetals like PdTe' [19, 20]. Our calculations suggest that this node is from +the spin-orbital texture of the bulk Dirac crossing indicated in Fig. S10. Even with SOC, this bulk +Dirac cone is gapless and protects the node near Z.. Our tight-binding calculations further confirm +that the location of the nodal point should be close to the projection of this bulk Dirac cone in the +momentum space [Fig. S11], consistent with experimental data. Thus, we demonstrate a SOC +enforced bulk-nodal gap structure in PdTe along with a fully gapped Fermi arc surface state. +Recent works [29, 30] suggest that for nodal bulk pairing in Dirac semimetal with C( rotational +symmetry, high order or Majorana hinge modes can exist. It will be very interesting to observe +these high order hinge modes experimentally. Similar states could also appear in PdTe with C# +rotational symmetry, even though details may be different. 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Lett. 120, 067003 (2018). + + +Acknowledgement + +The authors thank D. Lu and M. Hashimoto at Beamline 5-2 of the Stanford Synchrotron Radiation +Lightsource (SSRL) at the SLAC National Accelerator Laboratory, CA, USA for support. The +authors acknowledge enlightening discussions with X. Wu. Work at Princeton University is +supported by the Gordon and Betty Moore Foundation (Grants No. GBMF4547 and No. +GBMF9461; M. Z. H.). The ARPES work is supported by the United States Department of Energy +(US DOE) under the Basic Energy Sciences program (Grant No. DOE/BES DE-FG-02- +05ER46200; M. Z. H.). Materials characterization and the study of topological quantum properties +are supported by the U.S. Department of Energy, Office of Science, National Quantum Information +Science Research Centers, Quantum Science Center and Princeton University. Use of the Stanford +Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the +U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract +No. DE-AC02-76SF00515. Laser ARPES measurements in University of Tokyo are supported by +Grants-in-Aid for Scientific Research (KAKENHI) (Grants No. JP19H01818 and No. +JP19H00651) from the Japan Society for the Promotion of Science (JSPS) and by JSPS KAKENHI +on Innovative Areas “Quantum Liquid Crystals” (Grants No. JP19H05826). We thank HZB for + + + + +8 + +the allocation of synchrotron radiation beamtime at the U125-2-PGM beamline of BESSY II. J. +S.-B. acknowledges financial support from the Impuls-und Vernetzungsfonds der Helmholtz- +Gemeinschaft under grant No. HRSF-0067. Crystal growth and characterization (R. C., J. B., R. +J.) are supported by NSF DMR-1504226. STEM characterization is performed with the use of +Princeton University’s Imaging and Analysis Center, which is partially supported by the Princeton +Center for Complex Materials (PCCM), a National Science Foundation (NSF)-MRSEC program +(DMR-2011750). T.-R. C. is supported by the Young Scholar Fellowship Program from the +Ministry of Science and Technology (MOST) in Taiwan, under a MOST grant for the Columbus +Program MOST110-2636-M-006-016, National Cheng Kung University, Taiwan, and National +Center for Theoretical Sciences, Taiwan. This work is supported partially by the MOST, Taiwan, +grant MOST107-2627-E-006-001. This research is supported in part by Higher Education Sprout +Project, Ministry of Education to the Headquarters of University Advancement at National Cheng +Kung University (NCKU). J. Z. and S. G. are supported as part of the Institute for Quantum Matter, +an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, +Basic Energy Sciences under Award No. DE-SC0019331. The work at Northeastern University is +supported by the Air Force Office of Scientific Research under award number FA9550-20-1-0322, +and it benefits from the computational resources of Northeastern University's Advanced Scientific +Computation Center (ASCC) and the Discovery Cluster. T. A. C. acknowledges the support of the +National Science Foundation Graduate Research Fellowship Program (DGE-1656466). I. B. +acknowledges the generous support of the Special Postdoctoral Researchers Program, RIKEN +during the late stages of this work. M. Z. H. acknowledges support from Lawrence Berkeley +National Laboratory and the Miller Institute of Basic Research in Science at the University of +California, Berkeley in the form of a Visiting Miller Professorship. M. Z. H. also acknowledges +support from the U.S. Department of Energy, Office of Science, National Quantum Information +Science Research Centers, Quantum Science Center. + +X. P. Y., Y. Z., S. M., T. A. C., and R. C. contributed equally to this work. + + + + + + + + + + + + + + + + + + + + + + + +9 + +Figures + + + +FIG. 1. Superconductivity and crystal structure of PdTe. (a) The crystal structure of PdTe. Blue is +Pd atom and cyan is Te atom. (b) Bulk and surface Brillouin zones of PdTe. High symmetry points +are marked. (c) Temperature-dependence zero-field electrical resistivity of PdTe at low +temperature between 2 and 10 K. The dashed line is a guide for the eye. (d) Calculated bulk +electronic structure of PdTe with spin-orbit coupling included. The bulk Dirac crossing is marked +by the circle. The Fermi level is adjusted according to the experimental data. (e) Side surface +scanning transmission electron microscope (STEM) image of PdTe. The unit cell is superimposed +on top of the STEM image. (f) Semi-infinite Green’s function surface calculations along the high +symmetry directions on the (010) Pd-terminated surface. + + + +(a) +(b) +(c) +kz +1.5 +A +Zi +H +U +cm) +1 +un) +M +K +0.5 +y +Y +p +Tc = 4.5 K +......... +C +2 +4 +6 +8 +10 +T (K) +a +(d) +(e) +(f) +Te Pz +Te Pa + Py +1 +(010) side surface +0.5 +0.5 +(eV) +(eV) +0 +中 +0 +出 +出 +-0.5 +1 nm +I +K M +A +H L +Y +r +A +z + + +10 + + +FIG. 2. Dirac crossing and Fermi arc state in PdTe above T". (a) ARPES constant energy contour +at binding energy of 100 meV corresponding to the Dirac crossing on the Te-terminated (010) +surface. Red dotted lines indicate ARPES dispersion cuts 1-3 in (c-e). Purple dots represent the +projection of the Dirac cone and the yellow dotted lines connecting these dots are Fermi arc surface +states. (b) Calculated constant energy contour of (a). (c) ARPES dispersion map (i) along cut 1 in +(a). Black arrow is the Fermi arc state. Corresponding semi-infinite Green’s function surface +calculations including (ii) and excluding (iii) surface states contributions. (d) ARPES dispersion +map (i) and corresponding DFT calculations (ii, iii) along cut 2 in (a). Black arrow in (i) indicates +the surface state connecting to the bulk Dirac crossing. (e) ARPES dispersion map (i) and +corresponding DFT calculations (ii, iii) along cut 3 in (a). Black arrow in (i) represents the bulk +Dirac crossing. + +(a) +(c) +2 +LARPES +(eV) +0 +0 +-0.2 +Fermi arc +-0.2 +0.5 +.F +E-0.4 +-0.4 +Cut 1 +0 +3 +-0.6 +-0.6 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +kx (A") +kx (A") +-0.5 +(d) +Fermi arc +.ARPES +0 +M +0 +-0.2 +0.5 +1.5 +-0.2 +1 +Fermi arc +Fermi +kz (A-") +-0.4 +arc +(b) +Cut 2 +E -0.6 +ii +-0.6 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +0.5 +kx (A") +kx (A") +(e) +0 +0 +(eV) +Fermi arc, +-0.2 +-0.2 +Dirac +Dirac +-0.5 +point +-0.4 +point +-0.4 +出 +Cut 3 +i +-0.6 +-0.6 +1 +0 +0.5 +1.5 +0 +0.5 +1 +0 +0.5 +1 +1 +kz (A") +kz (A") +kz (A) + + +11 + + +FIG. 3. Spin polarization of the Fermi arc state above T". (a) ARPES dispersion map of the Fermi +arc state (same position as cut 1 in Fig. 2(a)). The yellow dotted lines represent the two momenta +selected for spin-resolved photoemission measurements. (b-c) Spin-resolved intensity and +polarization along the in-plane direction (k% direction) for k1 and k2. + + +FIG. 4. Fully gapped surface states and gapless bulk node in the superconducting state of PdTe. +(a) Comparison between laser ARPES Fermi surface map (right) and slab calculation (left) for the +Pd termination in PdTe. Several points marked by red dots (red stars) along the Fermi arc (on the +bulk states) are selected for superconducting gap measurements. ∅ defines the angles used in (b- +c). (b-c) Symmetrized energy distribution curves (EDCs) corresponding to the red dots and stars +in (a) below and above the critical temperature. The specific momentum locations of EDCs are +marked by the Fermi surface angles defined in (a). All symmetrized EDCs reveal the opening of + +(b) +c) +(a) +0.6- +0.6- +Intensity (arb. units) +units) +0.5 +k1 +K2 +0.4 +Intensity (arb. +0.4 +0.00- +0.3 +0.3 +0.2 - +k1 +0.2- +k2 +(eV) +-0.05- +0.1- +0.1 +0.0- +0.0 +-0.10- +-0.2 +0.0 +-0.2 +0.0 +0.2 - +0.2 - +Polarization +Polarization +-0.15- +0.1- +0.1 +0.0 +0.0 +0.1 +0.1 +-0.4-0.20.0C +0.2 +0.4 +0.2 +0.2 +k (A-1) +-0.2 +0.0 +-0.2 +0.0 +E-E, (eV) +E-E, (eV)(a) +e +55° +0 +LT +5K +0.8 +3.5 +3.5 +7K +44° +units) +8 +立 +0 +36° +Intensity (arb. I +0.6 +3 +3 +units) +units) +26° +0.4 +2 +2.5 +0.3 +0 +-0.3 +Intensity (arb. +(arb. +0° +0.2 +kx (A") +Intensity +node +2 +2 +(d) +36° +21° +near Z +FS +0.5 +1.5 +44° +1.5 +30° +(A") +units) +Intensity +0 +Z- +55° +7K +42° +(arb. ++ +-0.5 +5K +LT +4K +5K +Fermiarc +0.5 +2K +bulk +7K +0.5 +fit +.5 +0 +0.5 +11.5 +-2 +0 +2 +-2 +0 +2 +-4 +-2 +0 +2 +4 +E-E- (meV) +E-E= (meV) +E-E= (meV) + + +12 + +the superconducting gap below the critical temperature. Solid black lines represent fits to the +Dynes’ function. Experimental Fermi level position is determined by measuring a gold reference. +(d) VUV ARPES Fermi surface map on the (010) surface. Laser ARPES Fermi surface map with +much smaller momentum range is embedded on the VUV ARPES result. High symmetry points +are marked in red. The location of the gapless node is indicated by the red square. (e) Temperature +dependence of the EDC corresponding to the red square marked in (d) showing a node in the bulk +band structure below the critical temperature. LT stands for the base temperature (~ 2 K) of our +laser ARPES setup. (f) Symmetrized EDCs in (e) indicating the existence of a nodal point. + diff --git a/rNAzT4oBgHgl3EQfcfwN/content/tmp_files/load_file.txt b/rNAzT4oBgHgl3EQfcfwN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d56056ce8064702d4372f18e7361fd195cc5908 --- /dev/null +++ b/rNAzT4oBgHgl3EQfcfwN/content/tmp_files/load_file.txt @@ -0,0 +1,1084 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf,len=1083 +page_content='Coexistence of bulk-nodal and surface-nodeless Cooper pairings in a superconducting Dirac semimetal Xian P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yang1*, Yigui Zhong2, Sougata Mardanya3, Tyler A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Cochran1, Ramakanta Chapai4, Akifumi Mine2, Junyi Zhang5*, Jaime Sánchez-Barriga6, 7, Zi-Jia Cheng1, Oliver J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Clark6, Jia- Xin Yin8, Joanna Blawat4, 9, Guangming Cheng10, Ilya Belopolski1, Tsubaki Nagashima2, Najafzadeh Sahand2, Shiyuan Gao5, Nan Yao10, Arun Bansil11, Rongying Jin4, 9, Tay-Rong Chang3, 12, 13, Shik Shin2, 14, 15, Kozo Okazaki2, 15, 16, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zahid Hasan1, 10, 17† 1Laboratory for Topological Quantum Matter and Advanced Spectroscopy (B7), Department of Physics, Princeton University, Princeton, NJ 08544, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 3Department of Physics, National Cheng Kung University, Tainan 701, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 5Institute for Quantum Matter and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 6Helmholtz-Zentrum Berlin für Materialien und Energie, Elektronenspeicherring BESSY II, Albert-Einstein Strasse 15, Berlin 12489, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 7IMDEA Nanoscience, C/ Faraday 9, Campus de Cantoblanco, Madrid 28049, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 8Department of Physics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 9Center for Experimental Nanoscale Physics, Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 10Princeton Institute for Science and Technology of Materials, Princeton University, Princeton, NJ 08544, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 11Department of Physics, Northeastern University, Boston, MA 02115, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 12Center for Quantum Frontiers of Research and Technology (QFort), Tainan 701, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 13Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 14Office of University Professor, University of Tokyo, Kashiwa, Chiba 277-8581, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 15Material Innovation Research Center, University of Tokyo, Kashiwa, Chiba 277-8581, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 16Trans-scale Quantum Science Institute, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 17Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' xiany@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='edu jzhan312@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='edu †mzhasan@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='edu 2 The interplay of nontrivial topology and superconductivity in condensed matter physics gives rise to exotic phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' However, materials are extremely rare where it is possible to explore the full details of the superconducting pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Here, we investigate the momentum dependence of the superconducting gap distribution in a novel Dirac material PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Using high resolution, low temperature photoemission spectroscopy, we establish it as a spin-orbit coupled Dirac semimetal with the topological Fermi arc crossing the Fermi level on the (010) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' This spin-textured surface state exhibits a fully gapped superconducting Cooper pairing structure below 𝐓𝐜 ~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Moreover, we find a node in the bulk near the Brillouin zone boundary, away from the topological Fermi arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' These observations not only demonstrate the band resolved electronic correlation between topological Fermi arc states and the way it induces Cooper pairing in PdTe, but also provide a rare case where surface and bulk states host a coexistence of nodeless and nodal gap structures enforced by spin- orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Superconducting topological materials are emerging as the new frontier of condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Despite increasing interest in this field, few candidates have been identified because it is very challenging to confirm topological superconductivity experimentally [1-22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' While the topological surface states of three-dimensional (3D) topological insulators (TIs) can host topological superconductivity below the critical temperature (T"), spin-momentum locked Fermi arcs in 3D Dirac semimetals could also induce topological superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Such a superconducting pairing between Fermi arcs with opposite spins and momenta has been proposed to reveal the nonlocal electronic correlations in topological semimetals [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' However, the superconducting state of Fermi arcs has rarely been explored experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Therefore, understanding the Cooper pairings of the Fermi arc states offers a new perspective for topological superconductors (TSCs) [23-27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Moreover, recent theoretical work predicts the existence of nodes from the bulk bands in superconducting semimetals [28-30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Thus, in superconducting semimetals that are also topological, surface Fermi arcs and nodal gap distribution of the bulk states can coexist, which in turn provide insight into the mechanism of unconventional and topological superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Angle-resolved photoemission spectroscopy (ARPES) is arguably the most powerful technique for probing the superconducting gap distribution in the momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Here, we use ARPES to demonstrate spin-orbit coupling (SOC) enforced surface-nodeless plus bulk-nodal pairings in a topological material PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' We first establish it as a 3D Dirac semimetal with a bulk Dirac point just below the Fermi level in its normal state with Fermi arc surface states on the (010) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Furthermore, our spin-resolved photoemission measurements indicate a polarized spin texture of these surface states in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Using high resolution ARPES, we observe a fully gapped Fermi arc below T", thus could potentially give rise to topological superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Moreover, we discover a node in the bulk bands near the Brillouin zone boundary below T".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Therefore, PdTe displays a nodal gap in the bulk but nodeless gap distribution from the nontrivial surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Our study demonstrates a rare and exotic unconventional superconductor to study the interplay between nontrivial topology and superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' PdTe crystalizes in the NiAs type hexagonal structure with space group P6#/mmc [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1(b) displays the Brillouin zone (BZ) where high symmetry points are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' For ARPES study, 3 we focus on the (010) surface [k$-k% surface in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Millimeter-sized PdTe single crystals were successfully synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Powder X-ray diffraction (XRD) patterns [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S1(a)] confirm that the sample has only a single phase with the hexagonal P6#/mmc (194) structure [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The XRD pattern from the largest surface of a single crystal shows only (00n) peaks without any impurity peak, indicating the (001) direction [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The high quality of our single crystals is further confirmed by core level photoemission measurements with clear Pd d core levels [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Moreover, we identify a sharp superconducting transition in the electrical resistivity of PdTe with the critical temperature T" ~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 K [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1(c)], in agreement with previous reports [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Like the well-established Dirac semimetal Na#Bi [34], there is a type I bulk band crossing near the Fermi level along the Γ-A direction [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1(d)] in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Given the threefold rotational symmetry around the c axis [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1(b)], this band crossing remains gapless when spin-orbit coupling is included [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Since the two bulk Dirac crossings along the Γ-A and A-Γ will project onto the same position on the (001) surface in momentum space, it is impossible to experimentally observe the Fermi arc states on this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' However, on the (010) surface, the two Dirac crossings are well separated in momentum, and the double Fermi arc surface states connecting the bulk Dirac points can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Therefore, here, we focus on the (010) surface [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Indeed, the bulk band constant energy contour at the energy of the Dirac point [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S2] clearly demonstrates the location of the Dirac crossings on (010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' To help understand the connection of the surface Fermi arc states with the Dirac point, we calculate the surface spectrum of a semi-infinite Green’s function projected on the (010), which displays the bulk Dirac cone as well as the Fermi arc surface states connecting to the Dirac crossing [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The connection of the surface Fermi arcs to the bulk Dirac points confirms PdTe as a Dirac semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Having identified the expected topology of PdTe, we systematically study its electronic structure and spin polarization at the (010) surface in its normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' We demonstrate the topological Dirac fermion along the Γ-A direction and the corresponding Fermi arc states associated with the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(a) shows the experimentally measured constant energy contour of PdTe at 100 meV binding energy corresponding to the binding energy of the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The constant energy contour displays two bulk Dirac points and two Fermi arcs connecting the two Dirac points in the Brillouin zone (BZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' This is consistent with DFT calculations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Overall, our ARPES data agree well with theoretical calculations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' To better demonstrate the topology, we show the energy dispersion along the k$ direction in cut 1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Several dispersive bands cross the Fermi level in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(c)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' A comparison between (ii) and (iii) suggests that the inner dispersive band is a surface state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(a), cut 1 should pass through the Fermi arc states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Therefore, we conclude that the inner band is the Fermi arc surface state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Photon energy dependent ARPES data also verify the surface nature of this Fermi arc state [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' For further confirmation, we focus on cut 2 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Since cut 2 passes through the bulk Dirac point as well as Fermi arc states, it can display the bulk-boundary correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Indeed, (ii) and (iii) indicate the Dirac crossing below the Fermi level is a bulk state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Moreover, the bands emerging from the Dirac states have a surface origin, as they only appear in surface calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Our ARPES data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(d)i clearly show the surface states from the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' To observe the bulk Dirac crossing, we choose another direction, cut 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Here, we can see the type I Dirac point through our data [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(e)i], which agrees well with DFT calculations [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(c)ii-iii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' By comparing ARPES and DFT in three cuts, we confirm the bulk Dirac nodes and the connecting Fermi arc surface states in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Apart from 4 the Fermi arc surface states, there are also many topologically nontrivial surface states near the Fermi level [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Next, we perform spin-resolved photoemission measurements to probe the spin polarization of the Fermi arc surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' We measure two energy distribution curves (EDCs) indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Since the two EDCs cut through the Fermi arc at the opposite side of Γ, they are expected to show reversed in-plane spin polarizations (along the k% direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The spin-resolved EDCs and spin polarization curves [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 3(b-c)] indeed display reversed spin polarizations, which confirm that the Fermi arc surface states are spin polarized in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The Fermi arcs might mix with bulk states, thus reducing the value of the experimentally measured spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' As a comparison, the other two spin components along the k$ and k& directions show negligible spin polarizations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' To precisely determine the fermiology of the Fermi arcs across the transition temperature, we utilize low-temperature, high-resolution laser ARPES [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Our experimental data clearly reveal the curved Fermi arc surface state around the center of the BZ, in excellent agreement with theoretical calculations, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The shape of the Fermi arc states is slightly different from vacuum ultraviolet (VUV) ARPES results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2 since laser ARPES measures a distinct surface termination [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S7], but the nontrivial topology remains the same for both terminations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Having established the fermiology with laser ARPES, we show the superconducting gap in the momentum space for both the Fermi arc surface states and the bulk bands in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The positions of the correspondingly extracted EDCs are defined by an angle ∅ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' For all the angles, the EDCs with the base temperature at around 2K show a small yet finite leading-edge shift compared to EDCs above the transition temperature [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' This finite edge shift across T" suggests a vanishing density of states at the Fermi level and thus displays the signature of a superconducting gap opening for both bulk and surface bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' For a better visualization, we symmetrize these EDCs with respect to the Fermi level in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(b-c), and they all display a clear gap opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Therefore, our low-temperature data visualize the formation of Cooper pairs inside a superconducting Fermi arc state in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' This fully gapped Fermi arc may possibly lead to a topological superconducting phase in this 3D Dirac semimetal like the topological surface states in some iron-based superconductors do [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Further theoretical works should address whether topological edge states can form in the potentially topological superconducting state in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' More intriguingly, despite the fully gapped bulk bands near the Fermi arc [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(c)], we find that the bulk states near Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' have a node in the superconducting state by exploring the bands close to the Brillouin zone boundary, away from the Fermi arc [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(e), the corresponding EDC displays no leading-edge shift across the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Additionally, the symmetrized EDCs don’t show any indication of a valley structure [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' In other words, in contrast with the clear superconducting gap structures in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(b-c), there is no sign of conventional superconductivity such as coherence peaks or a dip at the Fermi level in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Therefore, our measurements have uncovered a gapless node in the bulk bands, below the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Our results highlight a novel platform to study the interplay between superconductivity and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The spin-resolved ARPES data clearly demonstrate the spin-momentum locking of the Fermi arcs, which are fully gapped below T" as revealed by laser ARPES measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 5 Furthermore, the bulk bands have a node below T".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' So far, there are just a few photoemission studies on superconducting Dirac semimetal and none of them probes the superconducting Fermi arc states below T" [18-22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Challenges are due to either the Dirac crossing far below the Fermi level [19, 21] or difficulty in measuring the side surface such as the (010) surface to directly observe the Fermi arc states by ARPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Therefore, it is essential to search for an ideal 3D Dirac material system that (i) contains Dirac points near the Fermi level, (ii) has an accessible T" and (iii) can be cleaved for measuring superconducting Fermi arc states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' PdTe is such a candidate satisfying all the above requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Moreover, it provides a rare case to study not only superconducting surface states already hard to access in other materials, but also is an uncommon demonstration of a gapless bulk node in a superconducting Dirac semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=" The existence of a bulk node suggests the possibility of unconventional superconductivity in PdTe, thus distinguishing it from related superconducting semimetals like PdTe' [19, 20]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Our calculations suggest that this node is from the spin-orbital texture of the bulk Dirac crossing indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Even with SOC, this bulk Dirac cone is gapless and protects the node near Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='. Our tight-binding calculations further confirm that the location of the nodal point should be close to the projection of this bulk Dirac cone in the momentum space [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S11], consistent with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Thus, we demonstrate a SOC enforced bulk-nodal gap structure in PdTe along with a fully gapped Fermi arc surface state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Recent works [29, 30] suggest that for nodal bulk pairing in Dirac semimetal with C( rotational symmetry, high order or Majorana hinge modes can exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' It will be very interesting to observe these high order hinge modes experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Similar states could also appear in PdTe with C# rotational symmetry, even though details may be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Further theoretical and experimental studies are needed to verify the existence of potential high order hinge modes, thus leading to a better understanding of the complex intertwining of topology and superconductivity in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Reference [1] A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Hu, P-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Hor, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Ding, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Pan, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 11, 543 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Kohama, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Bareille, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Kuroda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Kondo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Okazaki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Sato, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Shin, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 15, 41 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Hor, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Checkelsky, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Roushan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Seo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zandbergen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yazdani, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Ong, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Cava, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 104, 057001 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Sasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Kriener, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Segawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yada, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Sato, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Ando, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 107, 217001 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 6 [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wray, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Xia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Xue, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Mu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zhang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Huang, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 31, 1901942 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Pan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Fang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Song, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Ma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Huang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Li, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Xue, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 15, 1046 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Fang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Liang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Shi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Pei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Xu, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Jia, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='- J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Xie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wei, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wang, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 15, 38 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Aggarwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 15, 32 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Huang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zhou, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zhang, B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Leng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Paulsen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Huang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' de Visser, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' B 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Riley, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Meevasana, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Fujii, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Vobornik, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Hoesch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Sasagawa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wahl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Bahramy, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' King, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 120, 156401 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Bahramy, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Clark, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Feng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Bawden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Riley, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Marković, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Mazzola, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Sunko, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Biswas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Cooil, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Jorge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wells, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Leandersson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Balasubramanian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Fujii, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Vobornik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Rault, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Hoesch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Okawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 17, 21 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Huang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Luo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Chen, G.' metadata={'source': 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Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Liu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' B 103, 155148 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Geng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Gao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Cui, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Feng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Song, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Luo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Arita, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Kumar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Schwier, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Shimada, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zhao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Weng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Shi, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Feng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' B 102, 205117 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Kitaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 44, 131 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} 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Lu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Hashimoto at Beamline 5-2 of the Stanford Synchrotron Radiation Lightsource (SSRL) at the SLAC National Accelerator Laboratory, CA, USA for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The authors acknowledge enlightening discussions with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Work at Princeton University is supported by the Gordon and Betty Moore Foundation (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' GBMF4547 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' GBMF9461;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The ARPES work is supported by the United States Department of Energy (US DOE) under the Basic Energy Sciences program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' DOE/BES DE-FG-02- 05ER46200;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Materials characterization and the study of topological quantum properties are supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center and Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' DE-AC02-76SF00515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Laser ARPES measurements in University of Tokyo are supported by Grants-in-Aid for Scientific Research (KAKENHI) (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' JP19H01818 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' JP19H00651) from the Japan Society for the Promotion of Science (JSPS) and by JSPS KAKENHI on Innovative Areas “Quantum Liquid Crystals” (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' JP19H05826).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' We thank HZB for 8 the allocation of synchrotron radiation beamtime at the U125-2-PGM beamline of BESSY II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' acknowledges financial support from the Impuls-und Vernetzungsfonds der Helmholtz- Gemeinschaft under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' HRSF-0067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Crystal growth and characterization (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=') are supported by NSF DMR-1504226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' STEM characterization is performed with the use of Princeton University’s Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials (PCCM), a National Science Foundation (NSF)-MRSEC program (DMR-2011750).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' is supported by the Young Scholar Fellowship Program from the Ministry of Science and Technology (MOST) in Taiwan, under a MOST grant for the Columbus Program MOST110-2636-M-006-016, National Cheng Kung University, Taiwan, and National Center for Theoretical Sciences, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' This work is supported partially by the MOST, Taiwan, grant MOST107-2627-E-006-001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' This research is supported in part by Higher Education Sprout Project, Ministry of Education to the Headquarters of University Advancement at National Cheng Kung University (NCKU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' are supported as part of the Institute for Quantum Matter, an Energy Frontier Research Center funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Department of Energy, Office of Science, Basic Energy Sciences under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' DE-SC0019331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=" The work at Northeastern University is supported by the Air Force Office of Scientific Research under award number FA9550-20-1-0322, and it benefits from the computational resources of Northeastern University's Advanced Scientific Computation Center (ASCC) and the Discovery Cluster." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' acknowledges the support of the National Science Foundation Graduate Research Fellowship Program (DGE-1656466).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' acknowledges the generous support of the Special Postdoctoral Researchers Program, RIKEN during the late stages of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' acknowledges support from Lawrence Berkeley National Laboratory and the Miller Institute of Basic Research in Science at the University of California, Berkeley in the form of a Visiting Miller Professorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' also acknowledges support from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=', and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 9 Figures FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Superconductivity and crystal structure of PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (a) The crystal structure of PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Blue is Pd atom and cyan is Te atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (b) Bulk and surface Brillouin zones of PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' High symmetry points are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (c) Temperature-dependence zero-field electrical resistivity of PdTe at low temperature between 2 and 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The dashed line is a guide for the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (d) Calculated bulk electronic structure of PdTe with spin-orbit coupling included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The bulk Dirac crossing is marked by the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The Fermi level is adjusted according to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (e) Side surface scanning transmission electron microscope (STEM) image of PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The unit cell is superimposed on top of the STEM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (f) Semi-infinite Green’s function surface calculations along the high symmetry directions on the (010) Pd-terminated surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (a) (b) (c) kz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 A Zi H U cm) 1 un) M K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 y Y p Tc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' C 2 4 6 8 10 T (K) a (d) (e) (f) Te Pz Te Pa + Py 1 (010) side surface 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 (eV) (eV) 0 中 0 出 出 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 1 nm I K M A H L Y r A z 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Dirac crossing and Fermi arc state in PdTe above T".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (a) ARPES constant energy contour at binding energy of 100 meV corresponding to the Dirac crossing on the Te-terminated (010) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Red dotted lines indicate ARPES dispersion cuts 1-3 in (c-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Purple dots represent the projection of the Dirac cone and the yellow dotted lines connecting these dots are Fermi arc surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (b) Calculated constant energy contour of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (c) ARPES dispersion map (i) along cut 1 in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Black arrow is the Fermi arc state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Corresponding semi-infinite Green’s function surface calculations including (ii) and excluding (iii) surface states contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (d) ARPES dispersion map (i) and corresponding DFT calculations (ii, iii) along cut 2 in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Black arrow in (i) indicates the surface state connecting to the bulk Dirac crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (e) ARPES dispersion map (i) and corresponding DFT calculations (ii, iii) along cut 3 in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Black arrow in (i) represents the bulk Dirac crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (a) (c) 2 LARPES (eV) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='2 Fermi arc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='F E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='4 Cut 1 0 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 kx (A") kx (A") 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 (d) Fermi arc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='ARPES 0 M 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='2 1 Fermi arc Fermi kz (A ") 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='4 arc (b) Cut 2 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='6 ii 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 kx (A") kx (A") (e) 0 0 (eV) Fermi arc, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='2 Dirac Dirac 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 point 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='4 point 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='4 出 Cut 3 i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='6 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 1 1 kz (A") kz (A") kz (A) 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Spin polarization of the Fermi arc state above T".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (a) ARPES dispersion map of the Fermi arc state (same position as cut 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The yellow dotted lines represent the two momenta selected for spin-resolved photoemission measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (b-c) Spin-resolved intensity and polarization along the in-plane direction (k% direction) for k1 and k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Fully gapped surface states and gapless bulk node in the superconducting state of PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (a) Comparison between laser ARPES Fermi surface map (right) and slab calculation (left) for the Pd termination in PdTe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Several points marked by red dots (red stars) along the Fermi arc (on the bulk states) are selected for superconducting gap measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' ∅ defines the angles used in (b- c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (b-c) Symmetrized energy distribution curves (EDCs) corresponding to the red dots and stars in (a) below and above the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The specific momentum locations of EDCs are marked by the Fermi surface angles defined in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' All symmetrized EDCs reveal the opening of (b) c) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='6 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' units) units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='5 k1 K2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='4 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='2 k1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='2 k2 (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (d) VUV ARPES Fermi surface map on the (010) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' Laser ARPES Fermi surface map with much smaller momentum range is embedded on the VUV ARPES result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' High symmetry points are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' The location of the gapless node is indicated by the red square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (e) Temperature dependence of the EDC corresponding to the red square marked in (d) showing a node in the bulk band structure below the critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' LT stands for the base temperature (~ 2 K) of our laser ARPES setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} +page_content=' (f) Symmetrized EDCs in (e) indicating the existence of a nodal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNAzT4oBgHgl3EQfcfwN/content/2301.01402v1.pdf'} diff --git a/rtFRT4oBgHgl3EQffTfy/content/2301.13576v1.pdf b/rtFRT4oBgHgl3EQffTfy/content/2301.13576v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2c4b4042bd8877d91f7d63eb6dd86bcc8c1ed86a --- /dev/null +++ b/rtFRT4oBgHgl3EQffTfy/content/2301.13576v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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In semantic communications, both transmitter and +receiver share some common knowledge, which can be used to +extract small-size information at the transmitter and recover +the original information at the receiver. Due to different design +purposes, security issues in semantic communications have two +unique features compared to standard bit-wise communications. +First, an attacker in semantic communications considers not +only the amount of stolen data but also the meanings of stolen +data. Second, an attacker in semantic communication systems +can attack not only semantic information transmission as done +in standard communication systems but also attacks machine +learning (ML) models used for semantic information extraction +since most of semantic information is generated using ML based +methods. Due to these unique features, in this paper, we present +an overview on the fundamentals and key challenges in the design +of secure semantic communication. We first provide various +methods to define and extract semantic information. Then, we +focus on secure semantic communication techniques in two areas: +information security and semantic ML model security. For each +area, we identify the main problems and challenges. Then, we +will provide a comprehensive treatment of these problems. In a +nutshell, this article provides a holistic set of guidelines on how to +design secure semantic communication systems over real-world +wireless communication networks. +Index Terms—Secure semantic communication, information +security, semantic ML model security. +I. INTRODUCTION +The development of smartphone processors enable edge +devices (i.e., mobile phones) to generate and process large- +scale image, video, and immersive extended reality data, which +will significantly increase network congestion [1]. Therefore, +it is necessary to design novel communication techniques to +support such large data-sized data transmission and processing. +Current research [1] studied the use of machine learning (ML) +tools, reflecting intelligent surface (RIS), millimeter wave +Zhaohui Yang and Zhaoyang Zhang are with the College of Informa- +tion Science and Electronic Engineering, Zhejiang University, Hangzhou +310027, China, and Zhejiang Provincial Key Lab of Information Process- +ing, Communication and Networking (IPCAN), Hangzhou 310007, China. +Zhaohui Yang is also with Zhejiang Lab, Hangzhou 31121, China. (e-mails: +yang zhaohui@zju.edu.cn and ning ming@zju.edu.cn). +Gaolei Li is with the School of Electronic Information and Electri- +cal Engineering, Shanghai Jiao Tong University, Shanghai, China. (e-mail: +gaolei li@sjtu.edu.cn) +Yang Yang is with the School of Information and Communication Engi- +neering, BUPT. (e-mail: yangyang01@bupt.edu.cn) +Mingzhe Chen is with the Department of Electrical and Computer Engi- +neering and Institute for Data Science and Computing, University of Miami, +Coral Gables, FL 33146 USA. (e-mail: mingzhe.chen@miami.edu) +(mmWave), and edge computing to improve network perfor- +mance. However, the performance of a network that exploits +these techniques will be limited by the Shannon capacity since +most of these techniques are trying to maximize edge devices’ +data rates so as to reach the Shannon capacity limit [2]. In +consequence, it is necessary to design novel communication +techniques that further improve network performance beyond +the Shannon capacity limit. +Semantic communication technique is a promising method +to overcome the Shannon capacity limit, which enables an +edge device to extract the meaning of large-sized data, called +semantic information hereinafter, and transmit only the se- +mantic information to the receiver instead of transmitting the +entire data [3]. Therefore, compared to current works that +only focus on the maximization of devices’ data rates, the +purpose of semantic communications is not only to maximize +each device’s data rate but also maximize the meanings that +the transmitted data can carry. Since semantic communication +is still in its infancy, it faces many challenges such as se- +mantic information definition, semantic information extraction, +semantic communication measurement, security issues, and +resilience. +Recently, a number of surveys and tutorials related to se- +mantic communications appeared in [3]–[6]. In particular, the +authors in [3]–[5] provided a comprehensive tutorial on the use +of information theory for semantic information representation +and semantic communication metric design. The authors in +[6] introduced an edge intelligence based semantic communi- +cation framework and present its implementation challenges. +However, none of these existing surveys and tutorials [3]–[6] +introduced the security problems in semantic communications. +Compared to attackers that only consider the amount of stolen +data in standard communication systems, attackers in semantic +communications have two unique features. First, an attacker +in semantic communications considers not only the amount of +stolen data but also the meanings of stolen data. For example, +if one attacker steals a large amount of data from a user but +does not obtain the target content/meanings from the stolen +data, we will consider that the attacker does not attack the user +successfully. Second, an attacker in semantic communication +systems can attack not only semantic information transmission +as done in standard communication systems but also attacks +ML models used for semantic information extraction since +most of semantic information is generated using ML based +methods. Due to these unique features of attackers in semantic +communications, it is necessary to provide an introduction +on the fundamentals and challenges of implementing secure +semantic communications. +arXiv:2301.01421v1 [cs.IT] 4 Jan 2023 + +2 +Semantic information +extraction +Central knowledge +base +…… +Local knowledge +base +Wireless transmission +Original large-size data +Small-size semantic +information +Common +knowledge +Private +knowledge +Receiver +Transmitter +Fig. 1. +Illustration of the basic structures of a semantic communication +system. +In this paper, we introduce fundamentals, solutions, and +challenges of designing secure semantic communication sys- +tems. In this context, we first introduce basic semantic com- +munication process. Then, we overview four methods to +define semantic information: a) autoencoder, b) information +bottleneck (IB), c) knowledge graph, and, d) probability graph, +and summarize their advantages, drawbacks, and applications. +We then introduce how attackers can attack semantic informa- +tion transmission and extraction, and explain the methods to +defense these attacks from the point views of both information +security and ML security. +II. FUNDAMENTALS OF SEMANTIC COMMUNICATIONS +In this section, we first the process of semantic communica- +tions. Then, we introduce four methods to model the semantic +information extracted from original data and explain their +differences, advantages, disadvantages, and applications. +A. Semantic Communication Process +The overall semantic communication process mainly in- +cludes three stages. In the first stage, the transmitter utilizes +ML tool to extract the small-size semantic information from +the original large-size data based on the central knowledge +base, as shown in Fig. 1. Then, the semantic information +is transmitted over wireless link in the second stage. In the +third stage, the receiver recovers the intended meaning behind +the semantic information based on its own local knowledge +base, which includes both common knowledge and private +knowledge. +B. Semantic Information Construction +1) Autoencoder: Semantic communication transmits se- +mantic messages, which refer to a sequence of well-formed +symbols learned from the “meaning” underlying source. Cor- +respondingly, the receiver aims at fully understanding the +“meaning” of the encoded semantic symbols. Therefore, ef- +fectively extracting the semantics of the source while ignoring +the redundant information plays a key role in semantic com- +munications. Due to the powerful representation and learning +capability, neural networks are typically employed to extract +semantics from the source. In particular, autoencoder is a type +of neural network used to learn efficient representation for +high-dimensional data, which can extract the most important +information and is thus particularly suitable for semantic +communications. In particular, autoencoder consists of an +encoder and a decoder. The encoder outputs encoded symbols +with much fewer dimensions compared to the source data, +since it only reserves the key information while discards the +insignificant parts of the data. Then, the decoder is used to +recover the original data from the low-dimensional symbols. +Using autoencoder to extract the semantic information has +following advantages. First, autoencoder can be implemented +based on various types of neural networks such as convolu- +tional neural network, transformer, and fully-connected neural +networks. Therefore, it is applicable to semantic communica- +tions of different source data including text, images, videos, +and multi-modal data. In addition, since the output of the +encoder has much less dimensions, the transmission efficiency +of semantic communications can be significantly improved. +Moreover, autoencoders can be trained in an self-supervised +way. However, there are also some key challenges of autoen- +coders. In particular, even though the coding generated by the +autoencoder can be efficiently understood by machines, it is +incomprehensible for humans, which seems to be contradicted +to the principle of semantic communications in a certain way. +Table I summarizes the advantages, disadvantages, and appli- +cations of using autoencoder to model semantic information. +2) Information Bottleneck: Since semantic communications +aim to only reserve the semantics, the essence of semantic +communications is a lossy compression problem. To solve this +type of problems, Claude Shannon has proposed fundamental +theory, i.e. rate-distortion theory, which solves the optimal +trade-off between compression and fidelity [7]. In particular, +rate-distortion theory aims to minimize the required rate under +a given distortion, which can be used to guide the training +process of semantic communication systems. However, one +problem of rate-distortion theory is that it needs to choose +a specific distortion function in advance, which will further +determine the extracted semantics. However, the choice of the +distortion function is not part of the theory. +To tackle this issue, information bottleneck (IB) principle +was proposed from the perspective of information theory, +which can be deemed as a generalization of rate-distortion +theory [8]. On the one hand, the distortion in IB principle +is measured by the mutual information between the encoded +semantic symbols and a target variable. In semantic commu- +nications, the target variable varies with the applications. For +instance, for an image classification task, the target variable is +the label of the source image, since we try to correctly classify +the source image. On the other hand, the rate in IB principle +is characterized by the mutual information between the source +and the encoded symbols, which indicates the number of bits +the encoded symbols used to represent the source. + +回3 +Knowledge +Graph +Probability +Graph +Autoencoder +Information +Bottleneck +Original large-size data +Semantic information +on +ck +Semantic Information Extraction +Fig. 2. Illustration of four types of semantic information extraction. +The advantage of IB is that it provides a specific theoretical +bound for minimizing the rate under given distortion. However, +in practice, the joint and marginal distributions of information +bits are challenging to obtain and, thus, the original IB cannot +be directly used to guide the training process of semantic +communications. Table I summarizes the advantages, disad- +vantages, and applications of using information bottleneck to +model semantic information. +3) Knowledge Graph: Since semantic information repre- +sented by the output of autoencoder is incomprehensible for +humans and does not have any physical meaning, next, we +introduce the use of knowledge graph for semantic information +representation. Since a knowledge graph consists of nodes and +edges, we use nodes to represent an object or a concept in +the original data and edges are used to represent the relations +between each pair of nodes. Each pair of nodes and their +relations are defined by a triple. Hence, a semantic information +modeled by a knowledge graph consists of multiple triples. +Different from other graph models, triples in knowledge graph +are determined by both original data that a user wants to +transmit and the knowledge that this user has to understand +the original data. Hence, the semantic information extracted +from the same original data by different users with different +knowledge may be different. +Using knowledge graph to model semantic information has +several key advantages. First, the extracted semantic informa- +tion is comprehensible for humans. Hence, a receiver may not +need to recover the original data when it receives a semantic +information since the semantic information represents the +similar meanings of original data. Second, one can manage +the data size of semantic information that consists of several +triples according to network conditions and resources. In +particular, when the network resources (i.e., bandwidth) are +limited, one can limit the number of triples in the semantic +information to meet communication service requirements (i.e., +delay). However, exploiting knowledge graphs for semantic +information also faces several challenges. First, all triples in +a semantic information are extracted by neural network based +methods. Therefore, the complexity and time of training these +neural networks must be considered and reduced when using +knowledge graph for energy limited devices (i.e., Internet-of- +things devices). Second, most of current researches assume +that all users have the same knowledge for triple extractions +which may not be practical. Hence, it is necessary to design +novel methods to model and generate unique knowledge +library for each user. Table I summarizes the advantages, +disadvantages, and applications of using knowledge graph to +model semantic information. +4) Probability Graph: The directional probability graph +can also be used to characterize the inherent information of +the transmitted information [9]. In the directional probability +graph, each vertex represents the semantic entity and the +edge stands for the probability of connection between these +two vertexes. Since multiple vertexes with high connection +probabilities among each other can be fused into a single +vertex, the new generated vertex can contain higher level +semantic information than the original vertexes. Probability +graph shows the probabilities among different entities, which +can be used to extract the corn semantic information with +overall high probability in the probability graph. +There are some advantages of extracting semantic informa- +tion with probability graph. First, different levels of semantic +information can be generated with using probability with +combing highly-rated low-level semantic entity into a high- +level semantic entity. Second, the probability graph can be +used to predict the incoming information of the receiver. +Through prediction and inference, the receiver side can adapt +its actions in advance. However, there are still some chal- +lenges using probability graph. Since the probability graph +can be learned with neural network and multi-level seman- +tic information extractions needs additional computation, the +complexity of constructing multi-level semantic information is + +hat +car2 +wear +behind +Infrontof +head +has +beside +carl +behind +man +tree +ridingon +under +park on +bicycle +along +street +beside +buildingEasyDnDifficultD, +Normall +Smartli +0.7 +0.3 +0.7 +0.3 +Low AgHigh A, +Do +0.3 +0.7 +Difficulty +Intelligence +D +0.8 +0.2 +LowS. +HighS, +Activity +0.9 +0.1 +l1 +0.3 +0.7 +Low G.Middle Gi +High G, +SAT +Do, lo +0.40 +0.40 +0.20 +Grade +Do, l1 +0.05 +0.15 +0.80 +NormalRo +Good Ri +Di, lo +0.90 +0.07 +0.03 +0.7 +0.3 +Di,l1 +0.60 +0.25 +0.15 +Normal R,Good R, +Go +0.85 +0.15 +Recommendation +Gi +0.45 +0.55 +letter +G2 +0.05 +0.95 +John is an intelligent student with low activity.The course if not difficult +this year and he gets good recommendation from the supervisor. +John intelligenteasy course +Probability Graphfortext informationextractionFeature Map +Source +Encoder +channel +